Sometimes extremely miscellaneous...
- 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.
mathematical art, Xah Lee.
- Anti-Euclidean Love Song.
- Anton's modest
little gallery of ray-traced 3d math.
- 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.
Batchelder's Stellated Dodecahedron T-shirt.
- BitArt spirolateral
self-running demo or construct new spirolaterals).
- 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
- 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.
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.
misconception regarding a cube, Paul Bourke. No, the Egyptian
pyramids were not formed by dropped giant cubes from space.
Gallery of Computation. Some kind of algorithmic art; I'm not sure
what algorithms were used to produce it but the results are pretty.
plots with trig functions.
Eric Weeks discovers a method of making interesting non-moiré patterns.
- Andrew Crompton.
Grotesque geometry, Tessellations, Lifelike Tilings, Escher style drawings,
Dissection Puzzles, Geometrical Graphics, Mathematical Art.
Anamorphic Mirrors, Aperiodic tilings, Optical Machines.
- 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.
- 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".
- Sylvie Donmoyer
geometry-inspired paintings including Menger sponges and
a behind-the-scenes look at Escher's Stars.
- Eight foxes.
Daily geometry problems.
- Electronic Geometry Models,
a refereed archive of interesting geometric examples and visualizations.
game, or whack-a-focus.
- Experiencing Geometry.
A poem by David Henderson.
- Fagnano's theorem.
This involves differences of lengths in an ellipse.
Joe Keane asks why it is unusual.
- Famous curve applet index.
Over fifty well-known plane curves, animated as Java applets.
- Adrian Fisher Maze Design
series of a gastropod. L. Zucca uses Fourier analysis to square
the circle and to make an odd spiral-like shape.
- Fractal fiber bundles.
Troy Christensen ponders origami on the fabric folds of spacetime.
- 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.
- Geek bodyart.
Geometric calculations for fitting your piercings.
metaphors in literature, K. Kovaka.
- 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.
problems involving circles and triangles, with proofs.
- The geometry
of ancient sites.
- Geometry in Hawaiian history and culture
- 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
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,
the land of the Incas,
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.
- Jean-Pierre Hébert - Studio.
Algorithmic and geometric art 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.
to write "computational geometry" in Japanese (or Chinese).
- Human Geometry
and Naked Geometry. The
human form as a building block of larger geometric figures, by Mike
- Images of geometry. From the geometry center graphics archives.
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.
Temple Geometry, Gordon Coale.
See also this
temple geometry tablet map.
Unfortunately Scientific American seems to have taken
down their (May 1998) article on the subject.
- Java lamp, S. M.
- 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.
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
page on Kakeya-Besicovitch constants, and
Weisstein's page on the Kakeya Needle Problem.
of Infinite Space. A new biography of H. S. M. Coxeter by Siobhan Roberts.
tile and Kurschak's theorem about the area of a circle-inscribed dodecagon.
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.
- Leaper tours.
Can generalized knights jump around generalized chessboards visiting each square once? By Ed Pegg Jr.
- Lego sextic.
Clive Tooth draws infinity symbols using lego linkages,
and analyzes the resulting algebraic variety.
- Limerick #18124:
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.
- Jim Loy's
geometry pages. With special emphases on geometric constructions
(and non-constructions such as angle
trisection) as well as many nice
Fine Art Studio Sacred Geometry Art.
Prints and paintings for sale of various geometric designs.
- Math Pages: Geometry
Math Professor: Geometry Resources on the Web.
- Meru Foundation appears to be
another sacred geometry site, with animated gifs of torus knots
and other geometric visualizations and articles.
- 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
- Mormon computational geometry.
by Impulse. Computational geometry applied to the simulation of
bowling allies and poolhalls.
- Mike Naylor's
Platonic solids, knots, fractals, and more.
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.
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.
- T. Nordstrand's
gallery of surfaces.
- 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.
- 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.
A short introduction to the geometry of perspective drawing.
- Phaistos disk
geometry. Claire Watson examines the patterns on a Mediterranean
kicking locus in rugby, Michael de Villiers.
other geometry papers.
- Plan for pocket-machining Austria, M. Held, Salzburg.
- 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.
Bill Casselman uses postscript to motivate a course
in Euclidean geometry.
See also his Coxeter group graph paper,
and Ed Rosten's
Beware, however, that postscript can not really represent
such basic geometric primitives as circles, instead approximating them
- 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.
- Puzzles. Discussions on the geometry.puzzles list,
collected by topic at the Swarthmore Geometry Forum.
- Quadrorhomb rotary engine
with chambers defined by the bars of a twelve-bar linkage rotating
around two nonconcentric axes.
- Random polygons.
Tim Lambert summarizes responses to a request for
a good random distribution on the n-vertex simple polygons.
- The rational and mathematical art of A/K/Rona
Richard M. Smith uses POVray to view complex geometric scenes.
- Reconstruction of a closed curve from its elliptic Fourier descriptor.
The ancient epicycle theory of planetary motion, animated in Java.
- Ruler and Compass.
Mathematical web site including special sections on the
- 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
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.
of Da Vinci's challenge.
A discussion of the symbology and design of this
- Sedona Sacred
Geometry Conference, Feb. 2004.
- Self-righting 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.
- 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.
- Shape metrics.
Larry Boxer and David Fry provide many bibliographic references
on functions measuring how similar two geometric shapes are.
- Sighting point.
John McKay asks, given a set of co-planar 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(n2) shapes, each the union of two disks, so doing this
naively and applying parametric search gives O(n4 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 LP-type algorithms [Matousek, Sharir, and Welzl, SCG 1992;
Amenta, Bern, and Eppstein,
Manifolds online e-book by Howard Iseri.
I'm not sure I see why this should be useful or interesting, but the
idea seems to be to define geometry-like structures (having objects
called points and lines that somehow resemble Euclidean points and lines)
that are non-uniform 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.
- Smoothly rolling
polygonal wheels and their roads, H. Serras, Ghent.
- Some images made by Konrad Polthier.
- Some pictures of symmetric tensegrities.
- 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
- Splitting the hair.
Matthew Merzbacher discusses how many times one can subdivide
a line segment by following certain rules.
kaleidoscope. Strange diatom-like shapes formed by varying the
parameters of a spline surface mesh refinement scheme outside their
supershapes. Paul Bourke generates a wide variety of interesting
shapes from a simple formula.
See also John
Whitfield's Nature article.
geometry, Richard Hawkins' digital archive. Animations and 3d
models of polyhedra and tensegrity structures. Very
- Define: Tangent.
- Tensegrity zoology.
A catalog of stable structures formed out of springs,
somehow forming a quantum theory of what used to be described as time.
- Three classical geek problems solved!
Hauke Reddmann, Hamburg.
- 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.
orthogonal surfaces, Matthias Weber.
to make abstract mosaics of famous faces.
- Unreal project.
Non-photorealistic rendering of mathematical objects,
Amenta, Duvall, and Rowley.
- Visual Mathematics,
journal and exhibitions relating art and math.
- The Vitruvian Man.
Connections between Leonardo's polygon-inscribed human figure and sacred
can we measure?
A gentle introduction to geometric measure theory.
- What seven straight lines in the plane are most important?
- Xah Lee's mathematics graphics gallery.
Damen Crop Circle Reconstructions. What is the geometry underlying
the construction of these large-scale patterns?
From the Geometry Junkyard,
and recreational geometry pointers.
Send email if you
know of an appropriate page not listed here.
from a common source file.