The Geometry Junkyard: All Topics

All Topics

This page collects in one place all the entries in the geometry junkyard.

Jan Abas' Islamic Patterns Page.
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 Ajima-Malfatti 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 prism-shaped 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 disk-packing on spheres.
Algorithmic vectorial geometry, French geometry e-textbook 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 500-year-old 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.
Escher-inspired 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.
Anti-Euclidean 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 ray-traced 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 13-square set of tiles into a 21-cube set.
Aperiodic space-filling 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.
Arc length surprise. The sum of the areas of the regions between a circular arc and the x-axis, and between the arc and the y-axis, 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.
Archimedean spiral extended into three dimensions, from the Mathematica graphics gallery.
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.
Arranging six squares. This Geometry Forum problem of the week asks for the number of different hexominoes, and for how many of them can be folded into a cube.
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 Jean-Louis Loday's paper on associahedron coordinates.
Associating the symmetry of the Platonic solids with polymorf manipulatives.
Astro-Logix 3d ball-and-stick 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.
Atoma polyhedral building set.
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. (For a more informative description of this sort of analysis, see Mathsoft's page on the subject).
David Bailey's world of tesselations. Primarily consists of Escher-like 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.
Basic crystallography diagrams, B. C. Taverner, Witwatersrand.
NebulaSearch: Polyomino.
Ned Batchelder's Stellated Dodecahedron T-shirt.
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, 1-10. Connelly had previously discovered non-convex 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 double-covered sphere, to illustrate their topological equivalence, together with several related animations.
Beyond the Möbius strip. Construction templates for paper models of Möbius-related three-dimensional surfaces.
BitArt spirolateral gallery (requires JavaScript to view large images, and Java to view self-running 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.
Books on polyhedra and polytopes. Collected by Tony Davie, St. Andrews U.
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.
Borromean paper clips.
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 "Heron-type" formula for the maximum area of a quadrilateral, Col. Sicherman's fave. He asks if it has higher-dimensional 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 non-equilateral triangle with three equally large inscribed squares. Is there a three-dimensional 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 sphere-packing 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 Cheng-Pleijel 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 compass-and-straightedge construction, dynamic geometry demonstrations and automatic theorem proving. Ulli Kortenkamp and Jürgen Richter-Gebert, 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 non-trivial knot or link in R³ 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 (self-crossing) polygon, that passes through every point of an n*n grid. I added a similar puzzle with circular arcs.
constant-width 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.
Contour plots with trig functions. Eric Weeks discovers a method of making interesting non-moiré patterns.
Convex Archimedean polychoremata, 4-dimensional analogues of the semiregular solids, described by Coxeter-Dynkin 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 high-dimensional polytopes into pieces with smaller diameter requires a number of pieces exponential in the dimension.
Counting polyforms, with links to images of various packing-puzzle solutions.
Covering the Aztec diamond with one-sided 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 NP-hard, 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.
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 n-dimensional cube, S. Finch, MathSoft, and Asymptotically efficient triangulations of the d-cube, 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 shows that any smooth, simple, closed curve in 3-space must have total curvature at least 4 pi.
Cut-the-knot logo. With a proof of the origami-folklore that this folded-flat overhand knot forms a regular pentagon.
Cyclocentric polyhedra. Nadia Sobin makes cast plaster and stone sculptures by warping the faces of regular polyhedra.
Dancing links. Don Knuth discusses implementation details of polyomino search algorithms.
Deconstructing Marty. Tom Beard and Dorking Labs analyze the Sierpinski-carpet-like 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 tetrahedron-octahedron space tiling depicted in Escher's "Flatworms".
Deltahedra, polyhedra with equilateral triangle faces. From Eric Weisstein's treasure trove of mathematics.
Dense sphere-packings in hyperbolic space.
Densest packings of equal spheres in a cube, Hugo Pfoertner. With nice ray-traced images of each packing. See also Martin Erren's applet for visualizing the sphere packings.
Design Science Toys.
Detecting the unknot in polynomial time, C. Delman and K. Wolcott, Eastern Illinois U.
DeVicci's Tesseract. Higher-dimensional 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.
Die-cast metal polyhedra available for sale from Pedagoguery Software.
Dilation-free 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 problem-of-the-month 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 sacred-geometry 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.
Dodecahedron measures, Paul Kunkel.
Domegalomaniahedron. Clive Tooth makes polyhedra out of his deep and inscrutable singular name.
Sylvie Donmoyer geometry-inspired paintings including Menger sponges and a behind-the-scenes 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 screen-saver 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 Poisson-Voronoi 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.
Edge-tangent 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 eight-point arrangement in which each perpendicular bisector passes through two other points. From Stan Wagon's PotW archive.
The 85 foldings of the Latin cross, E. Demaine et al.
Electronic Geometry Models, a refereed archive of interesting geometric examples and visualizations.
Ellipse game, or whack-a-focus.
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 R⁴ 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 2-unit sides and contradicting an earlier answer to Asimov's question.
The equivalence of two face-centered icosahedral tilings with respect to local derivability, J. Phys. A26 (1993) 1455. J. Roth dissects an aperiodic three-dimensional 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 seemingly-flat 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.
Escher-like 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 Blast-Ended 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 30-60-90 triangles. See also the mathpuzzle eternity page.
Euclid 3:16. Fun geometry T-shirt 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 120-cell! Free Windows+OpenGL+.Net software.
Exploring hyperspace with the geometric product. Thomas S. Briggs explains some four-dimensional shapes.
An extension of Napoleon's theorem. Placing similar isosceles triangles on the sides of an affine-transformed 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 minimum-perimeter 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 three-dimensional 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 well-known plane curves, animated as Java applets.
Dr. Fathauer's Encyclopedia of Fractal Tilings.
Chris Fearnley's 5 and 25 Frequency Geodesic Spheres rendered by POV-Ray.
Helaman Ferguson mathematical sculpture.
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 3-dimensional 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 four-way 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 space-filling polyhedra. And not the ones you're likely thinking of, either. Guy Inchbald, reproduced from Math. Gazette 80, November 1996.
Five-fold symmetry in crystalline quasicrystal lattices, Donald L. D. Caspar and Eric Fontano.
The flat torus in the three-sphere. 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 Conrad-Hartline 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 four-dimensional regular polytopes.
Four-dimensional visualization. Doug Zare gives some pointers on high-dimensional 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 spiral-like 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 beta-skeleton with high dilation. Beta-skeletons 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 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. Ray-traced 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.)
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 on-line 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 17-gon carved on it leads to some discussion of 17-gon construction and perfectly scalene triangles.
Gaussian continued fractions. Stephen Fortescue discusses some connections between basic number-theoretic algorithms and the geometry of tilings of 2d and 3d hyperbolic spaces.
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 Java-based 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 screensaver-like interactive images. See also his 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.
Geometric wire sculptures. Stainless steel dodecahedron and stella octangula shaped ornaments, adorned with faceted glass balls by Australian artist Leigh Boileau.
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 n-gons 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 Wang-tile-like hexagonal tiles, specified by matching rules on their edges. These systems are Turing-complete, 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. Custom-made for Clive Tooth by Bob Aurelius.
Glowing green rhombic triacontahedra in space. Rendered by Rob Wieringa for the May-June 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 n-gons 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 non-convex generalizations of Platonic solids) out of plastic "Polydron" pieces.
Green-haired geometric pre-hominids.
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 Hansen-Smith makes geometric art out of paper plates.
Happy cubes and other three-dimensional polyomino puzzles.
Happy Pentominoes, Vincent Goffin.
Harary's animal game. Chris Thompson asks about recent progress on this generalization of tic-tac-toe and go-moku in which players place stones attempting to form certain polyominoes.
George Hart's geometric sculpture.
Jean-Pierre 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 regular-polygon faces.
Hecatohedra. John Conway discusses the possible symmetry groups of hundred-sided 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 420-430BC, 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 X-ray 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 3-sphere into circles.
Houtrust Relief. Nice photo of a 3d version of one of Escher's bird-fish 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 n-line arrangement? Equivalently, how many opposite pairs of faces can an n-zone zonohedron have? It must be a number between n-1 and n(n-1)/2, but not all of those values are possible.
How many points can one find in three-dimensional space so that all triangles are equilateral or isosceles? One eight-point 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(n³) 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 previously-thought-impossible 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.
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 multi-parametric analysis.
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.
Hyperbolic shortbread. The Davis math department eats a Poincaré model of a tiling of the hyperbolic plane by 0-60-90 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 3-dimensional 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. Red-blue 3d visualizations produced with the virtual flower system.
Hyperdimensional Java. Several web applets illustrating high-dimensional 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 ray-traced 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 pre-natal consciousness in 1968. Primary-colored 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, space-fillers, 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 non-colinear 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 L-systems. 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 Bone-Roller'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 Escher-like 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 two-part 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 straight-line 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 Penrose Tiler, Geert-Jan van Opdorp. Shuxiang Zeng has written another Java applet to play with Penrose tiles.
Java pentomino puzzle solver, D. Eck, Hobart and William Smith Colleges.
Iwan Jensen counts polyominos (aka lattice animals), paths, and various related quantities.
Jenn open-source software for visualizing Cayley graphs of Coxeter groups as symmetric 4-dimensional polytopes.
Jiang Zhe-Ming's geometry challenge. A pretty problem involving cocircularity of five points defined by circles around a pentagram.
Jim ex machina. Escher-like 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 x-coordinate) 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 click-together 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 12-vertex complete graph as symmetrically as possible on a six-handle 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(n-2)/3.
Kadon Enterprises, makers of games and puzzles including polyominoes and Penrose tiles.
The Kakeya-Besicovitch 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 Kakeya-Besicovitch 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 Escher-like animated animal-form tilings.
kD-tree demo. Java applet by Jacob Marner.
Keller's cube-tiling conjecture is false in high dimensions, J. Lagarias and P. Shor, Bull. AMS 27 (1992). Constructs a tiling of ten-dimensional space by unit hypercubes no two of which meet face-to-face, contradicting a conjecture of Keller that any tiling included two face-to-face 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 equal-volume low-surface-area cells. Kelvin had conjectured that the truncated octahedron provided the optimal solution, but this turned out not to be true.
Richard Kenyon's Gallery of tilings by squares and equilateral triangles of varying sizes.
The Kepler Conjecture on dense packing of spheres.
Kepler-Poinsot Solids, concave polyhedra with star-shaped faces. From Eric Weisstein's treasure trove of mathematics. See also H. Serras' page on Kepler-Poinsot 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 Kneser-Poulsen 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. Energy-minimized 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 ray-traced pictures of a quartic surface with lots of symmetries.
Kurschak's tile and Kurschak's theorem about the area of a circle-inscribed dodecagon.
William Labbett's geometrical pictures, including some polyomino tilings.
Labyrinth tiling. This aperiodic substitution tiling by equilateral and isosceles triangles forms fractal space-filling labyrinths.
Landry Art, Escheresque tessellations, and balsa and paper polyhedra, including some prints, t-shirts, 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 self-programmed individual.
Largest 5-gon 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 n-square 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, rational-angled 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. Albrecht-Buehler.
Log-spiral 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 non-constructions 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 Cabri-géomètre dynamic geometry software system.
A magic geometric constant optimized by the Reuleaux triangle.
Magical transformations. Wil Laan animates several dissections and almost-dissections.
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.
Making your own set of Penrose rhombs, N. Casey.
3-Manifolds from regular solids. Brent Everitt lists the finite volume orientable hyperbolic and spherical 3-manifolds 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 still has some Escher-like tiling patterns on his home page, although his old Escheresque wallpaper group page seems to be irretrievably gone.
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 furniture-making 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 non-crossing equal-length line segments.
Math Made Easy: Geometry interactive video supplemental learning materials.
Math Pages: Geometry
Math, Penrose, Plato, and Pentagrams. Philip Marsh claims that Penrose's pentagonal tilings connect him to the numerology of the Pythagoreans.
The Math Professor: Geometry Resources on the Web.
Math Quilts.
Mathematica Graphics Gallery: Polyhedra
Mathematica Menger Sponge, Robert M. Dickau.
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.
The mathematics of polyominoes, K. Gong. Counts of k-ominoes, 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 3-manifold geometry at the Univ. of Illinois.
MatHSoliD Java animation of planar unfoldings of the Platonic and Archimedean polyhedra.
Max. non-adjacent vertices on 120-cell. Sci.math discussion on the size of the maximum independent set on this regular 4-polytope. Apparently it is known to be between 220 and 224 inclusive.
Maximizing the minimum distance of N points on a sphere, ray-traced by Hugo Pfoertner.
Maximum area cross-section 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 ray-traced 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.
Min-energy 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 half-grid 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 five-fold or six-fold 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 pie-cosahedron. 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.
Dave Molnar's research on non-Euclidean symmetry and long-range order, Penrose and substitution tilings, L-systems, and cellular automata.
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 unit-length 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.
N-dimensional 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, anti-prisms, 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 inside-out 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 no-three-in-line 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.
Non-Euclidean games implemented in Shockwave by students in an advanced high school geometry class: projective-plane asteroids, hyperbolic double-torus minesweeper, and cubical fruitarian snake.
Non-Euclidean geometry with LOGO. A project at Cardiff, Wales, for using the LOGO programming language to help mathematics students visualise non-Euclidean 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 x-ray image.
Number patterns, curves, and topology