All Topics
This page collects in one place all the entries in the geometry junkyard.
 Acme Klein Bottle.
A topologist's delight, handcrafted in glass.
 Acute square triangulation.
Can one partition the square into triangles with all angles acute?
How many triangles are needed, and what is the best angle bound possible?
 Adventitious geometry.
Quadrilaterals in which the sides and diagonals form
more rational angles with each other than one might expect.
Dave Rusin's known math pages include
another article on the same problem.
 Adventures among the toroids. Reference to a book on polyhedral tori by B. M. Stewart.
 The
Aesthetics of Symmetry, essay and design tips by Jeff Chapman.
 1st
and 2nd AjimaMalfatti points. How to pack three circles in a
triangle so they each touch the other two and two triangle sides. This
problem has a curious history, described in Wells' Penguin Dictionary
of Curious and Interesting Geometry: Malfatti's original (1803)
question was to carve three columns out of a prismshaped block of
marble with as little wasted stone as possible, but it wasn't until 1967
that it was shown that these three mutually tangent circles are never
the right answer.
See also
this Cabri geometry page,
the MathWorld
Malfatti circles page, and the Wikipedia
Malfatti circles page.
 Algorithmic
mathematical art, Xah Lee.
 Algorithmic packings
compared. Anton Sherwood looks at deterministic rules for
diskpacking on spheres.
 Algorithmic
vectorial geometry,
French geometry etextbook by J.P. Jurzak.
 Algorithms for
coloring quadtrees.
 Are all triangles
isosceles?
A fallacious proof from K. S. Brown's math pages.
 Allegria
fractal and mathematically inspired jewelry.
 Ancient
Islamic Penrose Tiles. Peter Lu uncovers evidence that the
architects of a 500yearold Iranian shrine used Penrose tiling to lay
out the decorative patterns on its archways. From Ivars Peterson's
MathTrek.
 Angle
trisection, from the geometry forum archives.
 Animated proof
of the Pythagorean theorem, M. D. Meyerson, US Naval Academy.
 Animation of the
fast Fourier transform of a Menger Sponge.
 Escherinspired animorphic art
by Kelly Houle, including "impossible figures" such as
linked Penrose
tribars.
 Anna's pentomino page. Anna Gardberg makes pentominoes out of sculpey and agate.
 AntiEuclidean Love Song.
 Antipodes.
Jim Propp asks whether the two farthest apart points,
as measured by surface distance, on a symmetric convex body
must be opposite each other on the body.
Apparently this is open even for rectangular boxes.
 Anton's modest
little gallery of raytraced 3d math.
 Aperiodic
colored tilings, F. Gähler.
Also
available in postscript.
 Aperiodic
tiling and Penrose tiles, Steve Edwards.
 An aperiodic set of Wang cubes, J. UCS 1:10 (1995).
Culik and Kari describe how to increase the dimension of sets of
aperiodic tilings, turning a 13square set of tiles into a 21cube set.
 Aperiodic spacefilling tiles:
John Conway describes a way of
glueing two prisms together to form a shape that tiles space only
aperiodically.
Ludwig Danzer speaks at NYU on
various aperiodic 3d tilings including Conway's
biprism.
 Apollonian Gasket,
a fractal circle packing formed by packing smaller circles into each
triangular gap formed by three larger circles.
From MathWorld.
 Applications of shapes of constant width.
A Reuleaux triangle doesn't quite drill out a square hole (it leaves
rounded corners) but a different and lesssymmetric constantwidth shape
based on an isosceles right triangle can be used to do so. This web page
also discusses coin design, cams, and rotary engines, all based on
curves of constant width; see
also discussion
on Metafilter.
 Arc
length surprise. The sum of the areas of the regions between a
circular arc and the xaxis, and between the arc and the yaxis,
does not depend on the position of the arc!
From Mudd Math Fun Facts.
 Archimedean solids:
John Conway describes some
interesting maps among the Archimedean
polytopes.
Eric Weisstein lists
properties
and pictures of the Archimedean solids.
 Are most manifolds hyperbolic? From Dave Rusin's known math pages.
 Area of
hyperbolic triangles. From the Geometry Center's
Java gallery of interactive
geometry
 Area of the Mandelbrot set.
One can upper bound this area by filling the area around the set by disks,
or lower bound it by counting pixels; strangely, Stan Isaacs notes,
these two methods do not seem to give the same answer.
 Art, Math,
and Computers  New Ways of Creating Pleasing Shapes, C. Séquin,
Educator's TECH Exchange, Jan. 1996.
 The
Art and Science of Tiling.
Penrose tiles at Carleton College.
 Art
at the 2005 Joint Mathematics Meetings, including many geometric models.
 Art
of the Tetrahedron. And by "Art" he means "Arthur". Arthur
Silverman's geometric sculpture, from Ivars Peterson's MathTrek.
 ASCII Menger sponge, W. Taylor.
 Associahedron
and Permutahedron.
The associahedron represents the set of triangulations of a hexagon,
with edges representing flips; the permutahedron represents the set of
permutations of four objects, with edges representing swaps.
This strangely asymmetric view of the associahedron (as an animated gif)
shows that it has some kind of geometric relation with the permutahedron:
it can be formed by cutting the permutahedron on two planes.
A more symmetric view is below.
See also a
more detailed description of the associahedron
and
JeanLouis Loday's paper
on associahedron coordinates.
 Associating the
symmetry of the Platonic solids with polymorf manipulatives.
 AstroLogix
3d ballandstick geometric construction kit.
 The Atomium, structure formed
for Expo 1958 in the form of nine spheres, representing an iron
crystal. The world's largest cube?
 Atlas of oriented knots and links, Corinne Cerf extends previous lists of all small knots and links,
to allow each component of the link to be marked by an orientation.
 On the average height of jute crops in the
month of September. Vijay Raghavan points out an obscure reference
to average case analysis of the Euclidean traveling salesman problem.
 David Bailey's
world of tesselations.
Primarily consists of Escherlike drawings but also includes
an interesting section about Kepler's work on polyhedra.
 Henry Baker's
hypertext version of HAKMEM
includes a
dissection of square and hexagon, depicted below.
 Balanced ternary
reptiles, Cantor's
hourglass reptile, spiral
reptile, stretchtiles,
trisection of India, the
three Bodhi problem,
and other Fractal tilings by
R. W. Gosper.
 Bamboo C.O.R.P.S.. Durable bamboo
models of the Platonic and Archimedean polyhedra, offered for sale.
 Basic crystallography diagrams, B. C. Taverner, Witwatersrand.
 Ned
Batchelder's Stellated Dodecahedron Tshirt.
 Beezer's PlayDome.
Rob Beezer makes truncated icosahedra out of old automobile tires.
 The bellows
conjecture, R. Connelly, I. Sabitov and A. Walz in Contributions to
Algebra and Geometry , volume 38 (1997), No.1, 110. Connelly had
previously discovered
nonconvex polyhedra which are flexible (can move through a continuous
family of shapes without bending or otherwise deforming any faces);
these authors prove that in any such example, the volume remains
constant throughout the flexing motion.
 Belousov's Brew.
A recipe for making spiraling patterns in chemical reactions.
 Bending a
soccer ball mathematically. Michael Trott animates morphs between a
torus and a doublecovered sphere, to illustrate their topological
equivalence, together with several related animations.
 BitArt spirolateral
gallery (requires JavaScript to view large images, and Java to view
selfrunning demo or construct new spirolaterals).
 Blocking
polyominos. R. M. Kurchan asks, for each k, what is the smallest
polyomino such that k copies can form a "blocked" configuration
in which no piece can be slid free of the others, but in which any
subconfiguration is not blocked.
 Border
pattern gallery. Oklahoma State U. class project displaying examples
of the seven types of symmetry (frieze groups) possible for
linear patterns in the plane.
 Borromean rings don't exist.
Geoff Mess relates a proof that
the Borromean ring configuration
(in which three loops are tangled together but no pair is linked)
can not be formed out of circles.
Dan Asimov discusses some related higher dimensional questions.
Matthew
Cook conjectures the converse.
 Are
Borromean links so rare?
S. Javan relates the history of the links and describes
various generalizations with more than three rings.
For more history and symbolism of the Borromean rings,
see Peter Cromwell's
web site.
 Bounded degree triangulation.
Pankaj Agarwal and Sandeep Sen ask for triangulations of convex polytopes
in which the vertex or edge degree is bounded by a constant or polylog.
 Box in a box.
What is the smallest cube that can be put inside another cube
touching all its faces?
There is a simple solution, but it seems difficult to prove its correctness.
The solution and proof are even prettier in four dimensions.
 Boy's surface:
Wikipedia,
MathWorld,
Geometry
Center,
and an
asymmetric animated gif from the Harvard zoo.
 Brahmagupta's formula.
A "Herontype" formula for the maximum area of a quadrilateral,
Col. Sicherman's fave. He asks if it has higherdimensional
generalizations.
 Breaking Bonds.
Geometric sculpture by Stephen Luecking combining buckyball, hexagon,
and amorphous shapes of carbon molecules.
 A
Brunnian link. Cutting any one of five links allows the remaining
four to be disconnected from each other, so this is in some sense a
generalization of the Borromean rings. However since each pair of links
crosses four times, it can't be drawn with circles.
 Buckyballs. The truncated icosahedron
recently acquired new fame and a new name when chemists discovered that
Carbon forms molecules with its shape.
 Buffon's needle.
What is the probability that a dropped needle lands on a crack on a
hardwood floor?
From Kunkel's mathematics
lessons.
 Building a better beam detector.
This is a set that intersects all lines through the unit disk.
The construction below achieves
total length approximately 5.1547, but better bounds were previously known.
 Krystyna Burczyk's Origami Gallery  regular polyhedra.
 The business card Menger
sponge project. Jeannine Mosely wants to build a fractal cube out of
66048 business cards. The MIT Origami Club has already made a smaller version of the same shape.
 Oliver
Byrne's 1847 edition of Euclid, put online by UBC.
"The first six books of the Elements of Euclid, in which coloured
diagrams and symbols are used instead of letters for the greater ease of
learners."
 Calabi's triangle constant, defining the unique nonequilateral triangle
with three equally large inscribed squares.
Is there a threedimensional analogue?
From MathSoft's favorite constants pages.
 Canonical
polygons.
Ronald Kyrmse investigates grid polygons in which all side lengths are
one or sqrt(2).
 Cardahedra.
Business card polyhedral origami.
 Carnival triangles.
A factoid about similar triangles inspired by a trigonometric identity.
 Catalogue of lattices, N. J. A. Sloane, AT&T Labs Research.
See also Sloane's spherepacking and lattice theory publications.
 Cellular
automaton run on Penrose tiles, D. Griffeath.
See also Eric
Weeks' page on cellular automata over quasicrystals.
 Centers of maximum matchings.
Andy Fingerhut asks, given a maximum (not minimum) matching of six
points in the Euclidean plane, whether there is a center point
close to all matched edges (within distance a constant times the length
of the edge). If so, it could be extended to more points via Helly's theorem.
Apparently this is related to communication network design.
I include a response I sent with a proof (of a constant worse than the
one he wanted, but generalizing as well to bipartite matching).
 Chaotic tiling
of two kinds of equilateral pentagon, with
30degree symmetry, Ed Pegg Jr.
 The charged particle
model: polytopes and optimal packing of p points in n dimensional spheres.
 The
ChengPleijel point. Given a closed plane curve and a height H,
this point is the apex of the minimum surface area cone of height H over
the curve. Ben Cheng demonstrates this concept with the help of a Java applet.
 The chromatic number of the plane.
Gordon Royle and Ilan Vardi summarize what's known about
the famous open problem of how many colors are needed to color
the plane so that no two points at a unit distance apart get the same color.
See also
another article from Dave Rusin's known math pages.
 Cinderella
multiplatform Java system for compassandstraightedge construction,
dynamic geometry demonstrations and
automatic theorem proving.
Ulli Kortenkamp and Jürgen RichterGebert, ETH Zurich.
 Circle fractal
based on repeated placement of two equal tangent circles within each
circle of the figure.
One could also get something like this by inversion, starting with three
mutually tangent circles, but then the circles at each level of the
recursion wouldn't all stay the same size as each other.
 Circle
packing and discrete complex analysis. Research by
Ken Stephenson including pictures, a bibliography, and downloadable circle packing
software.
 Circle packings.
Gareth McCaughan describes the connection between collections
of tangent circles and conformal mapping. Includes some pretty postscript
packing pictures.
 Circles in ellipses.
James Buddenhagen asks for the smallest ellipse that contains two
disjoint unit circles.
Discussion continued in a thread on
three
circles in an ellipse.
 Circular
coverage constants. How big must N equal disks be in order to
completely cover the unit disk? What about disks with sizes in
geometric progression? From MathSoft's favorite constants pages.
 Circular quadrilaterals.
Bill Taylor notes that if one connects the opposite midpoints
of a partition of the circle into four chords, the two line segments
you get are at right angles. Geoff Bailey supplies an elegant proof.
 Circumcenters of triangles.
Joe O'Rourke, Dave Watson, and William Flis
compare formulas for computing
the coordinates of a circle's center from three boundary points,
and higher dimensional generalizations.
 Circumference/perimeter
of an ellipse: formula(s). Interesting and detailed survey of
formulas giving accurate approximations to this value, which can not be
expressed exactly as a closed form formula.
 Circumnavigating
a cube and a tetrahedron, Henry Bottomley.
 Clusters and
decagons, new rules for using overlapping shapes to construct
Penrose tilings. Ivars Peterson, Science News, Oct. 1996.
 Colinear points on knots.
Greg Kuperberg
shows that a nontrivial knot or link in R^{3}
necessarily has four colinear points.
 Coloring line arrangements. The graphs
formed by overlaying a collection of lines require three, four, or five colors,
depending on whether one allows three or more lines to meet at a point,
and whether the lines are considered to wrap around through infinity.
Stan Wagon
asks similar questions for unit circle arrangements.
 Common
misconception regarding a cube, Paul Bourke. No, the Egyptian
pyramids were not formed by dropped giant cubes from space.
 Complex
polytope. A diagram representing a complex polytope, from
H. S. M. Coxeter's home
page.
 Complex
regular tesselations on the Euclid plane, Hironori Sakamoto.
 Complexification
Gallery of Computation. Some kind of algorithmic art; I'm not sure
what algorithms were used to produce it but the results are pretty.
 a
computational approach to tilings. Daniel Huson investigates the
combinatorics of periodic tilings in two and three dimensions, including
a classification of the tilings by shapes topologically equivalent to
the five Platonic solids.
 Computer art
inspired by M. C. Escher and V. Vasarely, H. Kuiper.
 Conceptual proof that
inversion sends circles to circles, G. Kuperberg.
 Connect the dots.
Ed Pegg asks how many sides are needed in a (selfcrossing) polygon,
that passes through every point of an n*n grid.
I added a similar puzzle with circular arcs.
 constantwidth
shapes and
Reuleaux
triangle from Eric Weisstein's treasure trove.
 Constructing Boy's surface out of paper and tape.
 Constructing a regular pentagon inscribed in a circle, by straightedge and compass.
Scott Brodie.
 Contortion
Engineering. Some Escherlike impossible figures from Offworld
Press.
 Contour
plots with trig functions.
Eric Weeks discovers a method of making interesting nonmoiré patterns.
 Convex
Archimedean polychoremata, 4dimensional analogues of the
semiregular solids, described by CoxeterDynkin diagrams
representing their symmetry groups.
 Cool math: tessellations
 A Counterexample to Borsuk's Conjecture, J. Kahn and G. Kalai,
Bull. AMS 29 (1993). Partitioning certain highdimensional polytopes
into pieces with smaller diameter requires a number of pieces
exponential in the dimension.
 Counting
polyforms, with links to images of various packingpuzzle solutions.
 Covering
the Aztec diamond with onesided tetrasticks,
A. Wassermann.
 Covering points by rectangles.
Stan Shebs discusses the problem of finding a minimum number of
copies of a given rectangle that will cover all points in some set,
and mentions an application to a computer strategy game.
This is NPhard, but I don't know how easy it is to approximate;
most related work I know of is on optimizing the rectangle size for a cover
by a fixed number of rectangles.
 Cranes, planes, and cuckoo clocks.
Announcement for a talk on mathematical origami by Robert Lang.
 Crocheted Seifert surfaces by Matthew Wright. George Hart, Make Magazine.
 Andrew Crompton.
Grotesque geometry, Tessellations, Lifelike Tilings, Escher style drawings,
Dissection Puzzles, Geometrical Graphics, Mathematical Art.
Anamorphic Mirrors, Aperiodic tilings, Optical Machines.
 Crop
circles: theorems in wheat fields. Various hoaxers make geometric
models by trampling plants.
 Crumpling
paper: states of an inextensible sheet.
 Crystallographic
topology. C. Johnson and M. Burnett of Oak Ridge National Lab use
topological methods to understand and classify the symmetries of the
lattice structures formed by crystals. (Somewhat technical.)
 Crystallography
now, tutorial on the seventeen plane symmetry groups by
George Baloglou.
 CSE
logo. This java applet allows interactive control of a rotating
collection of cubes.
 Cube
Dissection. How many smaller cubes can one divide a cube into?
From Eric Weisstein's
treasure trove of mathematics.
 Cube triangulation.
Can one divide a cube into congruent and disjoint tetrahedra?
And without the congruence assumption,
how many higher dimensional simplices are needed to triangulate a hypercube?
For more on this last problem, see
Triangulating
an ndimensional cube, S. Finch, MathSoft,
and
Asymptotically efficient
triangulations of the dcube, Orden and Santos.
 The
Curlicue Fractal, Fergus C. Murray.
 Curvature of crossing convex curves.
Oded Schramm considers two smooth convex planar curves crossing at at
least three points, and claims that the minimum curvature of one is at
most the maximum curvature of the other. Apparently this is related
to conformal mapping. He asks for prior appearances of this problem
in the literature.
 Curvature of knots.
Steve Fenner proves
the FaryMilnor
theorem that any smooth, simple, closed curve in 3space must have
total curvature at least 4 pi.
 Cuttheknot logo.
With a proof of the origamifolklore that this foldedflat overhand
knot forms a regular pentagon.
 Dancing links.
Don Knuth discusses implementation details of polyomino search algorithms.
 Deconstructing
Marty. Tom Beard and Dorking Labs
analyze the Sierpinskicarpetlike geometry of New Zealand fractal
artist Martin Thompson's works.
 Dehn
invariants of hyperbolic tiles. The Dehn invariant is one way
of testing whether a Euclidean polyhedron can be used to tile space.
But as Doug Zare describes, there are hyperbolic tiles
with nonzero Dehn invariant.
 Delaunay and regular triangulations.
Lecture by Herbert Edelsbrunner, transcribed by Pedro Ramos and Saugata
Basu. The regular triangulation has been popularized by Herbert as the
appropriate generalization of the Delaunay triangulation to collections
of disks.
 Delaunay triangulation and points of
intersection of lines. Tom McGlynn asks whether the DT of a line
arrangement's vertices must respect the lines; H. K. Ruud shows that the
answer is no.
 Delta Blocks.
Hop David discusses ideas for manufacturing building blocks based on
the tetrahedronoctahedron space tiling depicted in Escher's "Flatworms".
 Deltahedra, polyhedra with equilateral triangle faces. From Eric Weisstein's treasure trove of mathematics.
 Dense spherepackings in hyperbolic space.
 Densest
packings of equal spheres in a cube, Hugo Pfoertner.
With nice raytraced images of each packing.
See also Martin
Erren's applet for visualizing the sphere packings.
 Design Science Toys.
 Dérivés
de l'hexagone. Art by Jerome Pierre based on modifications to the edges in a hexagonal tiling of the plane.
 Detecting the unknot in polynomial time,
C. Delman and K. Wolcott, Eastern Illinois U.
 DeVicci's Tesseract.
Higherdimensional generalizations of Prince Rupert's cube,
from MathSoft's favorite constants pages.
 Diamond theory.
Steven Cullinane studies the symmetries of the shapes formed by
splitting each square of a grid into dark and light triangles.
 Dictionary of Combinatorics,
Joe Fields, U. Illinois at Chicago.
 Diecast metal polyhedra
available for sale from Pedagoguery Software.
 Dilationfree planar graphs.
How can you arrange n points so that the set of all lines between them
forms a planar graph with no extra vertices?
 Disjoint
triangles. Any 3n points in the plane can be partitioned into n
disjoint triangles. A. Bogomolny gives a simple proof and discusses
some generalizations.
 Dissection challenges.
Joshua Bao asks for some dissections of squares into other figures.
 Dissection and dissection tiling.
This page describes problems of partitioning polygons
into pieces that can be rearranged to tile the plane.
(With references to publications on dissection.)
 Dissection
problemofthemonth from the Geometry Forum.
Cut squares and equilateral triangles into pieces and rearrange them to
form each other or smaller copies of themselves.
 A dissection puzzle. T. Sillke asks for dissections of two heptominoes into squares, and of a square into similar triangles.
 Dissections. From Eric Weisstein's treasure trove of mathematics.
 Dissections de polygones, réguliers ou non réguliers.
Various polygon dissections, animated in CabriJava.
 Dissections:
Plane & Fancy, Greg Frederickson's dissection book.
Greg also has a list of more links to
geometric
dissections on the web.
 Distances on
the surface of a rectangular box, illustrated using colored wavefronts
in this Java applet by Henry Bottomley.
 DNA, apocalypse, & the end of the mystery. A sacredgeometry analysis of "the geometric pattern of the heavenly city which is the template of the New Jerusalem".
 Do buckyballs fill hyperbolic space?
 Dodecafoam.
A fractal froth of polyhedra fills space.
 Dodecahedral
melon and other fruit
polyhedra, by Vi Hart.
 Dodecahedron
measures, Paul Kunkel.
 Domegalomaniahedron.
Clive Tooth makes polyhedra out of his
deep and inscrutable singular name.
 Sylvie Donmoyer
geometryinspired paintings including Menger sponges and
a behindthescenes look at Escher's Stars.
 Double
bubbles. Joel Hass investigates shapes formed by soap films
enclosing two separate regions of space.
 The downstairs half bath.
Bob Jenkins decorated his bathroom with ceramic and painted pentagonal tiles.
 A
dream about sphere kissing numbers.
 Dreamscope
screensaver module makes patterns with various Kaleidoscopic symmetries.
 DUST
software for visualization of Voronoi diagrams, Delaunay triangulations,
minimum spanning trees, and matchings, U. Köln.
 Dynamic
formation of PoissonVoronoi tiles. David Griffeath constructs
Voronoi diagrams using cellular automata.
 The Dynamic Systems
and Technology Project at Boston Univ. offers several Java applets
and animations of fractals and iterated function systems.
 Edgetangent polytope illustrating Koebe's
theorem that any planar graph can be realized as the set of tangencies
between circles on a sphere. Placing vertices at points having those
circles as horizons forms a polytope with all edges tangent to the sphere.
Rendered by POVray.
 An
eightpoint arrangement in which each perpendicular bisector passes
through two other points.
From Stan Wagon's
PotW archive.
 Eight foxes.
Daily geometry problems.
 The
85 foldings of the Latin cross, E. Demaine et al.
 Einstein's origami
snowflake game. Rick Nordal challenges folders to make a sequence of geometric
shapes with a single sheet of origami paper as quickly as possible.
 Electronic Geometry Models,
a refereed archive of interesting geometric examples and visualizations.
 Ellipse
game, or whackafocus.
 Elliptical
billiard tables, H. Serras, Ghent.
 Embedding
the hyperbolic plane in higher dimensional Euclidean spaces.
D. Rusin summarizes what's known; the existence of an isometric
immersion into R^{4} is apparently open.
 Enumeration of
polygon triangulations and other combinatorial representations of
the Catalan numbers.
 Equiangular
spiral. Properties of Bernoulli's logarithmic 'spiralis mirabilis'.
 An
equilateral dillemma. IBM asks you to prove that the only triangles
that can be circumscribed around an equilateral triangle, with their
vertices equidistant from the equilateral vertices, are themselves equilateral.
 Equilateral
pentagons. Jorge Luis Mireles Jasso investigates these polygons
and dissects various polyominos into them.
 Equilateral
pentagons that tile the plane, Livio Zucca.
 Equilateral
triangles. Dan Asimov asks how large a triangle will fit into a
square torus; equivalently, the densest packing of equilateral triangles
in the pattern of a square lattice.
There is only one parameter to optimize, the angle of the triangle to
the lattice vectors; my answer
is that the densest packing occurs when
this angle is 15 or 45 degrees, shown below.
(If the lattice doesn't have to be square, it is possible to get density
2/3; apparently this was long known, e.g. see Fáry,
Bull. Soc. Math. France 78 (1950) 152.)
Asimov also asks for the smallest triangle that will always cover at least
one point of the integer lattice, or equivalently a triangle
such that no matter at what angle you place copies of it on an integer lattice,
they always cover the plane; my guess is that the worst angle is parallel
and 30 degrees to the lattice, giving a triangle with 2unit sides
and contradicting an earlier answer to Asimov's question.
 The
equivalence of two facecentered icosahedral tilings with respect to
local derivability, J. Phys. A26 (1993) 1455. J. Roth dissects an
aperiodic threedimensional tiling involving zonohedra into another
tiling involving tetrahedra and vice versa.
 Equivalents of the parallel postulate.
David Wilson quotes a book by George Martin, listing 26 axioms
equivalent to Euclid's parallel postulate.
See also Scott Brodie's proof of equivalence with the Pythagorean theorem.
 Erich's
Packing Page. Erich Friedman enjoys packing geometric shapes into
other geometric shapes.
 Escher for real and
beyond
Escher for real.
Gershon Elber uses layered manufacturing systems to build 3d models of
Escher's illusions. The trick is to make some seeminglyflat surfaces
curve towards and away from the viewplane.
 Escher and the
Droste effect. Mathematical analysis of Escher's "Print Gallery".
 Escher
in the classroom, Jill Britton.
 Escher in the Palace.
The official web site of the Escher museum in The Hague.
 Escherlike tilings of interlocking animal
and human figures, by various artists.
 M. C. Escher: Artist
or Mathematician?
 Escher Fish. Silvio Levy's tessellation of the Poincare model of the hyperbolic plane by fish in M.C. Escher's style.
From the Geometry Center archives.
 Escher patterns,
Yoshiaki Araki.
 Escher
sphere construction kit.
 Escher's
buildings in origami.
 Escher's
combinatorial patterns, D. Schattschneider, Elect. J. Combinatorics.
 Escherian BlastEnded Skrewts.
 Escherization.
How to find a periodic tile as close as possible to a given shape?
Craig S. Kaplan, U. Washington.
 Eternity puzzle
made from "polydrafters", compounds of 306090 triangles.
See also the
mathpuzzle eternity page.
 Euclid 3:16.
Fun geometry Tshirt sighting, from Izzycat's blog. I want one.

Euclid's
Elements, animated in Java by David Joyce. See also
Ralph Abraham's
extensively illustrated
edition,
and this
manuscript excerpt from a copy in the Bodleian library made in the year 888.
 Examples,
Counterexamples, and Enumeration Results for Foldings and Unfoldings
between Polygons and Polytopes,
Erik D. Demaine, Martin L. Demaine, Anna Lubiw, Joseph O'Rourke,
cs.CG/0007019.
 Expansions
geometric pattern creation techniques by John S. Stokes III.
 Experiencing Geometry.
A poem by David Henderson.
 Explore the 120cell!
Free Windows+OpenGL+.Net software.
 Exploring
hyperspace with the geometric product. Thomas S. Briggs explains
some fourdimensional shapes.
 An
extension of Napoleon's theorem. Placing similar isosceles
triangles on the sides of an affinetransformed regular polygon results
in a figure where the triangle vertices lie on another regular polygon.
Geometer's sketchpad animation by John Berglund.
 Fagnano's
problem of inscribing a minimumperimeter triangle within another
triangle, animated in Java by A. Bogomolny.
See also part II, part III,
and a reversed version.
 Fagnano's theorem.
This involves differences of lengths in an ellipse.
Joe Keane asks why it is unusual.
 All the fair dice.
Pictures of the polyhedra which can be used as dice,
in that there is a symmetry taking any face to any other face.
 Fake dissection.
An 8x8 (64 unit) square is cut into pieces
which (seemingly) can be rearranged to form a 5x13 (65 unit) rectangle.
Where did the extra unit come from?
Jim Propp asks about possible threedimensional generalizations.
Greg
Frederickson supplies one.
See also
Alexander
Bogomolny's dissection of a 9x11 rectangle into a 10x10 square and
Fibonacci
bamboozlement applet.
 Famous curve applet index.
Over fifty wellknown plane curves, animated as Java applets.
 Robert
Fathauer's Compendium of Fractal Tilings.
 Robert Fathauer's Escherlike tesselation art.
 Chris Fearnley's 5 and 25 Frequency Geodesic Spheres rendered by POVRay.
 Helaman Ferguson mathematical sculpture.
 Fermat's spiral and the line between Yin and Yang.
Taras Banakh, Oleg Verbitsky, and Yaroslav Vorobets argue that the ideal
shape of the dividing line in a YinYang symbol is formed, not from two
semicircles, but
from Fermat's
spiral.
 Michael Field's gallery
of symmetric chaos images.
See his home page
for more links to pages on dynamics, symmetry, and chaos.
 Figure eight knot / horoball diagram.
Research of A. Edmonds into the symmetries of knots,
relating them to something that looks
like a packing of spheres.
The MSRI Computing Group uses
another horoball
diagram as their logo.
 Filling space with unit circles. Daniel
Asimov asks what fraction of 3dimensional space can be filled by a
collection of disjoint unit circles. (It may not be obvious that this
fraction is nonzero, but a standard construction allows one to construct
a solid torus out of circles, and one can then pack tori to fill space,
leaving some uncovered gaps between the tori.) The geometry center has
information in several places on this problem, the best being an
article
describing a way of filling space by unit circles (discontinuously).
 Find
all polytopes. Koichi Hirata's web software for finding all ways of
gluing the edges of a polygon so that it can fold into a convex polytope.
 Finding
the wood by the trees. Marc van Kreveld studies strategies by which
a blind man with a rope could map out a forest.
 Adrian Fisher Maze Design
 Fisher Pavers.
A convex heptagon and some squares produce an interesting fourway
symmetric tiling system.
 Ephraim Fithian's
geometry web page. Teaching activities, test previews, and some
Macintosh game software.
 Five circle theorem.
Karl Rubin and Noam Elkies asked for a proof that a certain construction
leads to five cocircular points. This result was subsequently discovered
by Allan Adler and Gerald Edgar to be essentially the same as a theorem
proven in 1939 by F. Bath.
 Five
Platonic solids and a soccerball.
 Five
spacefilling polyhedra. And not the ones you're likely thinking of,
either.
Guy Inchbald, reproduced from Math. Gazette 80, November 1996.
 Fivefold symmetry in crystalline quasicrystal lattices, Donald L. D. Caspar and Eric Fontano.
 Flat
equilateral tori. Can one build a polyhedral torus in which all
faces are equilateral triangles and all vertices have six incident
edges? Probably not but this physical model comes close.
 The
flat torus in the threesphere. Thomas Banchoff animates the
Hopf fibration.
 Flatland:
A Romance of Many Dimensions.
 Flexagons.
Folded paper polyiamonds which can be "flexed" to show different sets of
faces. See also Harold
McIntosh's flexagon papers,
including copies of the original 1962 ConradHartline papers,
also
mirrored on Erik Demaine's website.
 Flexible polyhedra. From Dave Rusin's known math pages.
 Folding
geometry. Wheaton college course project on modular origami.
 The
Four Color Theorem.
A new proof by Robertson, Sanders, Seymour, and Thomas.
 Four dice hypercube visualization.
 The Fourth
Dimension. John Savard provides a nice graphical explanation of the
fourdimensional regular polytopes.
 Fourdimensional visualization.
Doug Zare gives some pointers on highdimensional visualization
including a description of an interesting chain of successively higher
dimensional polytopes beginning with a triangular prism.
 Fourier
series of a gastropod. L. Zucca uses Fourier analysis to square
the circle and to make an odd spirallike shape.
 4x4x4 Soma
Cube problem.
 Fractal
analysis of Jackson Pollock's abstract paintings.
 The fractal art of
Wolter Schraa. Includes some nice reptiles and sphere packings.
 Fractal bacteria.
 A fractal betaskeleton with high dilation.
Betaskeletons are graphs used, among other applications, in predicting
which pairs of cities should be connected by roads in a road network.
But if you build your road network this way, it may take you a long time
to get from point a to point b.
 Fractal
broccoli. Photo by alfredo matacotta.
See also this French page.
 Fractal fiber bundles.
Troy Christensen ponders origami on the fabric folds of spacetime.
 The
fractal gallery tour: Sierpinski tetrahedron
 Fractal geometry and complex bases.
Publications and software by W. Gilbert.
 Fractal instances of the traveling salesman problem, P. Moscato, Buenos Aires.
 Fractal knots, Robert Fathauer.
 Fractal patterns formed by repeated inversion of circles:
Indra's Pearls
Inversion graphics gallery, Xah Lee.
Inversive circles, W. Gilbert, Waterloo.
 Fractal patterns
in the real world, Ian Stewart.
 Fractal
planet and fractal
landscapes. Felix Golubov makes random triangulated polyhedra in Java
by perturbing the vertices of a recursive subdivision.
 Fractal
reptiles and other
tilings by IFS
attractors, Stewart Hinsley.
 Fractal
resources. A collection of web links by John Mathews.
 Fractal
skewed web. Sierpinski tetrahedron by Mary Ann Conners.
 Fractal tilings.
 Fractals.
The spanky fractal database at Canada's national meson research facility.
 Fractals by da duke.
Raytraced Menger sponges and Sierpinski gaskets.
 Fractiles,
multicolored magnetic rhombs with angles based on multiples of pi/7.
 Frequently asked questions about spheres. From Dave Rusin's known math pages.
 Erich Friedman's dissection puzzle.
Partition a 21x42x42 isosceles triangles into six smaller triangles,
all similar to the original but with no two equal sizes.
(The link is to a drawing of the solution.)
 Frustro, a font made of Escherian impossible figures.
 Fun with Fractals and
the Platonic Solids. Gayla Chandler places models of polyhedra and
polyhedral fractals such as the Sierpinski tetrahedron in scenic outdoor
settings and photographs them there.
 Gallery of interactive online geometry.
The Geometry Center's collection includes programs for generating
Penrose tilings, making periodic drawings a la Escher in the Euclidean
and hyperbolic planes, playing pinball in negatively curved spaces,
viewing 3d objects, exploring the space of angle geometries, and
visualizing Riemann surfaces.
 Gaudí's
geometric models. From the Gaudí museum in
Parc Güell, Barcelona.
 Gauss' tomb. The story that he asked
for (and failed to get) a regular 17gon carved on it leads to some
discussion of 17gon construction and perfectly scalene triangles.
 Gaussian
continued fractions.
Stephen Fortescue discusses some connections between basic
numbertheoretic algorithms and the geometry of tilings
of 2d and 3d hyperbolic spaces.
 Gavrog, a Java package for
visualizing 2d and 3d ornamental patterns with high degrees of symmetry.
 Gecko Stone
interlocking concrete pavers in geometric and animal shapes,
designed by John August.
 Geek bodyart.
Geometric calculations for fitting your piercings.
 Generating
Fractals from Voronoi Diagrams, Ken Shirriff, Berkeley and Sun.
 Geodesic dome
design software. Now you too can generate triangulations of the sphere.
Freeware for DOS, Mac, and Unix.
 Geodesic
math. Apparently this means links to pages about polyhedra.
 Geometria
Javabased software for constructing and measuring polyhedra
by transforming and slicing predefined starting blocks.
 Geometric
Arts. Knots, fractals, tesselations, and op art.
Formerly Quincy
Kim's World of Geometry.
 Geometric
Dissections by Gavin Theobald.
 Geometric graph coloring problems
from "Graph Coloring Problems" by Jensen and Toft.
 Geometric
Java applets by Fergus C. Murray
produce screensaverlike interactive images.
See also his
noninteractive
mathematical animations and
stills.
 Geometric
metaphors in literature, K. Kovaka.
 Geometric probability question.
What is the probability that the shortest paths between three random
points on a projective plane form a contractible loop?
 Geometric paper folding. David Huffman.
 Geometric
probability constants. From MathSoft's favorite constants pages.
 Geometric topology preprint server.
 A
Geometrical Picturebook of finite and combinatorial geometries,
B. Polster, to be published by Springer.
 Géometriés non euclidiennes.
Description of several models of the hyperbolic plane and
some interesting hyperbolic constructions.
From the Cabri geometry site.
(In French.)
 Geometrinity, geometric sculpture by Denny North.
 Geometry of alphabets.
Sacred geometry wackiness from the Library of Halexandria.
Something about how the first verse of Genesis forms a dodecahedron, or
a flower, or maybe a candlestick, somehow leading to squared circles,
spiraling shofars, and circumscribed tetrahedra.
 Geometry
problems involving circles and triangles, with proofs.
Antonio Gutierrez.
 Geometry,
algebra, and the analysis of polygons. Notes by M. Brundage on a
talk by B. Grünbaum on vector spaces formed by planar
ngons under componentwise addition.
 The geometry
of ancient sites.
 Geometry
corner with Martin Gardner.
He describes some problems of cutting polygons into similar and
congruent parts. From the
MAT 007 I News.
 Geometry
forum discussion on the Reuleux triangle and its ability to drill
out (most of) a square hole.
 Geometry in Hawaiian history and culture
 Geometry and the Imagination in Minneapolis.
Notes from a workshop led by Conway, Doyle, Gilman, and Thurston.
Includes several sections on polyhedra, knots, and symmetry groups.
 Geometry
turned on  making geometry dynamic. A book on the use of
interactive software in teaching.
 GeoProof interactive
geometry software including automated theorem proving methods.
 Gerard's pentomino page.
 Ghost
diagrams, Paul Harrison's software for finding tilings with
Wangtilelike hexagonal tiles, specified by matching rules on their
edges. These systems are Turingcomplete, so capable of forming all
sorts of complex patterns; the web site shows binary circuitry, fractals,
1d cellular automaton simulation, Feynman diagrams, and more.
 Glass
dodecahedron. Custommade for Clive Tooth by Bob Aurelius.
 Glowing green rhombic triacontahedra in space.
Rendered by Rob Wieringa for the MayJune 1997 Internet Ray Tracing Competition.
 The golden bowls
and the logarithmic spiral.
 The golden ratio in an equilateral triangle.
If one inscribes a circle in an ideal hyperbolic triangle,
its points of tangency form an equilateral triangle
with side length 4 ln phi!
One can then place horocycles centered on the ideal triangle's vertices
and tangent to each side of the inner equilateral triangle.
From the Cabri geometry site. (In French.)
 The
golden section and geometry. Somehow leading to questions like how
many stars there are on the US flag.
 Golden rectangles, Curtis McMullen.
 A
golden sales pitch. Julie Rehmeyer dissects the myth of the golden
ratio in classical art and describes some new uses for it in commerce.
 The golden
section and Euclid's construction of the dodecahedron, and
more
on the dodecahedron and icosahedron,
H. Serras, Ghent.
 Golden
spiral flash animation, Christian Stadler.
 Goldene
Schnittmuster. Article in German on Penrose tiling and related topics.
 Golygons,
polyominoes with consecutive integer side lengths.
See also the Mathworld
Golygon page.
 Gömböc, a
convex body in 3d with a single stable and a single unstable point of
equilibrium. Placed on a flat surface, it always rights itself; it may
not be a coincidence that some tortoise shells are similarly shaped.
See also Wikipedia, Metafilter, New
York Times.
 Graham's hexagon, maximizing the ratio of area to diameter.
You'd expect it to be a regular hexagon, right? Wrong.
From MathSoft's favorite constants pages.
See also
Wolfgang
Schildbach's java animation of this hexagon and similar ngons for larger values of n.
 The Graph of the Truncated Icosahedron and the Last Letter of Galois,
B. Kostant, Not. AMS, Sep. 1995.
Group theoretic mathematics of buckyballs.
See also J. Baez's
review of Kostant's paper.
 Graphite
with growth spirals on the basal pinacoids. Pretty pictures of
spirals in crystals. (A pinacoid, it turns out, is a plane parallel to
two crystallographic axes.)
 Polyhedra 
homage to U. A. Graziotti.
 Great
math programs. Xah Lee reviews mathematical software, focusing on
educational Macintosh applications. Includes sections on geometric
visualization, fractals, cellular automata, and geometric puzzles.
 Great
triambic icosidodecahedron quilt,
made by Mark Newbold and Sarah Mylchreest with the aid of
Mark's hyperspace star polytope slicer.
 Greek mathematics and its modern heirs.
Manuscripts of geometry texts by Euclid, Archimedes,
and others, from the Vatican Library.
 Melinda
Green's geometry page. Green makes models of regular sponges
(infinite nonconvex generalizations of Platonic solids) out of plastic
"Polydron" pieces.
 Greenhaired geometric prehominids.
 Greg's
favorite math party trick. A nice visual proof of van Aubel's
theorem, that equal perpendicular line segments connect the opposite
centers of squares exterior to the sides of any quadrilateral.
See also Wikipedia,
MathWorld,
Geometry from
the land of the Incas,
interactive
Java applet.
 Grid subgraphs.
Jan Kristian Haugland looks for sets of lattice points that induce
graphs with high degree but no short cycles.
 Rona
Gurkewitz' Modular Origami Polyhedra Systems Page.
With many nice images from two modular origami books by
Gurkewitz, Simon, and Arnstein.
 Hales,
Honeybees, and Hexagons. Thomas Hales proves the optimality of
bees' hexagonal honeycomb structure. Ivars Peterson, Science News Online.
 Ham
Sandwich Theorem: you can always cut your ham and two slices of
bread each in half with one slice, even before putting them together
into a sandwich.
From Eric Weisstein's treasure trove of mathematics.
 Bradford
HansenSmith makes geometric art out of paper plates.
 Happy cubes and other threedimensional polyomino puzzles.
 Happy Pentominoes, Vincent Goffin.
 Harary's animal game. Chris Thompson
asks about recent progress on this generalization of tictactoe and
gomoku in which players place stones attempting to form certain polyominoes.
 George Hart's
geometric sculpture.
 JeanPierre Hébert  Studio.
Algorithmic and geometric art site.
 Hebesphenomegacorona
onna stick in space! Space Station Science picture of
the day. In case you don't remember what a hebesphenomegacorona is, it's
one of the Johnson solids: convex polyhedra with regularpolygon faces.
 Hecatohedra.
John Conway discusses the possible symmetry groups of hundredsided polyhedra.
 Hedronometry.
Don McConnell discusses equations relating the angles and face areas
of tetrahedra. See also McConnell's hedronometry site.
 Helical geometry.
Ok, renaming a hyperbolic paraboloid a "helical right triangle"
and saying that it's "a revolutionary foundation for new knowledge"
seems a little cranky but there are some interesting pictures of shapes
formed by compounds of these saddles.
 Hippias'
Quadratrix, a curve discovered around 420430BC, can be used to
solve the classical Greek problems of squaring the circle, trisecting
angles, and doubling the cube.
Also described in
St.
Andrews famous curves index,
Xah's special curve index,
Eric
Weisstein's treasure trove, and
H. Serras'
quadratrix page.
 Heesch's problem. How many times can a shape
be completely surrounded by copies of itself, without being able to tile
the entire plane? W. R. Marshall and C. Mann have recently made
significant progress on this problem using shapes formed by indenting
and outdenting the edges of polyhexes.
 Heilbronn
triangle constants. How can you place n points in a square
so that all triangles formed by triples of points have large area?
 Helical Gallery.
Spirals in the
work of M. C. Escher
and in Xray observations of the sun's corona.
 Heptomino
Packings.
Clive Tooth shows us all 108 heptominos, packed into a 7x9x12 box.
 Hermite's constants.
Are certain values associated with dense lattice packings of spheres
always rational?
Part of Mathsoft's
collection of
mathematical constants.
 Hero's
Formula for the area of a triangle in terms of its side lengths.
Mark Dominus explains.
 High school
buckyball art.
Kerry Stefancyk, Allison Cahill, and Jessica Smith make polyhedral
models out of stained glass.
 Hilbert's
3rd Problem and Dehn Invariants.
How to tell whether two polyhedra can be dissected into each other.
See also Walter
Neumann's paper connecting these ideas with problems of
classifying manifolds.
 Hinged dissections of polyominoes
 Hinged kite mirror dissection.
General techniques for cutting any polygon into pieces that can be
unfolded and refolded to form the polygon's mirror image.
 Chuck Hoberman's Unfolding Structures.
 Hollow
pyramid tetrasphere puzzle.
 Holyhedra.
Jade Vinson solves a question of John Conway on the existence of
finite polyhedra all of whose faces have holes in them
(the Menger sponge provides
an infinite example).
 Hopf fibration.
R. Kreminski,
the U.
Sheffield maths dept., and
MathWorld
explain and animate the partition of a 3sphere
into circles.
 Houtrust Relief.
Nice photo of a 3d version of one of Escher's birdfish textures, on the
wall of a water purifying plant in The Netherlands.
The same photographer has several
other Escher photos including one of Metamorphoses in the
Hague post office.
 How many intersection points
can you form from an nline arrangement?
Equivalently, how many opposite pairs of faces can an nzone
zonohedron have?
It must be a number between n1 and n(n1)/2,
but not all of those values are possible.
 How many
points can one find in threedimensional space so that all triangles
are equilateral or isosceles?
One eightpoint solution is formed by placing three points
on the axis of a regular pentagon.
This problem seems related to the fact that
any planar point set forms O(n^{7/3})
isosceles triangles; in three dimensions, Theta(n^{3}) are possible
(by generalizing the pentagon solution above). From Stan Wagon's
PotW archive.
 How to construct a golden rectangle, K. Wiedman.
 How to fold a piece
of paper in half twelve times. Britney Gallivan took on this
previouslythoughtimpossible task as a high school science project,
worked out an accurate mathematical model of the requirements,
and used that model to complete the task.
 How
to write "computational geometry" in Japanese (or Chinese).
 D. Huson's favorite hyperbolic tiling.
 George
Huttlin's Puzzle Page. Some ramblings in the world of polyominoes
and hexiamonds.
 Human Geometry
and Naked Geometry. The
human form as a building block of larger geometric figures, by Mike
Naylor.
 Hyacinthos
triangle geometry mailing list.
 Hyper hyper! Extra dimensions
 HypArr,
software for modeling and visualizing convex polyhedra and plane
arrangements,
now seems to be incorporated as a module in a larger
Matlab library for multiparametric analysis.
 Hyperbolic crochet coral
reef, the Institute for Figuring.
Daina Taimina's technique for crocheting yarn into hyperbolic surfaces
forms the basis for an exhibit of woolen undersea fauna and flora.
 Hyperbolic
games. Freeware multiplatform software for games such as Sudoku on
hyperbolic surfaces, intended as a way for students to gain familiarity
with hyperbolic geometry. By Jeff Weeks.
 Hyperbolic geometry. Visualizations and animations including
several pictures of hyperbolic tessellations.
 Hyperbolic Knot.
From Eric Weisstein's treasure trove of mathematics.
 Hyperbolic packing of convex bodies.
William Thurston answers a question of
Greg Kuperberg, on
whether there is a constant C such that every convex body in the
hyperbolic plane can be packed with density C. The answer is no  long
skinny bodies can not be packed efficiently.
 The HyperSphere, from an Artistic point of View,
Rebecca Frankel.
 Hyperbolic
shortbread. The Davis math department eats a Poincaré model
of a tiling of the hyperbolic plane by 06090 triangles.
 The
hyperbolic surface activity page. Tom Holroyd describes hyperbolic
surfaces occurring in nature, and explains how to make a paper model of
a hyperbolic surface based on a tiling by heptagons.
 Hyperbolic Tessellations, David Joyce, Clark U.
 Hyperbolic tiles.
John Conway answers a question of Doug Zare on the polyhedra
that can form periodic tilings of 3dimensional hyperbolic space.
 A hyperboloid
in Kobe, Japan, in the 1940s.
 Hyperbolic and
spherical tiling gallery, Bernie Freidin.
 Hyperbolic Geometry using Cabri
 Hyperbolic
planar tessellations, image gallery of many regular and semiregular
tilings by Don Hatch.
 Hypercube's Home Page. Speculations on the fourth dimension collected by Eric Saltsman.
 Hypercube fun.
John Atkeson finds a nice recursive drawing pattern for high dimensional
hypercubes in two dimensional planes.
 Hypercube game.
Experience the fourth dimension with an interactive, stereoscopic
java animation of the hypercube.
 Hypercube visualization,
Drew Olbrich.
 Hypercubes
in hyper perspective. Redblue 3d visualizations produced with the
virtual flower system.
 Hyperdimensional Java.
Several web applets illustrating highdimensional concepts, by
Ishihama Yoshiaki.
 HyperGami program for unfolding polyhedra, also described in
this
article from the American Scientist.
 HyperGami gallery. Paper polyhedral penguins, pinapples, pigs, and more.
 Hypergami polyhedral playground.
Rotatable wireframe models of platonic solids and of the penguinhedron.
 Hyperspace star
polytope slicer, Java animation by Mark Newbold.
 Hyperspace structures. Exploring the fourth dimension.
 Hyperspheres. Eric Weisstein calculates volumes and surface areas of hyperspheres, which curiously reach a maximum for dimensions around 5.257 and 7.257 respectively.
 Hyperspheres,
hyperspace, and the fourth spatial dimension. M. R. Feltz
views the universe as a closed cosmic hypersphere.
 The icosahedron, the great icosahedron, graph designs, and Hadamard matrices. Notes by M. Brundage from a talk by M. Rosenfeld.
 Icosamonohedra,
icosahedra made from congruent but not necessarily equilateral triangles.
 Ideal
hyperbolic polyhedra
raytraced by Matthias Weber.
 Images of geometry. From the geometry center graphics archives.
More
images, from
"Interactive
Methods for Visualizable Geometry", A. Hanson, T. Munzner, and G. Francis.
 Imagine, Geometry.
Starting with visions of prenatal consciousness in 1968.
Primarycolored animations of platonic solids
turn your brain cells into puffed, expanded dodecahedra.
 Improved
dense packing of equal disks in a square, D. Boll et al., Elect. J. Combinatorics.
 Guy Inchbald's
polyhedra pages.
Stellations, hendecahedra, duality, spacefillers, quasicrystals, and more.
 In
plane sight. Equilateral triangle visibility problem from Andy
Drucker. See also
here.
 Infect.
Eric Weeks generates interesting colorings of aperiodic tilings.
 Infinite
families of simplicial arrangements.
 Information on
Pentomino Puzzles and Information on Polyominoes, from
F. Ruskey's Combinatorial Object Server.
 The Institute for
Figuring's online exhibit on hyperbolic space.
 Integer distances.
Robert Israel gives a nice proof (originally due to Erdös) of the
fact that,
in any noncolinear planar point set in which all distances
are integers, there are only finitely many points.
Infinite sets of points with rational distances are known,
from which arbitrarily large finite sets of points with integer
distances can be constructed; however it is open whether there are even
seven points at integer distances in general position
(no three in a line and no four on a circle).
 IFS and Lsystems.
Vittoria Rezzonico grows fractal broccoli and Sierpinski pyramids.
 Interactive Delaunay triangulation and Voronoi diagrams:
VoroGlide, Icking, Klein, Köllner, Ma, Hagen.
D. Watson, CSIRO, Australia.
Baker et al., Brown U.
Paul Chew, Cornell U.
 Interactive
fractal polyhedra, Evgeny Demidov.
 Interconnection Trees. Java minimum spanning tree implementation, Joe Ganley, Virginia.
 Interlocking puzzle pieces and other geometric toys.
 Interlocking Puzzles LLC
are makers of hand crafted hardwood puzzles including burrs,
pentominoes, and polyhedra.
 The International
BoneRoller's Guild ponders the
isohedra:
polyhedra that can act as fair dice, because all faces are symmetric to
each other.
 Intersecting cube diagonals.
Mark McConnell asks for a proof that, if a convex polyhedron
combinatorially equivalent to a cube has three of the four
body diagonals meeting at a point, then the fourth one meets
there as well. There is apparently some connection to toric varieties.
 Intriguing
tessellations.
Marjorie Rice's Escherlike art.
 Inversive geometry. Geometric transformations of circles, animated with CabriJava.
 Investigating
Patterns: Symmetry and Tessellations.
Companion site to a middle school text by Jill Britton,
with links to many other web sites involving symmetry or tiling.
 Irrational
tiling by logical quantifiers. LICS proceedings cover art by Alvy Ray Smith, based on the Penrose tiling.
 Irreptiles.
Karl Scherer and Erich Friedman generalize the concept of a reptile
(tiling of a shape
by smaller copies of itself) to allow the copies to have different scales.
See also
Karl Scherer's twopart irreptile puzzle.
 Islamic
geometric art.
 Isoperimetric
polygons. Livio Zucca groups grid polygons by their perimeter
instead of by their area. For small integer perimeter the results are
just polyominos but after that it gets more complicated...
 The isoperimetric problem for pinwheel tilings.
In these aperiodic tilings (generated by a substitution system involving
similar triangles) vertices are connected by paths almost as good
as the Euclidean straightline distance.
 Isosceles
pairs. Stan Wagon asks which triangles can be dissected into
two isosceles triangles.
 Isotiles,
workbook on the shapes that can be formed by combining isosceles
triangles with side lengths in the golden ratio.
 Jacqui's
Polyomino Workshop.
Activities associated with polyominoes, aimed at the level of primary
(or elementary) school mathematics.
 Japanese
Temple Geometry, Gordon Coale.
See also this
clickable
temple geometry tablet map.
Unfortunately Scientific American seems to have taken
down their (May 1998) article on the subject.
 Japanese Triangulation Theorem. The sum of inradii in a triangulation of a
cyclic polygon doesn't depend on which triangulation you choose!
Conversely, any polygon for which this is true is cyclic.
 Java
applets on mathematics, Walter Fendt.
 Java gallery of geometric algorithms, Z. Zhao, Ohio State U.
 Java lamp, S. M.
Christensen.
 Java quadric surface raytracer, P. Flavin.
 Java pentomino puzzle solver, D. Eck, Hobart and William Smith Colleges.
 Iwan Jensen
counts polyominos (aka lattice animals), paths, and various related quantities.
 Jenn
opensource software for visualizing Cayley graphs of Coxeter groups
as symmetric 4dimensional polytopes.
 Jiang ZheMing's
geometry challenge. A pretty problem involving cocircularity of five
points defined by circles around a pentagram.
 Jim ex machina.
Escherlike tessellations by Jim McNeill.
 Joe's Cafe.
Java applets for creating images of iteration systems
a la Field and Golubitsky's "Symmetry in Chaos".
 Johnson Solids, convex polyhedra with regular faces. From Eric Weisstein's
treasure trove of mathematics.
 Jordan sorting. This is the problem of
sorting (by xcoordinate) the intersections of a line with a simple
polygon. Complicated linear time algorithms for this are known (for
instance one can triangulate the polygon then walk from triangle to
triangle); Paul Callahan discusses an alternate algorithm based on the
dynamic optimality conjecture for splay trees.
 Jovo Click 'n Construct.
Plastic clicktogether triangular, square, and pentagonal tiles for
building models of polyhedra and polygonal tilings.
Includes a mathematical model
gallery
showing examples of shapes constructable from Jovo.
 K12
on G6. Carlo Séquin investigates how to draw a 12vertex
complete graph as symmetrically as possible on a sixhandle surface
(the minimum genus surface on which it can be drawn without crossings).
 Sándor Kabai's
mathematical graphics, primarily polyhedra and 3d fractals.
 Kabon Triangles. How many disjoint triangles can you make out of n line segments? From Eric Weisstein's treasure trove of mathematics.
According to Toshi Kato, these should actually be called Kobon
triangles, after Kobon Fujimura in Japan;
Kato also tells me that Mr. Saburo Tamura proved a bound of
F(n) <n(n2)/3.
 Kadon Enterprises,
makers of games and puzzles including polyominoes and Penrose tiles.
 The
KakeyaBesicovitch problem.
Paul Wellin describes this famous
problem of rotating a needle in a planar set of minimal area. As it
turns out the area can be made arbitrarily close to zero. See also
Steven Finch's
page on KakeyaBesicovitch constants, and
Eric
Weisstein's page on the Kakeya Needle Problem.
 Kaleidoscope
geometry, Ephraim Fithian.
 Kaleidotile
software for visualizing tilings of the sphere, Euclidean plane, and
hyperbolic plane.
 Kali,
software for making symmetrical drawings based on any of the 17 plane
tiling groups.
 Robert
F. Kauffman's fractal and Escherian art, with
Escherlike animated animalform tilings.
 kDtree
demo. Java applet by Jacob Marner.
 Keller's cubetiling conjecture is false in high dimensions,
J. Lagarias and P. Shor, Bull. AMS 27 (1992).
Constructs a tiling of tendimensional space by unit hypercubes
no two of which meet facetoface, contradicting a
conjecture
of Keller
that any tiling included two facetoface cubes.
 Aaron Kellner Linear Sculpture.
Art in the form of geometric tangles of metal and wood rods.
 Kelvin conjecture counterexample.
Evelyn Sander forwards news about the discovery by Phelan and Weaire
of a better way to partition space
into equalvolume lowsurfacearea cells.
Kelvin had conjectured that the truncated octahedron provided the optimal
solution, but this turned out not to be true.
See also Ruggero Gabbrielli's comparison of equalvolume partitions and
JavaView
foam models.
 Richard
Kenyon's Gallery of tilings by squares and equilateral triangles of
varying sizes.
 The Kepler Conjecture on dense packing of spheres.
 KeplerPoinsot
Solids, concave polyhedra with starshaped faces. From Eric
Weisstein's treasure trove of mathematics. See also
H. Serras'
page on KeplerPoinsot solids.
 King
of Infinite Space. A new biography of H. S. M. Coxeter by Siobhan Roberts.
 Kissing
numbers. Eric Weisstein lists known bounds on the kissing numbers
of spheres in dimensions up to 24.
 Kleinian Groups.
Rather incomprehensible exposition of hyperbolic symmetry, but plenty of pretty pictures.
 The
KneserPoulsen Conjecture.
Bezdek and Connelly solve an old problem about pushing disks together.
 Knight's Move Tessellations.
Dan Thomasson looks at tilings with polygons that can be traced out by
knight moves on a chessboard.
 Knight's tour
art,
Dan Thomasson.
 Knot art. Keith and Fran Griffin.
 Knot pictures. Energyminimized smooth and polygonal knots, from the
ming
knot evolver, Y. Wu, U. Iowa.
 Knotology.
How to form regular polyhedra from folded strips of paper?
 KnotPlot.
Pictures of knots and links, from Robert Scharein at UBC.
 Knots on the Web,
P. Suber. Includes sections on knot tying and knot art as well as knot
theory.
 Mike Kolountzakis' publications include several recent papers on lattice tiling.
 Kummer's
surface. Nice raytraced pictures of a quartic surface with lots of
symmetries.
 Kurschak's
tile and Kurschak's theorem about the area of a circleinscribed dodecagon.
 Labyrinth tiling.
This aperiodic substitution tiling by equilateral and isosceles triangles
forms fractal spacefilling labyrinths.
 Landry Art,
Escheresque tessellations, and balsa and paper polyhedra,
including some prints, tshirts, and models available for purchase.
 The
Landscape of Geometry Terms.
Musical and typographic presentation of geometric nomenclature.
 Language Generator Tool and
Die Lab. Tennis ball theorems, hourglass theorems, and cellular
hierarchies. From a truly selfprogrammed individual.
 Largest
5gon in a square, or more interestingly smallest equilateral
pentagon inscribed in a square.
Posting to sci.math by Rainer Rosenthal.
 Lattice
animal constant. What is the asymptotic behavior of the number
of nsquare polyominos, as a function of n?
From MathSoft's favorite constants pages.
 Lattice pentagons.
The vertices of a regular pentagon are not the subset of any lattice.
 Layered graph drawing.
 Laying
Track. The combinatorics and topology of Brio train layouts. From
Ivars Peterson's MathTrek.
 Leaper tours.
Can generalized knights jump around generalized chessboards visiting each square once? By Ed Pegg Jr.
 Tom
Lechner's Sculptures. Lechner makes geometric models from wood,
water, plexiglass, and steel.
 Lego
Pentominos, Eric Harshbarger. He writes that the hard part was
finding legos in enough different colors.
See also his
Lego
math puzzles
and pentominoes
pages.
 Lego sextic.
Clive Tooth draws infinity symbols using lego linkages,
and analyzes the resulting algebraic variety.
 Lenses,
rationalangled equilateral hexagons can tile the plane in various
interesting patterns. See also Jorge Mireles' nice
lens
puzzle applet: rotate decagons and stars to get the pieces into the
right places.
 Mathematical imagery by Jos Leys.
Knots, Escher tilings, spirals, fractals, circle inversions, hyperbolic
tilings, Penrose tilings, and more.
 LightSource
sacred geometry software.
 Limerick #18124:
affine geometry.
 Line
designs for the computer. Jill Britton brings to the web material
from John Millington's 1989 book on geometric patterns formed by
stitching yarn through cardboard. The Java simulation of a Spectrum
computer running Basic programs is a little (ok a lot) clunky, and froze
Mozilla when I tried it, but there's also plenty of interesting static content.
 Line fractal.
Java animation allows user control of a fractal formed by repeated
replacement of line segments by similar polygonal chains.
 Links2go: Polyhedra
 LiveCube polycube puzzle building
toy.
 Logical art and the
art of logic, pentomino art, philosophy, and DOS software,
G. AlbrechtBuehler.
 Logspiral tiling,
and other radial
and spiral tilings, S. Dutch.
 Looking at
sunflowers. In this abstract of an undergraduate research paper,
Surat Intasang investigates the spiral patterns formed by sunflower seeds,
and discovers that often four sets of spirals can be discerned,
rather than the two sets one normally notices.
 Louis Bel's povray galleries:
les
polyhèdres réguliers,
knots,
and
more knots.
 Jim Loy's
geometry pages. With special emphases on geometric constructions
(and nonconstructions such as angle
trisection) as well as many nice
Cinderella animations.
 Lunatic's
guide to polyhedra.
 M203 Cabri Page. Wilson Stothers explains the geometry of conic sections
using the Cabrigéomètre dynamic geometry software system.
 Magazine Puzzle
Fun. Fifteen years of back issues of an Argentine magazine about
pentominoes (in English).
 A magic geometric constant optimized by the Reuleaux triangle.
 Magical
transformations. Wil Laan animates several dissections and almostdissections.
 MagicTile.
Klein's quartic meets the Rubik's cube, by Roice Nelson.
 Maille Weaves.
Different repetitive patterns formed by linked circles along a plane in space,
as used for making chain mail. Along with some linear patterns for
jewelry chains.
 Making a Sierpinski pyramid with Maple, S. Sutherland, Stony Brook.
 3Manifolds from regular solids.
Brent Everitt lists the finite volume orientable hyperbolic and
spherical 3manifolds obtained by identifying the faces of regular solids.
 Manipula
Math with Java. Interactive applets to help students grasp the
meaning of mathematical ideas.
 A map of all
triangles and the search for the ideal acute scalene triangle, Robert Simms.
 Maple
polyhedron gallery.
 Marcin Malinowski's Escherlike tesselations.
 The Margulis Napkin Problem.
Jim Propp asked for a proof that the perimeter of a flat origami
figure must be at most that of the original starting square.
Gregory Sorkin provides a simple example showing that on the contrary,
the perimeter can be arbitrarily large.
 Marius
Fine Art Studio Sacred Geometry Art.
Prints and paintings for sale of various geometric designs.
 Martin's pretty
polyhedra. Simulation of particles repelling each other on the
sphere produces nice triangulations of its surface.
 Materialized Mathematical
Models.
Jan de Koning exercises his furnituremaking skills by making
wood, plastic, stone and steel polyhedra.
 Match
sticks in the summer. Ivars Peterson discusses the graphs
that can be formed by connecting vertices by noncrossing equallength
line segments.
 Math Made Easy:
Geometry interactive video supplemental learning materials.
 Math Pages: Geometry
 The
Math Professor: Geometry Resources on the Web.
 Math
Quilts.
 Mathematica Graphics Gallery: Polyhedra
 Mathematica
Menger Sponge, Robert M. Dickau.
 Mathematical balloon
twisting. Vi Hart makes polyhedra and polyhedral tangles from balloons.
 Mathematical
lego sculptures and Escher Lego, Andrew Lipson.
 Mathematical
origami, Helena Verrill. Includes constructions of a shape with
greater perimeter than the original square, tessellations, hyperbolic
paraboloids, and more.
 A
mathematical theory of origami. R. Alperin defines fields of
numbers constructible by origami folds.
 Mathematically
correct breakfast. George Hart describes how to cut a single bagel
into two linked Möbius strips. As a bonus, you get more surface
area for your cream cheese than a standard sliced bagel.
 The
mathematics of polyominoes, K. Gong. Counts of kominoes,
Macintosh polyomino software, and more links.
 Mathematics
in John Robinson's symbolic sculptures. Borromean rings, torus
knots, fiber bundles, and unorientable geometries.
 Mathenautics. Visualization of 3manifold geometry at the Univ. of Illinois.
 MatHSoliD
Java animation of planar unfoldings of the Platonic and Archimedean polyhedra.
 Max.
nonadjacent vertices on 120cell. Sci.math discussion on the size
of the maximum independent set on this regular 4polytope.
Apparently
it is known to be between 220 and 224 inclusive.
 Maximizing the
minimum distance of N points on a sphere, raytraced by Hugo Pfoertner.
 Maximum
area crosssection of a hypercube.
 Maximum convex hulls of connected systems of segments and of polyominoes. Bezdek, Brass, and Harborth place bounds on the convex area needed to contain a polyomino.
 Measurement
sample. Ed Dickey advocates teaching about sphere packings and
kissing numbers to high school students as part of a
teaching
strategy involving manipulative devices.
 Mengermania!
 Menger
Cubes, Peter C. Miller.
Including some animated ray traces and a discussion of eliminating
irrelevant internal surfaces prior to rendering.
 Menger sponge
floating in space. Everyone and his brother makes raytraced
fractals with unlikely backgrounds nowadays, but Cliff Pickover was
there first.
 Meru Foundation appears to be
another sacred geometry site, with animated gifs of torus knots
and other geometric visualizations and articles.
 MicroGeometry
volume and surface area calculator for the Palm.
 Midpoint
triangle porism.
Two nested circles define a continuous family of triangles having endpoints
on the outer circle and edge midpoints on the inner circle.
A similar
porism works for quadrilaterals and, seemingly,
higher
order polygons.
Geometer's sketchpad animations by John Berglund.
 Minenergy
configurations of electrons on a sphere, K. S. Brown.
 The MindBlock.
Reassemble a chessboard cut into twelve interlocking polyominos.
 Minesweeper
on Archimedean polyhedra, Robert Webb.
 A minimal
domino tiling. How small a square board can one fill with dominos in
a way that can't be separated into two smaller rectangles?
From Stan Wagon's
PotW archive.
 A
minimal winter's tale. Macalester College's snow sculpture of
Enneper's surface wins second place at Breckenridge.
 Minimax elastic bending energy of sphere eversions.
Rob Kusner, U. Mass. Amherst.
 Minimize
the slopes. How few different slopes can be formed by the lines
connecting 881 points?
From Stan Wagon's
PotW archive.
 Minimizing
surface area to volume ratio in a cube.
 Maximum volume
arrangements of points on a sphere, Hugo Pfoertner.
 Miquel's
pentagram theorem on circles associated with a pentagon.
With annoying music.
 Miquel's six
circles in 3d.
Reinterpreting a statement about intersecting circles to be about
inscribed cuboids.
 Mirror Curves.
Slavik Jablan investigates patterns formed by crisscrossing a curve around points in a regular grid, and finds examples of these patterns in
art from various cultures.
 Mirrored room illumination.
A summary by Christine Piatko of the old open problem of, given
a polygon in which all sides are perfect mirrors, and a point source
of light, whether the entire polygon will be lit up.
The answer is no if smooth curves are allowed.
See also Eric Weisstein's page on the
Illumination
Problem.
 Miscellaneous
polyomino explorations. Alexandre Owen Muniz
looks at double polyomino tilings that simultaneously cover all
halfgrid edges, magic polyominoes, and more.
 Mitre Tiling.
Ed Pegg describes the discovery of the versatile tiling system
(with Adrian Fisher and Miroslav Vicher), also discussing many
other interesting tilings including a tile that can fill the plane with
either fivefold or sixfold symmetry.
 Möbius
at the Shopping Mall. Topological sculpture as public seating. From MathTrek.
 Modeling
mollusc shells with logarithmic spirals, O. Hammer, Norsk
Net. Tech. Also includes a list of logarithmic spiral links.
 Models
of Mathematical Machines at the University Museum of Natural Science
and Scientific Instruments of the University of Modena.
Main exhibit is in Italian but there is an English
preface
and
htm.
 Models of Small Geometries.
Burkard Polster draws diagrams of combinatorial configurations such as the
Fano plane and Desargues' theorem (shown below) in an attempt to capture the
mathematical beauty of these geometries.
 Modular
piecosahedron. Turkey Tek makes geometric models out of pecan pie.
 Modularity in art.
Slavik Jablan explores connections between art, tiling, knotwork, and
other mathematical topics.
 Moebius
transformations revealed. Video by Douglas N. Arnold and Jonathan
Rogness explaining 2d Moebius transformations in terms of the motions of
a 3d sphere. See also MathTrek.
 Monge's
theorem and Desargues' theorem, identified.
Thomas Banchoff relates these two results,
on colinearity of intersections of external tangents to disjoint circles,
and of intersections of sides of perspective triangles, respectively.
He also describes generalizations to higher dimensional spheres.
 More
hyperbolic tilings and software for creating them, J. Mount.
 Mormon computational geometry.
 Moser's Worm.
What is the smallest area shape (in a given class of shapes) that
can cover any unitlength path?
Part of Mathsoft's
collection of
mathematical constants.
 Mostly modular origami. Valerie Vann makes polyhedra out of folded paper.
 Movies
by Impulse. Computational geometry applied to the simulation of
bowling allies and poolhalls.
 Museum of
harmony and golden section.
 Mutations and knots.
Connections between knot theory and dissection of hyperbolic polyhedra.
 My face on a Voronoi Diagram.
 Ndimensional cubes, J. Bowen, Oxford.
 Natural neighbors.
Dave Watson supplies instances where shapes from
nature are (almost) Voronoi polygons. He also has a page
of related references.
 Mike Naylor's
ASCII art.
Platonic solids, knots, fractals, and more.
 Nested
Klein bottles. From the London Science Museum gallery, by way of Boing
Boing. Topological glassware by Alan Bennett.
 Netlib polyhedra.
Coordinates for regular and Archimedean polyhedra,
prisms, antiprisms, and more.
 New
directions in aperiodic tilings, L. Danzer, Aperiodic '94.
 A
new Masonic interpretation of Euclid's 47th Problem. Confused about
why those wacky Freemasons care so much about the Pythagorean Theorem,
Bro. Jeff Peace proposes the existence of a different Euclid
and a different 47th problem more related to theology than geometry.
 New
perspective systems, by Dick Termes,
an artist who paints insideout scenes on spheres which
give the illusion of looking into separate small worlds.
His site also includes an unfolded dodecahedron example
you can print, cut, and fold yourself.
 Mark Newbold's
Rhombic Dodecahedron Page.
 Nine.
Drew Olbrich discovers the associahedron by evenly spacing nine points
on a sphere and dualizing.
 No cubed cube.
David Moews offers a cute proof that no cube can be divided into smaller
cubes, all different.
 The
nothreeinline problem.
How many points can be placed in an n*n grid with no three
on a common line? The solution is known to be between 1.5n and
2n. Achim Flammenkamp discusses some new computational results
including bounds on the number of symmetric solutions.
 NonEuclidean games implemented in Shockwave by students in an
advanced high school geometry class:
projectiveplane asteroids,
hyperbolic
doubletorus minesweeper, and
cubical
fruitarian snake.
 NonEuclidean
geometry with LOGO. A project at Cardiff, Wales, for using the LOGO
programming language to help mathematics students visualise
nonEuclidean geometry.
 Nonorthogonal polyhedra built from rectangles.
Melody Donoso and Joe O'Rourke answer an open question of Biedl, Lubiw,
and Sun.
 Non
periodic tiling of the plane.
Including Penrose tiles, Pinhweel tiling, and more. Paul Bourke.
 Nontrivial
convexity. Ed Pegg asks about partitions of convex regions into
equal tiles, other than the "trivial" ones in which some rotational or
translational symmetry group relates all the tile positions to each other.
See also Miroslav
Vicher's page on nontrivial convexity
 T. Nordstrand's
gallery of surfaces.
 Not. AMS
Cover, Apr. 1995. This illustration for an article on geometric
tomography depicts objects (a cuboctahedron and warped rhombic
dodecahedron) that disguise themselves as regular tetrahedra
by having the same width function or xray image.
 Number patterns,
curves, and topology, J. Britton.
Includes sections on the golden ratio, conics, Moiré patterns,
Reuleaux triangles, spirograph curves, fractals, and flexagons.
 Objects that cannot be taken apart with two hands.
J. Snoeyink, U. British Columbia.
 Occult correspondences of the Platonic solids.
Some random thoughts from
Anders
Sandberg.
 Occurrence of
the conics.
Jill Britton explains how the different conic curves can all be formed
by slicing the same cone at different angles, and finds many examples of
them in technology and nature.
 Octacube.
Stainless steel 3d model of the 24cell (one of the six regular
polytopes in four dimensions), by Adrian Ocneanu, installed as a
sculpture in the Penn State Math Department. Includes also a shockwave
flythrough of the model.
 Odd rectangles for L_{4n+2}.
Phillippe Rosselet shows that any Lshaped (4n+2)omino can tile a rectangle
with an odd side.
 Odd squared distances. Warren Smith considers point sets
for which the square of each interpoint distance is an odd integer.
Clearly one can always do this with an appropriately scaled regular simplex;
Warren shows that one can squeeze just one more point in,
iff the dimension is 2 (mod 4).
Moshe Rosenfeld has published a related paper in Geombinatorics
(vol. 5, 1996, pp. 156159).
 Oops
image warper, based on three simple geometric transformations.
 Open problems:
Demaine  Mitchell 
O'Rourke open problems project
From Jeff Erickson, Duke U.
From Jorge Urrutia, U. Ottawa.
From the 2nd MSI Worksh. on Computational Geometry.
From
SCG '98.
 Optimal
illumination of a sphere. An interesting variation on the problem of
equally spacing points, by Hugo Pfoertner.
 The Optiverse.
An amazing 6minute video on how to turn spheres inside out.
 Origami: a study in symmetry. M. Johnson and B. Beug, Capital H.S.
 Origami & math,
Eric Andersen.
 Origami
mathematics, Tom Hull, Merrimack.
 Origami
Menger Sponge
built from Sonobe modules by K. & W. Burczyk.
 Origami polyhedra. Jim Plank makes geometric constructions by
folding paper squares.
 Origami proof of the Pythagorean theorem,
Vi Hart.
 Origami Tesselations.
Geometric paperfolding by Eric Gjerde.
 Origami
tessellations and
paper mosaics, Alex Bateman.
 Origamic tetrahedron.
The image below depicts a way of making five folds in a 234 triangle,
so that it folds up into a tetrahedron. Toshi Kato asks if you can fold
the triangle into a tetrahedron with only three folds.
It turns out that there is a unique solution, although many
tetrahedra can be formed with more folds.
 The
Origami Lab. New Yorker article on Robert Lang's origami mathematics.
 Orthogonal discrete knots.
Hew Wolff asks questions about the minimum total length, or the minimum volume of a rectangular box, needed to form different knots as threedimensional polygons using only integerlength axisparallel edges.
 Ozbird Escherlike tessellations
by John Osborn, including several based on Penrose tilings.
 Ozzigami tessellations,
papercraft, unfolded peelnstick glitter Platonic solids, and more.
 Packing
circles in circles and circles on a sphere,
Jim Buddenhagen.
Mostly about optimal packing but includes also some nonoptimal spiral
and pinwheel packings.
 Packing
circles in the hyperbolic plane, Java animation by
Kevin Pilgrim illustrating the effects of changing radii in the
hyperbolic plane.
 Packing Ferrers Shapes.
Alon, Bóna, and Spencer show that one can't cover very much of an n by p(n)
rectangle with staircase polyominoes (where p(n) is the number of these
shapes).
 Packing
polyominoes.
Mark Michell investigates the problem of arranging pentominoes into
rectangles of various (noninteger) aspect ratios,
in order to saw the largest possible pieces from a given size piece of
wood.
 Packing
pennies in the plane, an illustrated proof of Kepler's conjecture in
2D by Bill Casselman.
 Packing rectangles into similar rectangles.
A problem of the month from Erich Friedman's Math Magic site:
how small an aspectratior rectangle can contain n unitarea
aspectratior rectangles?
As you might hope for in a problem dealing with aspect ratios
of rectangles, the golden rectangle does show up, as one of the
breakpoints in the size function for packing five smaller rectangles.
 Packing
Tetrahedrons, and Closing in on a Perfect Fit. Elizabeth Chen and
others use experiments on hundreds of D&D dice to smash previous
records for packing density.
 Packings in Grassmannian spaces, N. Sloane, AT&T.
How to arrange lines, planes, and other lowdimensional spaces into
higherdimensional spaces.
 Packomania!
 A
pair of triangle centers, Vincent Goffin.
Do these really count as centers? They are invariant under translation
and rotation but switch places under reflection.
 Pairwise
touching hypercubes. Erich Friedman asks how to partition the unit cubes
of an a*b*cunit rectangular box into as many connected polycubes as
possible with a shared face between every pair of polycubes.
He lists both general upper and lower bounds as functions of a, b, and
c, and specific constructions for specific sizes of box.
I've seen the same question asked for ddimensional hypercubes
formed out of 2^d unit hypercubes; there is a lower bound of roughly
2^{d/2} (from embedding a 2*2^{d/2}*2^{d/2} box
into the hypercube)
and an upper bound of O(2^{d/2} sqrt d)
(from computing how many cubes must be in a polycube
to give it enough faces to touch all the others).
 Paper folding a 306090 triangle.
From the geometry.puzzles archives.
 Paperfolding
and the dragon curve. David Wright discusses the connections
between
the dragon fractal,
symbolic dynamics, folded pieces of paper, and
trigonometric sums.
 Paperforms.
John Vonachen uses laser cutters and spray paint to make and sell paper
models of polyhedra, stellated
polyhedra, polyhedral complexes, Sierpinski tetrahedra, etc.
 Paper
models of polyhedra.
 Pappus
on the Archimedean solids. Translation of an excerpt of a fourth century
geometry text.
 Paraboloid.
Raytraced image created to illustrate the
lifting transformation
used to relate Delaunay triangulation
with convex hulls in one higher dimension.
 Parallel pentagons.
Thomas Feng defines these as pentagons in which each diagonal
is parallel to its opposite side, and asks for a clean construction
of a parallel pentagon through three given points.
(He is aware of the obvious reduction via affine transformation to the
construction of regular pentagons, but finds that nonelegant.)
 Parquet
deformations.
Craig Kaplan involves continuous spatial transformations of one tiling to another.
 The Partridge
Puzzle. Dissect an (n choose 2)x(n choose 2)
square into 1 1x1 square, 2 2x2 squares, etc.
The 306090
triangle version of the puzzle is also interesting
 Patterns within
rhombic Penrose tilings. Stephen Collins' program "Bob" generates
these tilings and explores the patterns formed by geodesic walks in them.
 Pavages hyperboliques dans le modèle de Poincaré.
Animated with CabriJava. Includes separate pages on hyperbolic tilings
with regular polygons including squares, pentagons, and hexagons.
 The pavilion of polyhedreality.
George Hart makes geometric constructions from coffee stirrers and
dacron thread. Includes many pointers to
related web pages.
 Peek, software for visualizing highdimensional polytopes.
 Penguins
on the hyperbolic plane, Misha Kapovich.
See also his Escherlike
Crocodiles
on the Euclidean plane.
 Pennies in
a tray, Ivars Peterson.
 Penrose
mandala and fiveway Borromean rings.
 Penrose quilt on a
snow bank, M.&S. Newbold. See also
Lisbeth
Clemens' Penrose quilt.
 The penrose
tile and the golden mean: towards hyperdimensional intergeometry.
 Penrosetiled
lace doily.
 Penrosetiled swallow
 Penrose tiles
and how their visualization leads to strange looks from priests and
small children. Drew Olbrich.
 Penrose tiles and worse. This article from Dave Rusin's known math pages discusses the difficulty of correctly placing tiles in a Penrose tiling, as well as describing other tilings such as the pinwheel.
 Penrose
Tiles entry from E. Weisstein's treasure trove.
 Penrose tilings.
This fivefoldsymmetric tiling by rhombs or kites and darts
is probably the most well known aperiodic tiling.
 Perplexing
pentagons, Doris Schattschneider, from the Discovering Geometry
Newsletter.
A brief introduction to the problem of tiling the plane by pentagons.
 Pentagon
packing on a circle and on a sphere,
T. Tamai.
 Pentagonal coffee table with rhombic bronze casting related to the Penrose tiling, by Greg Frederickson.
 Pentagonal
Tessellations. John Savard experiments with substitution systems to
produce tilings resembling Kepler's.
 Pentagons that tile the plane, Bob Jenkins.
See also
Ed Pegg's page on
pentagon tiles.
 The
pentagram and the golden ratio. Thomas Green, Contra Costa College.
 Pentamini.
Italian site on pentominoes, by L. Zucca.
 Pentomino.
Extensive website on polyomino problems, developed by secondary school
students in Belgium. Includes regular prize contest problems involving
maximizing the area enclosed by polyominos in various ways.
 The Pentomino
Dictionary and other oulipian exercises,
G. EspositoFarèse. The twelve pentominoes
resemble letters; what words do they spell? Also includes sections on
"perecquian" configurations and a pentomino jigsaw puzzle.
 Pentomino
dissection of a square annulus. From Scott Kim's Inversions Gallery.
 Pentomino
projectofthemonth from the Geometry Forum. List the pentominoes;
fold them to form a cube; play a pentomino game.
See also proteon's polyomino cubeunfoldings and
Livio
Zucca's polyominocovered cube.
 Pento  A
Program to Solve the Pentominoes Problem. Sean Vyain.
 Pento
pentomino solving software from Amamas Software.
 Pentomino HungarIQa.
What happens to standard pentomino puzzles and games
if you use polyrhombs instead of polysquares?
 Pentomino
relationships.
A. Smith classifies pentomino packings according to their shared
subpatterns.
 Pentominoes,
expository paper by R. Bhat and A. Fletcher.
 Pentominoes  an introduction.
From the Centre for Innovation in Mathematics Teaching.
 Penumbral shadows of polygons
form projections of fourdimensional polytopes.
From the Graphics Center's graphics archives.
 Perron
Number Tiling Systems.
Mathematica software for computing fractals that tile the plane from
Perron numbers.
 Person polygons. Marc van Kreveld defines this interesting and
important class of simple polygons, and derives a linear time algorithm
(with a rather large constant factor) for recognizing a special case
in which there are many reflex vertices.
 The
Perspective Page.
A short introduction to the geometry of perspective drawing.
 Phaistos disk
geometry. Claire Watson examines the patterns on a Mediterranean
bronzeage artifact.
 Lorente Philippe's pentomino homepage. In French.
 Pi
curve. Kevin Trinder squares the circle using its involute spiral.
See also his quadrature
based on the 345 triangle.
 Pi and the Mandelbrot set.
 Pi
squared by six rectangle dissected into unequal integer squares
(or an approximation thereof) by Clive Tooth.
 Pick's Theorem.
Mark Dominus explains the formula for area of polygons with vertices in
an integer grid.
 Pictures of 3d and 4d regular solids, R. Koch, U. Oregon.
Koch also provides some
4D regular solid visualization applets.
 Pictures
of various spirals, Eric Weeks.
 Place
kicking locus in rugby, Michael de Villiers.
See also
Villiers'
other geometry papers.
 Plan for pocketmachining Austria, M. Held, Salzburg.
 Plane
color. How big can the difference between the numbers of black and
white regions in a twocolored line arrangement?
From Stan Wagon's
PotW archive.
 Plates
and crowns. Erich Friedman investigates the convex polygons that
can be dissected into certain pentagons and heptagons having all angles
right or 135 degrees.
 Plato, Fuller, and the three little pigs.
Paul Flavin makes tensegrity structures out of ball point pens and
rubber bands.
 The
Platonic solids. With Java viewers for interactive manipulation. Peter Alfeld, Utah.
 Platonic
solids and Euler's formula. Vishal Lama shows how the formula can be
used to show that the familiar five Platonic solids are the only ones
possible.
 Platonic
solids and quaternion groups, J. Baez.
 Platonic
solids transformed by Michael Hansmeyer using subdivisionsurface
algorithms into shapes
resembling radiolarans.
See also Boing Boing discussion.
 Platonic spheres.
Java animation, with a discussion of platonic solid classification,
Euler's formula, and sphere symmetries.
 Platonic
tesselations of Riemann surfaces, Gerard Westendorp.
 Platonic Universe,
Stephan Werbeck. What shapes can you form by gluing regular dodecahedra
facetoface?
 Pleats, twists, and
sliceforms. Some links to Richard Sweeney's fractal paperfolding
art, via dataisnature.
 Plexagons.
Ron Evans proposes to use surfaces made out of pleated hexagons as
modular construction units. Paul Bourke explains.
 Plücker coordinates.
A description by Bob Knighten of this useful and standard way
of giving coordinates to lines, planes, and higher dimensional
subspaces of projective space.
 Points on
a sphere. Paul Bourke describes a simple randomstart hillclimbing
heuristic for spreading points evenly on a sphere, with pretty pictures
and C source.
 Poly, Windows/Mac shareware
for exploring various classes of polyhedra including Platonic solids,
Archimedean solids, Johnson solids, etc. Includes perspective views,
Shlegel diagrams, and unfolded nets.
 The Poly Pages.
Andrew L. Clarke provides information and links on the various polyforms.
 PolyB
Unix software for enumerating lattice animals, Paul Janssens.
 Polycell.
George Olshevsky makes and sells polyhedra from colored cardstock.
 Polydron
patented polychromatic plastic polygons.
 Polyedergarten.
Ulrich Mikloweit makes polyhedral models out of colored typewriter
paper, cut into lace so you can see the internal structure.
 PolygonPat
Australian school program involving coloring in geometric patterns in
glazed terracotta.
 Polygon
power. How can one arrange six points to maximize the number of
simple polygons having all six points as vertices?
From Stan Wagon's
PotW archive.
See also Heidi Burgiel's
simple
ngon counter.
 Polygon Puzzle
open source polyomino and polyform placement solitaire game.
 Polygonal and polyhedral geometry. Dave Rusin, Northern Illinois U.
 Polygons as projections of polytopes.
Andrew Kepert answers a question of
George Baloglou on whether every planar figure formed by a convex
polygon and all its diagonals can be formed by projecting a
threedimensional convex polyhedron.
 Polygons,
polyhedra, polytopes, R. Towle.
 Polygons
with angles of different kgons.
Leroy Quet asks whether polygons formed by combining the angles of
different regular polygons can tile the plane.
The answer turns out to be related to
Egyptian fraction
decompositions of 1 and 1/2.
 Polyhedra.
Bruce Fast is building a library of images of polyhedra.
He describes some of the regular and semiregular polyhedra,
and lists names of many more including the Johnson solids
(all convex polyhedra with regular faces).
 Polyhedra Blender.
Mathematica software and Javabased interactive web gallery for what look like
Minkowski sums of polyhedra. If the inputs to the Minkowski
sums were line segments, cubes, or zonohedra, the results would be again
zonohedra, but the ability to supply other inputs allows more general
polyhedra to be formed.
 Polyhedra
collection, V. Bulatov.
 Polyhedra
exhibition.
Many regularpolyhedron compounds, rendered in povray by Alexandre Buchmann.
 Polyhedra
pastimes, links to teaching activities collected by J. Britton.
 Polyhedra
plaited with paper strips,
H. B. Meyer.
See also Jim
Blowers' collection of plaited polyhedra.
 A
polyhedral analysis. Ken Gourlay looks at the Platonic solids and
their stellations.
 Polyhedral nets and dissection.
David Paterson outlines an algorithm to search for minimal dissections.
 Polyhedral
solids. Raytraced images by Tom Gettys,
and a primer on constructing paper models.
 Polyhedron,
polyhedra, polytopes, ...  Numericana.
 Polyhedron challenge: cuboctahedron.
 Polyhedron man.
Nice article from Ivars Peterson's Mathland about George Hart and his
polyhedral art.
 Polyhedron web scavenger hunt
 Polyform and dissection puzzle links.
 Polyforms.
Ed Pegg Jr.'s site has many pages on tiling, packing, and related problems
involving polyominos, polyiamonds, polyspheres, and related shapes.
 Polyform
spirals.
 PolyGloss.
Wendy Krieger is unsatisfied with terminology for higher dimensional geometry
and attempts a better replacement.
Her geometry works
include some other material on higher dimensional polytopes.
 Polyiamond
exclusion. Colonel Sicherman asks what fraction of the triangles
need to be removed from a regular triangular tiling of the plane, in
order to make sure that the remaining triangles contain no copy of a
given polyiamond.
 Polyiamonds.
This Geometry Forum problem of the week asks whether a sixpoint star
can be dissected to form eight distinct hexiamonds.
 Polymorf
geometric construction set system created by Rick Engel.
 PolyMultiForms.
L. Zucca uses pinwheel tilers to dissect an illustration of the Pythagorean
theorem into few congruent triangles.
 Polyomino
applet, Wil Laan.
 Polyomino covers.
Alexandre Owen Muniz investigates the minimum size of a polygon that can
contain each of the nominoes.
 Polyomino
enumeration, K. S. Brown.
 Polyomino inclusion problem.
Yann David wants to know how to test whether all sufficiently large
polyominoes contain at least one member of a given set.
 Polyomino
problems and variations of a theme. Information about filling
rectangles, other polygons, boxes, etc., with dominoes, trominoes,
tetrominoes, pentominoes, solid pentominoes, hexiamonds, and whatever
else people have invented as variations of a theme.
 Polyomino tiling.
Joseph Myers classifies the nominoes up to n=15 according to how
symmetrically they can tile the plane.
 Polyominoes, figures formed from subsets
of the square lattice tiling of the plane. Interesting problems
associated with these shapes include finding all of them, determining
which ones tile the plane, and dissecting rectangles or other shapes
into sets of them. Also includes related
material on polyiamonds, polyhexes, and animals.
 Polyominoes
7.0 Macintosh shareware.
 Polyominoids,
connected sets of squares in a 3d cubical lattice.
Includes a Java applet as well as nonanimated description.
By Jorge L. Mireles Jasso.
 Polypolygon
tilings, S. Dutch.
 Polytopia
CDROMs on tessellations, polyhedra, honycombs, and polytopes.
 Polytope
movie page. GIF animations by Komei Fukuda.
 Poncelet's
porism, the theorem that if a polygon is simultaneously
inscribed in one circle and circumscribed in another, then there exists
an infinite family of such polygons, one touching each point of each
circle. From the secret blogging seminar.
 Popsicle
stick bombs, lashings and weavings in the plane, F. Saliola.
 Postscript
geometry.
Bill Casselman uses postscript to motivate a course
in Euclidean geometry.
See also his Coxeter group graph paper,
and Ed Rosten's
postscript doodles.
Beware, however, that postscript can not really represent
such basic geometric primitives as circles, instead approximating them
by splines.
 A presliced
triangle. Given a triangle with three lines drawn across it, how to
draw more lines to make it into a triangulation?
From Stan Wagon's
PotW archive.
 The
Pretzel Page. Eric Sedgwick uses animated movies of twisting pretzel knots
to visualize a theorem about Heegard splittings
(ways of dividing a complex topological space into two simple pieces).
 Primes
of a 14omino. Michael Reid shows that a 3x6 rectangle with a 2x2
bite removed can tile a (much larger) rectangle.
It is open whether it can do this using an odd number of copies.
 Prince
Rupert's Cube. It's possible to push a larger cube through a hole
drilled into a smaller cube. How much larger? 1.06065... From Eric
Weisstein's treasure trove of mathematics.
 Prince
Rupert's tetrahedra? One tetrahedron can be entirely contained in
another, and yet have a larger sum of edge lengths. But how much larger?
From Stan Wagon's
PotW archive.
 Prints by Robert
Fathauer. Escherlike interlocking animals form spiral tilings and
fractals.
 Programming for 3d
modeling, T. Longtin. Tensegrity structures, twisted torus space frames,
Moebius band gear assemblies, jigsaw puzzle polyhedra, Hilbert fractal helices,
herds of turtles, and more.
 Projective
Duality. This Java applet by F. Henle of Dartmouth demonstrates
three different incidencepreserving translations from points to lines
and vice versa in the projective plane.
 Project
X. "a shape that is homogenized, saturated with equalities, inanely
geometric, yet also irresolvable, paradoxical, UNHEALTHY"
 Prolog somacube puzzle program
 Proofs of Euler's Formula.
VE+F=2, where V, E, and F are respectively the numbers of
vertices, edges, and faces of a convex polyhedron.
 Proofs of the Pythagorean Theorem.
 Proteon's Puzzle Notes
WenShan Kao covers cubes with polyominos and polysticks, packs worms
into boxes, and studies giant tangram like puzzles.
 ProtoZone
interactive shockwave museum exhibits for exploring geometric concepts
such as symmetry, tiling, and wallpaper groups.
 Pseudospherical surfaces.
These surfaces are equally "saddleshaped" at each point.
 Publications on quasicrystals and aperiodic tilings, F. Gähler.
 Pushing
disks together. If unit disks move so their pairwise distances all
decrease, does the area of their union also decrease?
 Puzzle Fun,
a quarterly bulletin edited by R. Kurchan about polyominoes and
other puzzles.
 Puzzle World
gallery of handcrafted mechanical puzzles. Includes many geometric
toys and puzzles.
 Puzzles. Discussions on the geometry.puzzles list,
collected by topic at the Swarthmore Geometry Forum.
 Puzzles
with polyhedra and numbers,
J. Rezende.
Some questions about labeling edges of platonic solids with numbers,
and their connections with group theory.
 A Puzzling Journey To The Reptiles And Related Animals, and
New Mosaics.
Books on tiling by Karl Scherer.
 The
Puzzling World of Polyhedral Dissections.
Stewart T. Coffin's classic book on geometric puzzles,
now available in full text on the internet!
 Pythagoras' Haven.
Java animation of Euclid's proof of the Pythagorean theorem.
 Pythagorean
theorem by dissection,
part II,
and part III, Java Applets by A. Bogomolny.
 Pythagorean
tilings. William Heierman asks about dissections of rectangles
into dissimilar integersided right triangles.
 Quadrature.
Michael Rack finds what appears to be an accurate numerical
approximation to pi using compass and straightedge.
 Quadrorhomb rotary engine
with chambers defined by the bars of a twelvebar linkage rotating
around two nonconcentric axes.
 Quaquaversal
Tilings and Rotations. John Conway and Charles Radin describe a
threedimensional generalization of the pinwheel tiling, the mathematics
of which is messier due to the noncommutativity of threedimensional
rotations.
 Quark constructions.
The sun4v.qc Team investigates polyhedra that fit together
to form a modular set of building blocks.
 Quark
Park. An ephemeral outdoor display of geometric art, in Princeton,
New Jersey. From Ivars Peterson's MathTrek.
Quasicrystals
and aperiodic tilings, A. Zerhusen, U. Kentucky.
Includes a nice description of how to make 3d aperiodic tiles
from zometool pieces.
 A quasipolynomial bound for the diameter of graphs of polyhedra,
G. Kalai and D. Kleitman, Bull. AMS 26 (1992). A famous open conjecture in polyhedral
combinatorics (with applications to e.g. the simplex method in linear
programming) states that any two vertices of an nface polytope are
linked by a chain of O(n) edges. This paper gives the weaker bound
O(n^{log d}).
 Quasitiler image, E. Durand.
 Qubits, modular geometric building
blocks by architect Mark Burginger, inspired by Fuller's geodesic domes.
 Quicktime VR and mathematical visualization.
 Rabbit style object on geometrical solid.
Complete and detailed instructions
for this origami construction, in 3 easy steps and one difficult step.
 Peter
Raedschelder's page of Escherlike figures
 Rainbow
Sierpinski tetrahedron by Aécio de Féo Flora Neto.
 Ram's Horn
cardboard model of an interesting 3d spiral shape bounded by a helicoid
and two nested cones.
 Random domino tiling of an Aztec diamond
and other undergrad research on random tiling.
 Random spherical arc crossings.
Bill Taylor and Tal Kubo prove that if one takes two random geodesics
on the sphere, the probability that they cross is 1/8.
This seems closely related a famous problem on the probability
of choosing a convex quadrilateral from a planar distribution.
The minimum (over all possible distributions) of this probability
also turns out to solve a seemingly unrelated combinatorial
geometry problem, on the minimum
number of crossings possible in a drawing of the complete graph with
straightline edges:
see also "The
rectilinear crossing number of a complete graph
and Sylvester's four point problem of geometric probability",
E. Scheinerman and H. Wilf, Amer. Math. Monthly 101 (1994) 939943,
rectilinear
crossing constant, S. Finch, MathSoft, and
Calluna's pit,
Douglas Reay.
 Random polygons.
Tim Lambert summarizes responses to a request for
a good random distribution on the nvertex simple polygons.
 Rational
maps with symmetries.
Buff and Henriksen investigate rational functions invariant under
certain families of Möbius transformations, and use them to
generate symmetric Julia sets.
 The rational and mathematical art of A/K/Rona
 Rational
square. David Turner shows that a rectangle can only be dissected
into finitely many squares if its sides are in a rational proportion.
 Rational triangles.
This well known problem asks whether there exists a triangle with
the side lengths, medians, altitudes, and area all rational numbers.
Randall Rathbun provides some "near misses"  triangles in which
most but not all of these quantities are irrational.
See also Dan Asimov's question in geometry.puzzles
about integer rightangled tetrahedra.
 Raytrace
rendering.
Richard M. Smith uses POVray to view complex geometric scenes.
 Realization Spaces of 4polytopes are Universal,
G. Ziegler and J. RichterGebert, Bull. AMS 32 (1995).
 Realizing a
Delaunay triangulation. Many authors have written Java code for computing
Delaunay triangulations of points. But Tim Lambert's applet does the
reverse: give it a triangulation, and it finds points for which that
triangulation is Delaunay.
 Rec.puzzles archive: dissection problems.
 Rec.puzzles archive: coloring problems.
 Reconstruction of a closed curve from its elliptic Fourier descriptor.
The ancient epicycle theory of planetary motion, animated in Java.
 Rectangles divided
into (mostly) unequal squares, R. W. Gosper.
 Rectangular cartograms: the game.
Change the shape of rectangles (without changing their area) and group
them into larger rectangular and Lshaped units to fit them into a
given frame. Bettina Speckmann, TUE. Requires a browser with support for
Java SE 6.
 The
reflection of light rays in a cup of coffee or
the curves obtained with b^n mod p, S. Plouffe, Simon Fraser U.
(Warning: large animated gif. You may prefer the more wordy explanation at
Plouffe's other page on the same subject.)
 Regard
mathématique sur Bruxelles. Student project to photograph
city features of mathematical interest and model them in Cabri.
 Regular
4d polytope foldouts. Java animations by Andrew Weimholt.
Also includes some irregular polytops.
 Regular
polyhedra as intersecting cylinders.
Jim Buddenhagen exhibits raytraces of the shapes formed by
extending halfinfinite cylinders around rays from the center
to each vertex of a regular polyhedron.
The boundary faces of the resulting unions form
combinatorially equivalent complexes to those of the dual polyhedra.
 Regular
polytopes in higher dimensions. Russell Towle uses Mathematica to
slice and dice simplices, hypercubes, and the other highdimensional
regular polytopes.
See also
Russell's 4D star
polytope quicktime animations.
 Regular polytopes in Hilbert space.
Dan Asimov asks what the right definition of such a thing should be.
 Regular solids.
Information on Schlafli symbols, coordinates, and duals
of the five Platonic solids.
(This page's title says also Archimedean solids, but I don't see many of
them here.)
 Reproduction of
sexehexes. Livio Zucca finds an interesting fractal polyhex based on
a simple matching rule.
 Reptile
projectofthemonth from the Geometry Forum.
Form tilings by dividing polygons into copies of themselves.
 Research: spirals, Mícheál Mac an Airchinnigh. Presumably this connects to his thesis that "there is a geometry of curves which is
computationally equivalent to a Turing Machine".
 Resistance and
conductance of polyhedra. Derek Locke computes formulae for networks
of unit resistors in the patterns of the edges of the Platonic solids.
See also the section on resistors in the rec.puzzles faq.
 Reuleaux triangles. These curves of
constant width, formed by combining three circular arcs into an
equilateral triangle, can drill out (most of) a square hole.
 Reuleaux
triangle entry from
Kunkel's mathematics
lessons.
 Reuleaux
Wheel. From Mudd Math Fun Facts.
 Vittoria Rezzonico's
Java applets. Hypercube and polyhedron visualization, and circle
inversion patterns. Requires both Java and JavaScript.
 Rhombic
spirallohedra, concave rhombusfaced polyhedra that tile space,
R. Towle.
 Rhombic
tilings. Abstract of Serge Elnitsky's thesis, "Rhombic tilings of
polygons and classes of reduced words in Coxeter groups". He also supplied the
picture below of a rhombically tiled 48gon, available with better color
resolution from his website.
 Rhombic triacosiohedron. Pretty model of
a nonconvex genus11 polyhedron with 300 congruent faces.
 Riemann Surfaces and the Geometrization of 3Manifolds,
C. McMullen, Bull. AMS 27 (1992).
This expository (but very technical) article outlines Thurston's
technique for finding geometric structures in 3dimensional topology.
 Rigid
regular rgons.
Erich Friedman asks how many unitlength bars are needed in a
barandjoint linkage network to make a unit regular polygon rigid.
What if the polygon can have nonunitlength edges?
 Rob's
polyhedron models, made with the help of his program
Stella.
 Robinson Friedenthal polyhedral explorations.
Geometric sculpture.
 Roger's Connection.
Magnetic construction toy, scientific exploration tool,
executive desk toy, magnet learning tool, architectural design tool,
artistic sculpture system, manual dexterity training, and much more!
(Make geometric shapes out of steel balls and magnettipped plastic tubes.)
See also Simon Fraser's
Roger's Connection gallery.
 Sascha Rogmann's hyperbolic geometry page
 Rolling
polyhedra. Dave Boll investigates Hamiltonian paths on (duals of)
regular polyhedra.
 Rolling with Reuleaux from Ivars Peterson's MathLand.
 Romain
triangle theorem.
An analogue of the Pythagorean theorem for triangles in which one angle
is twice another.
 Rombix geometric puzzle based on dissections of regular polygons into joined pairs of rhombi.
 The rotating caliper graph.
A thrackle used in "Average Case Analysis of Dynamic Geometric Optimization"
for maintaining the width and diameter of a point set.
 Rotating
4cube applet, Bernd Grave Jakobi.
For the Germanchallenged, Drehen starts the rotation and the other
controls change the axis and speed of rotation.
 Rotating
zonohedron. This truncated rhombic dodecahedron forms
the logo of the T. U. Berlin
Algorithmic and Discrete Math. Group.
 Rubik's Cube
Menger Sponge, Hana Bizek.
 Rubik's
hypercube. 3x3x3x3 times as much
puzzlement. Windows software from Melinda Green and Don Hatch,
now also available as Linux executable and C++ source.
 Rudin's
example of an unshellable triangulation.
In this subdivision of a big tetrahedron into small tetrahedra,
every small tetrahedron has a vertex interior to a face of the big
tetrahedron, so you can't remove any of them without forming a hole.
Peter Alfeld, Utah.
 The RUG FTP origami archive
contains several papers on mathematical origami.
 Ruler and Compass.
Mathematical web site including special sections on the
geometry of
polyhedrons and
geometry
of polytopes.
 Ruler and compass construction of the Fibonacci numbers and other
integers, by
David and Ken Sloan, Dan Litchfield and Dave Goldenheim,
Domingo Gómez Morín,
and an 1811 textbook.
 Russian math olympiad problem on lattice
points.
Proof that, for any five lattice points in convex position,
another lattice point is on or inside
the inner pentagon of the fivepoint star they form.
 Sacred Geometry. Mystic insights into the
"principle of oneness underlying all geometry",
mixed with occasional outright falsehoods
such as the suggestion that dodecahedra and icosahedra arise in
crystals. But the illustrative diagrams are ok, if you just
ignore the words... For more mystic diagrams, see
The Sacred
Geometry Coloring Book.
 Sacred geometry,
new discoveries linking the great pyramid to the human form.
Charles Henry finds faces in raytraces of reflecting spheres.
 Sacred geometry discovery.
 Sangaku problem.
The incenters of four triangles in a cyclic quadrilateral form a rectangle.
Animated in Shockwave by Antonio Gutierrez.
 Santa Fe Ribbon,
painting by Connie Simon featuring a rhombic Penrose tiling.
 Santa Rosa
Menger Cube made by Tom Falbo and helpers at Santa Rosa Junior College
from 8000 1inchcubed oak blocks.
 Satellite
constellations. Sort of a dynamic version of a sphere packing
problem: how to arrange a bunch of satellites so each point of the
planet can always see one of them?
 Sausage
Conjecture. L. Fejes Tóth conjectured that, to minimize the volume of the convex hull of
hyperspheres in five or more dimensions, one should line them up in a row.
This has recently been solved for very high dimensions
(d > 42) by Betke and Henk
(see also Betke et al., J. Reine Angew. Math. 453 (1994) 165191
and the
MathWorld
Sausage Conjecture Page).
 The
Schläfli Double Six.
A lovely photoessay of models of this configuration,
in which twelve lines each meet five of thirty points.
Unfortunately only the first page seems to be archived...
(This site also referred to
related configurations involving 27 lines meeting either 45 or
135 points, but didn't describe any mathematical details.
For further descriptions of all of these, see Hilbert and
CohnVossen's "Geometry and the Imagination".)
 Oded
Schramm's mathematical picture gallery primarily concentrating in
square tilings and circle packings, many forming fractal patterns.
 In search of the ideal knot.
Piotr Pieranski applies an iterative shrinking heuristic to find the
minimum length unitdiameter rope that can be used to tie a given knot.
 Seashell spirals.
Xah Lee examines the shapes of various real seashells, and offers prize
money for formulas duplicating them.
 Secrets
of Da Vinci's challenge.
A discussion of the symbology and design of this
interlockedcirclepattern puzzle.
 Sedona Sacred
Geometry Conference, Feb. 2004.
 Selfaffine tiles, J. Lagarias and Y. Wang, DIMACS.
Mathematics of a class of generalized reptiles.
 Selfaffine tiles.
Marina Khibnik computes the convex hulls and boundary dimensions of
fractal tiles such as the twin dragon and fractal red cross.
 Selfdual
maps, Don McConnell.
 Selfrighting shapes.
Figures with only one stable and one unstable equilibrium, when placed
on a level surface. Surprisingly, they look much like certain kinds of turtles.
Julie J. Rehmeyer in MathTrek.
 Selftrapping random walks,
Hugo Pfoertner.
 Semiregular
tilings of the plane, K. Mitchell, Hobart and William Smith Colleges.
 Sensitivity analysis for traveling salesmen,
C. Jones, U. Washington.
Still a good title, and now the geometry has been made more
entertaining with Java and VRML.
 Sets of points with many halving lines.
Coordinates for arrangements of 14, 16, and 18 points for which
many of the lines determined by two points split the remaining points
exactly in half. From my 1992
tech. report.
 Seven
circle theorem, an applet illustrating the fact that if six circles
are tangent to and completely surrounding a seventh circle, then
connecting opposite points of tangency in pairs forms three lines that
meet in a single point, by Michael Borcherds.
Other applets by Borcherds
include Poncelet's
porism, a similar porism with an ellipse and a parabola and with two ellipses,
and more generally with two conics of variable type.
 757530 triangle dissection.
This isosceles triangle has the same area as a square with side length
equal to half the triangle's long side. Ed Pegg asks for a nice dissection
from one to the other.
 Shape metrics.
Larry Boxer and David Fry provide many bibliographic references
on functions measuring how similar two geometric shapes are.
 Shapes of constant width.
 Sierpinski carpet on the sphere.
From Curtis McMullen's
math gallery.
 Sierpinski
cookies. Actually more like Menger cookies, but whatever.
 Sierpinski
gaskets and Menger sponges, Paul Bourke.
Including stacks of coke cans, radio antennas, crumpled sponges, and more.
 Sierpinski Hamantaschen.
 Sierpinski gaskets and variations rendered by D. H. Hepting.
 Sierpinski
pentatope video by Chris Edward Dupilka. A fourdimensional analogue
of the Sierpinski triangle.
 Sierpinski pyramid.
C++ code for generating the Sierpinski tetrahedron.
 The Sierpinski Tetrahedron, everyone's
favorite three dimensional fractal.
Or is it a fractal?
 Sierpinski
tetrahedron. Awful Mathematica code used by Robert Dickau to
generate the following sequence of images.
 Sierpinski
tetrahedron animation (MSvideo format), Karl S. Frederickson.
 Sierpinski triangle reptile
based on a complex binary number system, R. W. Gosper.
 Sierpinski valentine from XKCD.
 Sighting point.
John McKay asks, given a set of coplanar points, how to find
a point to view them all from in a way that maximizes the
minimum viewing angle between any two points.
Somehow this is related to monodromy groups.
I don't know whether he ever got a useful response.
This is clearly
polynomial time: the decision problem can be solved by finding
the intersection of O(n^{2}) shapes, each the union of two disks, so doing this
naively and applying parametric search gives O(n^{4} polylog),
but it might be interesting to push the time bound further.
A closely related problem of
smoothing a triangular mesh by moving points one at a time to
optimize the angles of incident triangles can be solved in linear time
by LPtype algorithms [Matousek, Sharir, and Welzl, SCG 1992;
Amenta, Bern, and Eppstein,
SODA 1997].
 Rujith de Silva's
pentomino applet.
 Similar division.
Mineyuki Uyematsu, Michael Reid, and Ed Pegg ask for divisions of given shapes
into pieces, where all pieces must be similar to each other.
 A simple dodecahedron tiling puzzle.
Cover the dodecahedron's faces with pentagonal tetrominos.
 Simple
polygonizations. Erik Demaine explores the question of how many
different noncrossing traveling salesman tours an npoint set can have.
 The
Simplex: Minimal Higher Dimensional Structures.
D. Anderson.
 Simplex/hyperplane intersection.
Doug Zare nicely summarizes the shapes that can arise on intersecting
a simplex with a hyperplane: if there are p points on the hyperplane,
m on one side, and n on the other side, the shape is
(a projective transformation of)
a piterated cone over the product of m1 and n1 dimensional simplices.
 SingSurf
software for calculating singular algebraic curves and surfaces, R. Morris.
 Sixregular toroid.
Mike Paterson asks whether it is possible to make a torusshaped polyhedron
in which exactly six equilateral triangles meet at each vertex.
 Skewered lines.
Jim Buddenhagen notes that four lines in general position in R^{3}
have exactly two lines crossing them all, and asks how this generalizes
to higher dimensions.
 Sketchpad demo includes a Reuleaux triangle rolling between two
parallel lines.
 Sliced ball.
Raytraced image created to help
describe recent algorithms
for removing slivers from tetrahedral meshes.
 Sliceforms,
3d models made by interleaving two directions of planar slices.
 N.
J. A. Sloane's netlib directory includes many references and programs for
sphere packing and clustering in various models. See also his
list
of spherepacking and lattice theory publications.
 A small puzzle.
Joe Fields asks whether a certain decomposition into Lshaped
polyominoes provides a universal solution to dissections of pythagorean
triples of squares.
 SMAPO
library of polytopes encoding the solutions to optimization problems
such as the TSP.
 Smarandache
Manifolds online ebook by Howard Iseri.
I'm not sure I see why this should be useful or interesting, but the
idea seems to be to define geometrylike structures (having objects
called points and lines that somehow resemble Euclidean points and lines)
that are nonuniform in some strong sense: every Euclidean axiom
(and why not, every Euclidean theorem?) should be true at some point of
the geometry and false at some other point.
 The
smoothed octagon. A candidate for the symmetric convex shape that
is least able to pack the plane densely.
 Smoothly rolling
polygonal wheels and their roads, H. Serras, Ghent.
 Snakes.
What is the longest path of unitlength line segments, connected
endtoend with angles that are multiples of some fixed d,
and that can be covered by a square of given size?
 SnapPea, powerful software for computing geometric properties of
knot complements and other 3manifolds.
 sneJ made a
Mandelbrot set with sheet plastic and a laser cutter.
 Snowflake
reptile hexagonal substitution tiling (sometimes known as the Gosper
Island) rediscovered by NASA
and conjectured to perform visual processing in the human brain.
 Snub cube and dodecahedron.
Rob Moeser makes geometric constructions by carving broccoli stalks.
 Soap bubble 120cell
from the Geometry Center archives.
 Soap films and grid walks, Ivar Peterson.
A discussion of Steiner tree problems in rectilinear geometry.
 Soap films on knots. Ken Brakke, Susquehanna.
 Soccer
ball pictures,
spherical patterns generated by reflections that form rational angles to each
other.
 Soddy's Hexlet,
six spheres in a ring tangent to three others,
and Soddy's
Bowl of Integers, a sphere packing combining infinitely many hexlets,
from Mathworld.
 Soddy Spiral.
R. W. Gosper calculates the positions of a sequence of circles, each
tangent to the three previous ones.
 Sofa
movers' problem.
This wellknown problem asks for the largest area of a twodimensional
region that can be moved through a hallway with a rightangled bend.
Part of Mathsoft's
collection of
mathematical constants.
 Solid object which generates an anomalous picture.
Kokichi Sugihara makes models of Escherlike illusions from folded paper.
He has plenty more where this one came from, but maybe the others
aren't on the web.
 Solution
of ConwayRadinSadun problem.
Dissections of combinations of regular dodecahedra, regular icosahedra,
and related polyhedra into rhombs that tile space. By Dehn's solution to
Hilbert's third problem this is impossible for individual dodecahedra
and icosahedra, but Conway,
Radin, and Sadun showed that certain combinations could work.
Now Izidor Hafner shows how.
 Solution
to problem 10769. Apparently problems of coloring the points of a
sphere so that orthogonal points have different colors (or so that each
set of coordinate basis vectors has multiple colors) has some relevance
to quantum mechanics; see also papers
quantph/9905080 and
quantph/9911040
(on coloring just the rational points on a sphere), as well as this
fourdimensional construction
of an odd number of basis sets in which each vector appears an even
number of times, showing that one can't color the points on a
foursphere so that each basis set has exactly one black point.
 Solution
to the pentomino problem by pete@bignode.equinox.gen.nz, from the
rec.puzzles archives.
 Solving
the Petersen Graph Zome Challenge.
David MacMahon discovers that there is no way to make a
nonselfintersecting peterson graph with Zome tool.
Includes VRML illustrations.
 Soma cube
applet.
 The soma cube page and pentomino page, J. Jenicek.
 Some generalizations of the pinwheel tiling, L. Sadun, U. Texas.
 Some images made by Konrad Polthier.
 Some pictures of symmetric tensegrities.
 SpaceBric building blocks
and Windows software based on a tiling of 3d space by congruent
tetrahedra.
 Space Cubes
plastic geometric modeling puzzle based
on a rectangular Borromean link.
 Sphere distribution problems.
Page of links to other pages, collected by Anton Sherwood.
 Sphere packing and kissing numbers.
How should one arrange circles or spheres
so that they fill space as densely as possible?
What is the maximum number of spheres that can simultanously touch
another sphere?
 Spheres
and lattices. Razvan Surdulescu computes sphere volumes and
describes some lattice packings of spheres.
 Spheres with
colorful chickenpox. Digana Swapar describes an algorithm for
spreading points on a sphere to minimize the electrostatic potential,
via a combination of simulated annealing and conjugate gradient optimization.

Spherical
Julia set with dodecahedral symmetry
discovered by McMullen and Doyle in their work on
quintic equations and rendered by
Don Mitchell.
Update 12/14/00: I've lost the big version of this image and can't find
DonM anywhere on the net  can anyone help?
In the meantime, here's a link to
McMullen's
rendering.
 The sphericon,
a convex shape with one curved face and two semicircular edges that can
roll with a wobbling motion in a straight line.
See also
the
national curve bank sphericon page,
the MathWorld
sphericon page,
the Wikipedia sphericon page,
The
Differential Geometry of the Sphericon, and
building a
sphericon.
 Spidron,
a triangulated double spiral shape tiles the plane and various other
surfaces. With photos of related paperfolding experiments.
 Spira
Mirabilis logarithmic spiral applet by A. Bogomily.
 Spiral
generator, web form for creating bitmap images of colored
logarithmic spirals.
 Spiral in a liquid crystal film.
 Spiral
minaret of Samara.
 A spiral of squares with
Fibonaccinumber sizes, closely related to the golden spiral,
Keith Burnett.
See also his handpainted Taramundi
spiral.
 Spiral tea
cozy, Kathleen Sharp.
 Spiral tilings.
These similarity tilings are formed by applying the exponential function
to a lattice in the complex number plane.
 Spiral
tower. Photo of a building in Iraq, part of a web essay on the
geometry of cyberspace.
 Spiral
triangles, Eric Weeks.
 Spiraling
Sphere Models. Bo Atkinson studies the geometry of a solid of
revolution of an Archimedean spiral.
 Spirals. Mike
Callahan and Larry Shook use a spreadsheet to investigate the spirals
formed by repeatedly nesting squares within larger squares.
 Spirals
and other 2d curves,
Jan Wassenaar.
 Split square. How to subdivide a
square into two rectangular pieces, one of which circumscribes the
other?
 Spontaneous
patterns in disk packings, Lubachevsky, Graham, and Stillinger,
Visual Mathematics. A procedure for packing unit disks into square
containers produces large grains of hexagonally packed disks
with sporadic rattlers along the grain boundaries.
 Spring
into action. Dynamic origami. Ben Trumbore, based on a model by Jeff
Beynon from Tomoko Fuse's book Spirals.
 sqfig and sqtile,
software by Eric Laroche for generating polyominoes and polyomino
tilings.
 Square Knots. This article by Brian Hayes for American Scientist
examines how likely it is that a random
lattice polygon is knotted.
 Square wheels.
Bill Beatty describes how to make cubicallooking shapes that are round
enough to roll smoothly on an axle.
 Squared
squares and squared rectangles, thorough catalog by Stuart Anderson.
Erich Friedman
discusses several related problems on squared squares:
if one divides a square into k smaller squares,
how big can one make the smallest square?
How small can one make the biggest square?
How few copies of the same size square can one use?
See also
Robert
Harley's fourcolored squared square,
Mathworld's
perfect square dissection page, a
Geometry Forum
problem of the week on squared squares,
Keith Burnett's perfect square dissection
page,
and Bob Newman's
squared square drawing.
 Squares
are not diamonds. Izzycat gives a nice explanation of why
these shapes should be thought of differently, even though they're
congruent: they generalize to different things in higher dimensions.
 Squares on a Jordan curve.
Various people discuss the open problem of whether any Jordan curve
in the plane contains four points forming the vertices of a square,
and the related but not open problem of how to place
a square table level on a hilltop.
This is also in the
geometry.puzzles archive.
 Squaring
the circle. BNTR finds a pretty geometric visualization of Gregory's
Series for pi/4.
 Speculations on the
fourth dimension, Garrett Jones.
 Splitting the hair.
Matthew Merzbacher discusses how many times one can subdivide
a line segment by following certain rules.
 Stained glass
icosidodecahedron and rhombicosidodecahedron,
Helen & Liam Striker.
 Star
construction of shapes of constant width, animated in Java by A. Bogomolny.
 Stardust
Polyhedron Puzzles. This U.K. company sells unfolded polyhedral
puzzles and spacepacking shapes
(including a nice model of the
WeairePhelan spacefilling foam)
on cardstock, to cut out and build yourself.
 Starpage.
Artdeco paper models of stellated polyhedra, by
merrill.
 Stella and Stella4d,
Windows software for visualizing regular and semiregular polyhedra and
their stellations in three and four dimensions, morphing them into each other, drawing unfolded nets for
making paper models, and exporting polyhedra to various 3d design packages.
 Stellations
of the dodecahedron stereoscopically animated in Java by Mark Newbold.
 Sterescopic polyhedra
rendered with POVray by Mark Newbold.
 Steve's sprinklers.
An interesting 3d polygon made of copper pipe forms various symmetric 2d shapes
when viewed from different directions.
 Stomachion, a tangramlike shapeforming game based on a dissection of the square and studied by Archimedes.
 The
Story of the 120cell, John Stillwell, Notices of the AMS. History,
algebra, geometry, topology, and computer graphics of this
regular 4dimensional polytope.
 Wilson
Stothers' Cabri pages.
Geometric animations teaching projective conics,
hyperbolic geometry, and the Klein view of geometry as symmetry.
 Straighten
these curves. This problem from Stan Wagon's
PotW archive
asks for a dissection of a circle minus three lunes into a rectangle.
The ancient Greeks performed
similar
constructions
for certain
lunules
as an approach to
squaring the circle.
 Strange unfoldings of convex polytopes, Komei Fukuda, ETH Zurich.
 String figure
mathematics, or trivial knot theory.
 Structors.
Panagiotis Karagiorgis thinks he can get people to pay large sums of
money for exclusive rights to use fourdimensional regular polytopes
as building floor plans. But he does have some pretty pictures...
 Student of
Hyperspace. Pictures of 6 regular polytopes, E. Swab.
 Studio
modular origami, geometric paper art.
 Subdivision
kaleidoscope. Strange diatomlike shapes formed by varying the
parameters of a spline surface mesh refinement scheme outside their
normal ranges.
 Sums of square roots. A major bottleneck
in proving NPcompleteness for geometric problems is a mismatch between
the realnumber and Turing machine models of computation: one is good
for geometric algorithms but bad for reductions, and the other vice
versa. Specifically, it is not known on Turing machines how to quickly
compare a sum of distances (square roots of integers) with an integer or
other similar sums, so even (decision versions of) easy problems such as
the minimum spanning tree are not known to be in NP.
Joe O'Rourke
discusses an approach to this problem based on bounding the smallest
difference between two such sums, so that one could know how precise an
approximation to compute.
 Superliminal
Geometry. Topics include deltahedra, infinite polyhedra, and flexible
polyhedra.
 Supershapes
and 3d
supershapes. Paul Bourke generates a wide variety of interesting
shapes from a simple formula.
See also John
Whitfield's Nature article.
 Sylvester's theorem.
This states that any finite noncolinear point set has
a line containing only two points (equivalently, every zonohedron has a
quadrilateral face). Michael Larsen, Tim Chow, and Noam
Elkies discuss two proofs and a complexnumber generalization.
(They omit the very simple generalization from Euler's
formula: every convex polyhedron has a face of degree at most five.)
 SymmeToy,
windows shareware for creating paint patterns, symmetry roses,
tessellated art and symmetrically decorated 3D polyhedron models.
 Symmetries of torusshaped polyhedra
 Symmetry,
tilings, and polyhedra, S. Dutch.
 Symmetry and Tilings. Charles Radin, Not. AMS, Jan. 1995.
See also his
Symmetry
of Tilings of the Plane, Bull. AMS 29 (1993), which proves that the
pinwheel tiling is ergodic and can be generated by matching rules.
 Symmetry
in Threshold Design in South India.
 Symmetry web, an exploration of the symmetries of geometric figures.
 Synergetic
geometry, Richard Hawkins' digital archive. Animations and 3d
models of polyhedra and tensegrity structures. Very
bandwidthintensive.
 The Szilassi Polyhedron.
This polyhedral torus, discovered by
L.
Szilassi, has seven hexagonal faces, all adjacent to each other.
It has an axis of 180degree symmetry; three pairs of faces are congruent
leaving one unpaired hexagon that is itself symmetric.
Tom
Ace has more images as well as a downloadable unfolded pattern
for making your own copy.
See also Dave Rusin's page on
polyhedral
tori with few vertices and
Ivars'
Peterson's MathTrek article.
 Tales of the
dodecahedron, from Pythagoras to Plato to Poincaré. John
Baez, Reese Prosser Memorial Lecture, Dartmouth, 2006.
 Tangencies.
Animated compass and straightedge constructions of
various patterns of tangent circles.
 Tangencies
of circles and spheres. E. F. Dearing provides formulae for the
radii of Apollonian circles, and analogous threedimensional problems.
 Define: Tangent.
 Taprats
Java software for generating symmetric Islamicstyle star patterns.
 Tarquin
mathematical posters.
 The tea bag problem.
How big a volume can you enclose by two square sheets of paper
joined at the edges?
See also
the cubical teabag
problem.
 A teacher's guide to building the icosahedron as a class project
 Temari
dodecahedrally decorated Japanese thread ball.
See also Summer's
temari gallery for many more.
 Tensegrity zoology.
A catalog of stable structures formed out of springs,
somehow forming a quantum theory of what used to be described as time.
 Tessellated
polyhedra. Colored unfoldings of the Platonic solids, ready to be
printed, cut out, and folded, by Jill Britton.
 Tesselation
world of Makoto Nakamura.
 Tesselating
locking polyominos, Bob Newman.
 Tessellation
links, S. Alejandre.
 Tessellation
resources. Compiled for the Geometry Center by D. Schattschneider.
 Tessellations,
a company which makes Puzzellations puzzles, posters, prints, and
kaleidoscopes inspired in part by Escher, Penrose, and Mendelbrot.
 Tessellations, Periodic Drawings, Computer Graphics, Latticework, ...
William Chow likes Escherlike patterns of interlocking figure
and really really long web page titles.
 Tesseract
and
tesseractembedded
Möbius strip,
A. Bogomolny.
 Tetrahedra
packing. Mathematica implementation of the ChenEngelGlotzer packing
of space by regular tetrahedra, the densest known such packing to date.
 Tetrahedral
kite. A. Thyssen describes how to make Sierpinski tetrahedra out of
soda straws, kite strings, and plastic shopping bags.
 Tetrahedrons and spheres.
Given an arbitrary tetrahedron, is there a sphere tangent to each of its edges?
Jerzy Bednarczuk, Warsaw U.
 Tetrahedra classified
by their bad angles.
From "Dihedral bounds for mesh
generation in high dimensions".
 Tetrix. From Eric Weisstein's treasure trove.
 These two pictures by Richard Phillips
are from the nowdefunct maths with photographs website.
The chimney is (Phillips thinks) somewhere in North Nottinghamshire, England.
A similar collection of Phillips'
mathematical photos is now available on CDROM.
 30 computers.
Forrest McCluer makes polyhedral sculptures out of discarded electronics.
 This
is your brain on Tetris. Are pentominos really "an ancient Roman
puzzle"?
 Morwen Thistlethwait,
sphere packing, computational topology, symmetric knots,
and giant raytraced floating letters.
 Thoughts on the number six.
John Baez contemplates the symmetries of the icosahedron.
 Thrackles
are graphs embedded as a set of curves in the plane that cross each
other exactly once; Conway has conjectured that an nvertex
thrackle has at most n edges.
Stephan Wehner describes what is known about thrackles.
 Three classical geek problems solved!
Hauke Reddmann, Hamburg.
 Threecolor the Penrose tiling?
Mark Bickford asks if this tiling is always threecolorable.
Ivars
Peterson reports on a new proof by Tom Sibley and Stan Wagon
that the rhomb version of the tiling is 3colorable;
A proof of 3colorability for kites and darts
was recently published by Robert Babilon
[Discrete Mathematics 235(13):137143, May 2001].
This is closely related to my page on line
arrangement coloring, since every Penrose tiling is dual to
a "multigrid", which is just an arrangement of lines in parallel families.
But my page only deals with finite arrangements, while Penrose tilings are
infinite.
 Three cubes to one.
Calydon asks whether nine pieces is optimal for this dissection problem.
 3DGeometrie.
T. E. Dorozinski provides a gallery of images of 3d polyhedra,
2d and 3d tilings, and subdivisions of curved surfaces.
 3dXplorMath
Macintosh software for visualizing curves, surfaces, polyhedra,
conformal maps, and other planar and threedimensional mathematical objects.
 Threedimensional models based on the works of M. C. Escher
 The
three dimensional polyominoes of minimal area, L. Alonso and
R. Cert, Elect. J. Combinatorics.
 Three dimensional turtle talk description of a dodecahedron. The dodecahedron's description is "M40T72R5M40X63.435T288X296.565R5M40T72M40X63.435T288X296.565R4"; isn't that helpful?
 3D strange attractors and similar objects, Tim Stilson, Stanford.
 Three
nice pentomino coloring problems, Owen Muniz.
 Three spiral tattoos
from the Discover Magazine Science Tattoo Emporium.
 Three untetrahedralizable objects
 The Thurston Project: experimental differential geometry, uniformization and quantum field theory.
Steve Braham hopes to prove Thurston's uniformization conjecture
by computing flows that iron the wrinkles out of manifolds.
 Tic tac toe theorem.
Bill Taylor describes a construction of a warped
tic tac toe board from a given convex quadrilateral,
and asks for a proof that the middle quadrilateral
has area 1/9 the original. Apparently this is not
even worth a chocolate fish.
 Tilable
perspectives.
Patrick Snels creates twodimensional images which tile the plane to
form 3dlooking views including some interesting Escherlike warped
perspectives.
See also his even more Escherian tesselations page.
 TileDreams
Windows software for creating symmetric patterns with rhombi.
 A
tiling from ell. Stan Wagon asks
which rectangles can be tiled with an elltromino.
 Tiling plane
& fancy, Steven Edwards, SPSU.
 Tiling the infinite grid with finite clusters.
Mario Szegedy describes an algorithm for determining whether a (possibly
disconnected) polyomino will tile the plane by translation,
in the case where the number of squares in the polyomino is a prime
or four.
 Tiling the integers with one prototile.
Talk abstract by Ethan Coven on a onedimensional tiling problem on the
boundary between
geometry and number theory, with connections to factorization of finite
cyclic groups.
See also Coven's paper with Aaron Meyerowitz,
Tiling the integers
with translates of one finite set.
 Tiling problems.
Collected at a problem session at Smith College, 1993, by
Marjorie Senechal.
 The tiling puzzle games of
OOG. Windows and Java software for tangrams, polyominoes, and polyhexes.
 Tiling a
rectangle with the fewest squares. R. Kenyon shows that any
dissection of a p*q rectangle into squares (where p and q are integers
in lowest terms) must use at least log p pieces.
 Tiling rectangles
and half strips with congruent polyominoes, and
Tiling a square with eight congruent polyominoes, Michael Reid.
 Tiling
stuff. J. L. King examines problems of determining whether a given
rectangular brick can be tiled by certain smaller bricks.
 Tiling
transformer. Java applet for subdividing tilings (starting from a
square or hexagonal tiling) in various different ways.

Tiling the unit square with rectangles.
Erich Friedman
shows that the 5/6 by 5/6 square can always be tiled with 1/(k+1) by
1/(k+1) squares.
Will all the 1/k by 1/(k+1) rectangles, for k>0,
fit together in a unit square?
Note that the sum of the rectangle areas is 1.
Marc Paulhus can fit them into
a square of side 1.000000001: "An algorithm for packing squares",
J. Comb. Th. A 82 (1998) 147157,
MR1620857.
 Tiling with four cubes.
Torsten Sillke summarizes results and conjectures on
the problem of tiling 3dimensional boxes with a tile
formed by gluing three cubes onto three adjacent faces
of a fourth cube.
 Tiling with
notched cubes. Robert Hochberg and Michael Reid exhibit an unboxable
reptile: a polycube that can tile a larger copy of itself, but can't
tile any rectangular block.
 Tiling with polyominos.
Michael Reid summarizes results on the ability to cover
rectangles and other figures using polyominoes. See also
Torsten
Sillke's page of results on similar problems.
 Tiling dynamical systems.
Chris Hillman describes his research
on topological spaces in which each point represents a tiling.
 On
a tiling scheme by M. C. Escher, D. Davis, Elect. J. Combinatorics.
 Tilings.
Lecture notes from the Clay Math Institute, by Richard Stanley and
Federico Ardila, discussing polyomino tilings, coloring arguments for
proving the nonexistence of tilings, counting how many tilings a region
has, the arctic circle theorem for domino tilings of diamonds,
tiling the unit square with unitfraction rectangles, symmetry groups,
penrose tilings, and more. In only 21 pages, including the annotated
bibliography. A nice but necessarily concise introduction to the subject.
(Via Andrei Lopatenko.)
 Tilings of hyperbolic space.
 Tilings and visual symmetry, Xah Lee.
 Tim's Triangular Page.
 Tobi Toys
sell the
Vector Flexor, a flexible cuboctahedron skeleton, and
Foldaform,
an origami business card that folds to form a tetrahedron that can be
used as the building block for more complex polyhedra.
 Toilet
paper plagiarism. A big tissue company tries to rip off Sir Roger P.
 Touch3d, commercial
software for unfolding 3d models into flat printouts, to be folded back
up again for quick prototyping and mockups.
 Toroidal tile for tessellating threespace, C. Séquin, UC Berkeley.
 Totally Tessellated.
Mosaics, tilings, Escher, and beyond.
 A tour
of Archimedes' stomachion. Fan Chung and Ron Graham investigate
the number of different square solutions of this dissection puzzle.
 Toys from the Tech
Museum Store.
 The
tractrix and the pseudosphere, hyperbolic surfaces
modeled in Cabri.
 Transformational geometry.
Leslie Howe illustrates various plane symmetry types with Cabri animations.
 Traveling salesman problem and Delaunay
graphs. Mike Dillencourt and Dan Hoey revisit and simplify some
older work showing that the traveling salesman tour of a point set need
not follow Delaunay edges.
 Trefoil
knot stairs. Java animation of an Escherlike infinite stair construction,
intended as a Montreal metro station sculpture,
by Guillaume LaBelle.
 Triangle centers.
 Triangle geometry and the triangle book.
Steve Sigur's web site describing many important triangle centers and loci.
According to the site, he also has a book with John Conway on the
subject, coming soon.
 Triangle
table by Theo Gray, displaying the
Spieker Circle
of the 345 right triangle.
 Triangle tiling. Geom. Ctr. exhibit at the Science Museum of Minnesota.
 Triangle to a square.
David MacMillan asks geometry.puzzles about this dissection problem.
 Triangles and squares.
Slides from a talk I gave relating a simple 2d puzzle, Escher's drawings
of 3d polyhedra, and the combinatorics of 4d polytopes, via angles in
hyperbolic space. Warning: very large file (~8Mb).
For more technical details see
my
paper with Kuperberg and Ziegler.
 Federation Square.
This building in Melbourne uses the pinwheel
tiling as a design motif. Thanks to Khalad Karim for identifying it.
Photos by Dick Hess, scanned by Ed Pegg Jr.
See this Flickr
photopool for many more photos.
 Triangular
polyhex tilings. What is the smallest equilateral triangle that can
be tiled by a given polyhex?
 Triangulated
pig. M. Bern, Xerox.
 Triangulating 3dimensional polygons.
This is always possible (with exponentially many Steiner points)
if the polygon is unknotted, but NPcomplete if no Steiner points are allowed.
The proof uses gadgets in which quadrilaterals are
stacked like Pringles to form wires.
 Triangulation numbers. These classify the geometric structure of
viruses. Many viruses are shaped as simplicial polyhedra consisting of 12
symmetrically placed degree five vertices and more degree six vertices;
the number represents the distance between degree five vertices.
 Triangulations and arrangements.
Two lectures by Godfried Toussaint, transcribed by Laura Anderson and Peter
Yamamoto. I only have the lecture on triangulations.
 Triangulations with many different areas.
Eddie Grove asks for a function t(n) such that any nvertex convex polygon
has a triangulation with at least t(n) distinct triangle areas,
and also discusses a special case in which the vertices are points in a
lattice.
 Triply
orthogonal surfaces, Matthias Weber.
 Trisecting
an angle with origami. Julie Rehmeyer, MathTrek.
 The
trouble with five. Craig Kaplan explains why fivefold symmetry
doesn't work in regular plane tilings, but does work for the Penrose tiling.
 true_tile
mailing list for discussion of Euclidean and nonEuclidean tilings.
 Truncated
icosahedral symmetry. Explains why you might want to use a machined
aluminum buckyball as a gravitywave detector...
 Truncated
NanoOctahedron. Ned Seeman makes polyhedra out of DNA molecules.
 Truncated
Octahedra. Hop David has a nice picture of Coxeter's regular sponge
{6,44}, formed by leaving out the square faces from a tiling of space by truncated octahedra.
 Truncated
Trickery: Truncatering.
Some truncation relations among the Platonic solids and their friends.
 LunYi Tsai paints fine
art of foliatied 3manifolds, differentiable atlases, and other
topological structures.
 Tune's polyhedron models.
Sierpinski octahedra, stellated icosahedra, interlocking
zonohedrondissection puzzles, and more.
 Turkey
stuffing. A cube dissection puzzle from IBM research.
 Tuvel's
Polyhedra Page and
Tuvel's Hyperdimensional
Page.
Information and images on universal polyhedra and higher dimensional polytopes.
 27 lines on the Clebsch cubic, Matthias Weber.
 Twodistance sets.
Timothy Murphy and others discuss how many points one can have
in an ndimensional set, so that there are only two distinct
interpoint distances. The correct answer turns out to
be n^{2}/2 + O(n).
This
talk abstract by Petr Lisonek (and paper in JCTA 77 (1997) 318338)
describe some related results.
 270strut
tensegrity sphere. Jim Leftwich makes polyhedra out of dowels and
hairbands.
 Twothreeseven tiling of the hyperbolic plane
with lines that connect to give a fiery appearance.
From the Geometry Center archives.
 Typeface
Venus, Circle
Marilyn,
and Bubble
Mona. village9991
uses quadtrees
and superellipses
to make abstract mosaics of famous faces.
 Tysen
loves hexagons. And supplies ascii, powerpoint, and png graphics for
several styles of hexagonal grid graph paper.
 Ukrainian Easter Egg.
This zonohedron, computed by a Mathematica notebook I wrote, provides a lower bound for the complexity of the set of
centroids of points with approximate weights.
 UMass Gang
library of knots, surfaces, surface deformation movies, and
minimal surface meshing software.
 Unbalanced
anisohedral tiling.
Joseph Myers and
John Berglund find a polyhex that must be placed two different ways in
a tiling of a plane, such that one placement occurs twice as often as
the other.
 Unbeatable Tetris.
Java demonstration that this tetrominopacking game is a forced win
for the side dealing the tetrominoes.
 Uncyclopedia:
Geometry.
 Unfolding convex polytopes. From Jeff Erickson's geometry pages.
 Unfolding dodecahedron animation, Rick Mabry.
 Unfolding polyhedra.
A common way of making models of polyhedra is to unfold the faces into a
planar pattern, cut the pattern out of paper, and fold it back up.
Is this always possible?
 Unfolding convex polyhedra.
Catherine Schevon discusses whether it is always possible
to cut a convex polyhedron's edges so its boundary unfolds into a simple
planar polygon.
Dave Rusin's known math pages include
another article by J. O'Rourke on the same problem.
 Unfolding
some classes of orthogonal polyhedra,
Biedl, Demaine, Demaine, Lubiw, Overmars, O'Rourke, Robbins, and
Whitesides, CCCG 1998.
 Unfolding
the tesseract. Peter Turney lists the 261 polycubes that can be
folded in four dimensions to form the surface of a hypercube,
and provides animations of the unfolding process.
 Unfurling
crinkly shapes.
Science News discusses a recent result of Demaine, Connelly, and Rote,
that any nonconvex planar polygon can be continuously unfolded into
convex position.
 The uniform net
(10,3)a. An interesting crystal structure formed by packing square
and octagonal helices.
 Uniform
polychora.
A somewhat generalized definition of 4d polytopes,
investigated and classified by J. Bowers, the polyhedron dude.
See also the dude's pages on
4d polytwisters
and
3d
uniform polyhedron nomenclature.
 Uniform polyhedra.
Computed by Roman Maeder using a Mathematica
implementation of a method of Zvi Har'El.
Maeder also includes separately a picture of the
20 convex uniform polyhedra, and descriptions of the
59
stellations of the icosahedra.
 Uniform
polyhedra in POVray format, by Russell Towle.
 Uniform
polyhedra, R. Morris. Rotatable 3d java view of these polyhedra.
 An uninscribable 4regular polyhedron.
This shape can not be drawn with all its vertices on a single sphere.
 Uniqueness of focal points.
A focal point (aka equichord) in a starshaped curve is a point such that
all chords through the point have the same length.
Noam Elkies asks whether it is possible to have more than one focal point,
and Curtis McMullen discusses a generalization to nonstarshaped curves.
This problem has recently been put to rest by Marek Rychlik.
 Universal
coverage constants.
What is the minimum area figure of a given type that covers all
unitdiameter sets?
Part of Mathsoft's
collection of
mathematical constants.
 University of Arizona
mathematical models collection.
 Unreal project.
Nonphotorealistic rendering of mathematical objects,
Amenta, Duvall, and Rowley.
Here's another
unreal page.
 Unsolved
problems. Naoki Sato lists several conundrums from elementary
geometry and number theory.
 Variations of Uniform Polyhedra, Vince Matsko.
 Variations
on the theme of polyominos.
 Vasarely Design.
Hana Bizek makes geometric sculptures from Rubik's cubes.
 Vegreville,
Alberta, home of the world's largest easter egg.
Designed by Ron Resch, based on a technique he
patented
for folding paper or other flat construction
materials into flexible surfaces.
 A Venn diagram
made from five congruent ellipses. From F. Ruskey's Combinatorial
Object Server.
 Virtual Image,
makers of CDROMS of raytraced mathematical animation.
 Vision test.
Can you spot the hidden glide reflection symmetry lurking in
(the infinite continuation of) this pattern?
 Visual math,
the mathematical art of M. C. Escher.
 Visual Mathematics,
journal and exhibitions relating art and math.
 Visual
techniques for computing polyhedral volumes.
T. V. Raman and M. S. Krishnamoorthy use Zomebased ideas
to derive simple expressions for the volumes of the Platonic solids
and related shapes.
 Visualization
of a hyperbolic universe, Martin Bucher.
 Visualising
fractals in 3D. Sierpinski tetrahedron in Stonehenge, and a Menger sponge.
 Visualizing the hypersphere via 3d slices, and
other
fourdimensional thoughts by Jeff Fuquay.
 Visualization of the CarrilloLipman Polytope. Geometry arising from the simultaneous comparison of multiple DNA or protein sequences.
 The Vitruvian Man.
Connections between Leonardo's polygoninscribed human figure and sacred
temple geometry.
 Volume of a torus.
Paul Kunkel describes a simple and intuitive way of finding the formula for a
torus's volume by relating it to a cylinder.
 Volumes in
synergetics. Volumes of various regular and semiregular polyhedra,
scaled according to inscribed tetrahedra.
 Volumes of ideal hyperbolic hypercubes.
 Volumes of pieces of a dodecahedron.
David Epstein (not me!) wonders why parallel slices through the layers
of vertices of a dodecahedron produce equalvolume chunks.
 Voronoi Art.
Scott Sona Snibbe uses a retroreflective floor to display the Voronoi
diagram of people walking on it, exploring notions of personal space and
individualgroup relations.
Additional Voronoibased art is included in his
dynamic
systems series.
 Voronoi
diagram of a Penrose tiling (rhomb version), Cliff Reiter.
 Voronoi
diagrams at the Milwaukee Art Museum. Scott Snibbe's
artwork Boundary Functions,
as blogged by Quomodumque.
 Voronoi diagrams of lattices.
Greg Kuperberg
discusses an algorithm for constructing the Voronoi cells in a planar
lattice of points. This problem is closely related to some important
number theory: Euclid's algorithm for integer GCD's, continued
fractions, and good approximations of real numbers by rationals.
Higherdimensional generalizations (in which the Voronoi cells form
zonotopes) are much harder  one can find a basis of
short vectors using the wellknown LLL algorithm, but this doesn't
necessarily find
the vectors corresponding to Voronoi adjacencies. (In fact, according
to Senechal's Quasicrystals and Geometry, although the set
of Voronoi adjacencies of any lattice generates the lattice, it's not
known whether this set always contains a basis.)
 Voronoi
diagrams and ornamental design.
 The Voronoi Game.
Description, articles, references, and demonstration applet on
problems of competitive facility location, where two players place
sites in hopes of being nearest to as much area as possible.
See also
Crispy's
Voronoi game applet
and Dennis
Shasha's Voronoi game page.
 vZome
zometool design software for OS X and Windows.
(Warning, web site may be down on offhours.)
 Geometric sculpture by Elias
Wakan.
 Wallpaper groups. An illustrated guide to the 17 planar symmetry patterns.
See also Xah Lee's wallpaper group page.
 Wallpaper
patterns, R. Morris.
Kaleidoscopelike Java applet for making and transforming symmetric
tilings out of uploaded photos.
 Walt's toy
box. Walt Venables collects geometric toys, and uses them to help
design geodesic domes.
 The
Water Cube swimming venue at the 2008 Beijing Olympics uses the
WeairePhelan foam (a partition of 3d space into equalvolume cells with
the minimum known surface area per unit volume) as the basis of its structure.
 Waterman polyhedra,
formed from the convex hulls of centers of points near the origin in an
alternating lattice.
See also Paul
Bourke's Waterman Polyhedron page.
 Matthias
Weber's gallery of raytraced mathematical objects, such as minimal
surfaces floating in ponds.
 Wei and
Stan's Puzzle Selections, Key Press.
 Fr.
Magnus Wenninger, OSB, mathematician, builder of polyhedra.
 Westside Impressions
sells Escher Tshirts.
 What
can we measure?
A gentle introduction to geometric measure theory.
 What do you call a partially truncated rhombic dodecahedron? Doug Zare wants to know.
 What happens when you
connect uniformly spaced but not dyadic rational points along the Peano
spacefilling curve? R. W. Gosper illustrates the results.
 What is
arbelos you ask?
 What
is David Fowler making a Sierpinski tetrahedron out of? It looks
like toothpicks and marshmallows, or maybe pieces of styrofoam peanuts.
 What seven straight lines in the plane are most important?
 What to make with golf balls? Dale Seymour chooses a Sierpinski triangle and Sierpinski tetrahedron.
 When
can a polygon fold to a polytope? A. Lubiw and J. O'Rourke describe
algorithms for finding the folds that turn an unfolded paper model of a
polyhedron into the polyhedron itself. It turns out that the familiar
cross hexomino pattern for folding cubes can also be used to fold three other
polyhedra with four, five, and eight sides.
 Whimsical
rendering of a 4cube. Rick Mabry animates a 3d projection that has
a nice symmetrical 2d projection.
 Why doesn't Pick's theorem generalize?
One can compute the volume of a twodimensional polygon with integer
coordinates by counting the number of integer points in it and on its
boundary, but this doesn't work in higher dimensions.
 Why "snub cube"?
John Conway provides a lesson on polyhedron nomenclature and etymology.
From the geometry.research archives.
 Wingeom,
freeware Windows geometric construction software.
 Wonders of Ancient Greek Mathematics, T. Reluga.
This term paper for a course on Greek science includes sections
on the three classical problems, the Pythagorean theorem, the golden
ratio, and the Archimedean spiral.
 Wooden ballandstick models of Archimedean solids,
offered for sale by Dr. B's Science Basics.
 Wooden
polyhedra from Japan (but with English explanations). And more, in Japanese.
 Woolly
thoughts, mathematical knitwear.
 A word problem.
Group theoretic mathematics for determining whether a polygon formed out
of hexagons can be dissected into threehexagon triangles,
or whether a polygon formed out of squares can be dissected
into restrictedorientation triominoes.
 The world
of hyperbolic geometry, Colleen Robles.
 The
world's largest icosahedron. Jason Rosenfeld makes polyhedra
out of ten foot poles and shark fishing line.
 Worm
in a box. Emo Welzl proves that every curve of length pi can be
contained in a unit area rectangle.
 Vedder Wright
makes geometric models out of plastic forks.
 Joseph Wu's origami
page contains many pointers to origami in general.
 WWW spirograph.
Fill in a form to specify radii,
and generate pictures by rolling one circle around another.
For more pictures of cycloids, nephroids, trochoids,
and related spirograph shapes, see David Joyce's
Little Gallery of Roulettes.
Anu Garg
has implemented spirographs in Java.
 Xah Lee's mathematics graphics gallery.
 Xominoes.
Livio Zucca finds a set of markings for the edges of a square that lead
to exactly 100 possible tiles, and asks how to fit them into a 10x10 grid.
 Yantram sacred art toolbox.
Software for creating various kinds of symmetric fractal mandala.
 yukiToy. Shockwave plugin software for pushing around a few reddish spheres in
your browser window. But what exactly is the point?
(They're spheres, they don't have one, I guess.)
 Zef
Damen Crop Circle Reconstructions. What is the geometry underlying
the construction of these largescale patterns?
 Zillions
of Games: Pento.
 Al Zimmerman's
circlepacking contest. Cash prizes for finding the best packing of
circles with radii in arithmetic progression into a single larger circle.
 Zometool. The 31zone structural system for constructing
"mathematical models, from tilings to hyperspace projections, as well as
molecular models of quasicrystals and fullerenes, and architectural
space frame structures".
 Zometool truncated
icosahedron image from the A2Z science and learning store catalog.
This looks to me like a raytrace rather than a real model.
 Zonohedra
and cubic partial cubes. Connecting the geometric problem of
classifying simplicial line arrangements to the graphtheoretic one of
finding regular graphs that can be isometrically embedded on a cube.
 Zonohedra and zonohedrification. From George Hart's virtual polyhedron collection.
 Zonohedron. From Eric Weisstein's treasure trove of mathematics.
 Zonohedra and zonotopes. These centrally
symmetric polyhedra provide another way of understanding the
combinatorics of line arrangements.
 Zonohedron
Beta. A flexible polyhedron model made by Bathsheba
Grossman out of aluminum, stainless steel, and brass
(bronze optional). Also see the rest of Grossman's
geometric sculpture.
 Zonohedron generated by 30 vectors in a circle,
and another generated by 100 random vectors,
Paul Heckbert, CMU.
As a recent article in The Mathematica Journal explains,
the first kind of shape converges to a solid of revolution of a
sine curve. The second clearly converges to a sphere but Heckbert's example looks more like a
space potato.
 Zonotiles.
Russell Towle investigates tilings of zonogons (centrally symmetric
polygons) by smaller zonogons, and their relation to line arrangements,
with an implementation in Mathematica.
 Zonotopes.
Helena Verrill wonders in how many ways one can decompose a polygon into
parallelograms. The answer turns out to be equivalent to certain problems
of counting pseudoline arrangements.
 A zoo of surfaces.
 Frank Zubek's
Elusive Cube. Magnetic tetrahedra connect to form dissections of
cubes and many other shapes.
From the Geometry Junkyard,
computational
and recreational geometry pointers.
Send email if you
know of an appropriate page not listed here.
David Eppstein,
Theory Group,
ICS,
UC Irvine.
Semiautomatically
filtered
from a common source file.