- 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.
- 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 Albion College Menger Sponge.
- Algorithms for coloring quadtrees.
- Alice visits the fourth dimension. Stereoscopically animated cross sections of a hypercube, with German text.
- Bob Allanson's Polyhedra Page. Nice animated-GIF line art of the Platonic solids, Archimedean solids, and Archimedean duals.
- Almost research-related maths pictures. A. Kepert approximates superellipsoids by polyhedra.
- Alpha shapes gallery. Pulsating spherical globules depicting Edelsbrunner and Mucke's methods for finding shapes from point samples.
- Angle trisection, from the geometry forum archives.
- On angles whose squared trigonometric functions are rational, J. Conway, C. Radin, and L. Sadun. This somewhat technical paper on the theory of Dehn invariants (used to determine whether there exists a dissection from one polyhedron to another) makes the theory more computationally effective. It contains the fascinating observation that there should exist a dissection that combines pieces from a dodecahedron, icosahedron, and icosidodecahedron to form a single large cube. How many pieces are needed?
- 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.
- Ant on a block. If you walk along the surface of a 1x1x2 rectangular block, from one corner, where is the farthest point? You would think the opposite corner, right?
- 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 hecatohedra and other 3d math.
- Apartment F4 and other pictures of objects from finite geometry, by Andreas Schroth.
- Aperiodic colored tilings, F. Gähler. Also available in postscript.
- 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.
- Apollonius' angle trisection. Animated in Java.
- Archimedean polyhedra, Miroslav Vicher.
- 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.
- ARTiP: an automated rectangular tiling prover. This system uses a constraint-propagation algorithm, similar to Waltz' famous line-labeling technique, to automatically find dissections of planar regions into rectangles.
- ASCII Menger sponge, W. Taylor.
- Rolf Asmund's polyhedra page.
- 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).
- The average kissing number of sphere packings. Greg Kuperberg and Oded Schramm give upper and lower bounds strictly between 12 and 15 on the average kissing number of packings in which the spheres need not all be the same size.
- Duane Bailey's color postscript Penrose tiler
- 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, and the three Bodhi problem, R. W. Gosper.
- Basic crystallography diagrams, B. C. Taverner, Witwatersrand.
- Basic Research -- Combinatorial Geometry. A. Bachem, U. Koeln.
- 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.
- Bell's tetrahedral kite - history and plans. Alexander Graham Bell makes Sierpinski tetrahedra out of half-inch iron pipe.
- 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.
- 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.
- Box of Mirrors. Renderings of 3d reflection groups.
- 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.
- The Buckyball. Drawn in wireframe and tangent-circles views.
- Buckyball: a
C
_{60}molecule. Pretty pictures of truncated icosahedra. - 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.
- Building polyhedra, string art, and tessellation drawings, geometry lesson plans from Diana Coates.
- 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.
- 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.
- The California Math Show goes to Spain. Photo exhibit of various symmetric patterns found in the architecture of Granada.
- CalmPlex puzzles. Reassemble a chessboard cut into twelve interlocking polyominos.
- Canonical polygons. Ronald Kyrmse investigates grid polygons in which all side lengths are one or sqrt(2).
- Can't we make it non-Euclidean?
- Catalogue of lattices, N. J. A. Sloane, AT&T Labs Research. See also Sloane's sphere-packing and lattice theory publications.
- Cellular automata on hyperbolic tilings? Message to CAS mailing list from B. Borcic.
- Cellular automaton run on Penrose tiles, D. Griffeath.
- 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 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.
- Les cinq polyédres de Platon
- Cinderella multiplatform Java system for 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. In this brief talk abstract, Ken Stephenson mentions connections between circle packing and the classical geometry of analytic function theory. See his home page for more 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.
- 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.
- 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 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.
- Combinatorial complexity of spheres. Olivier Devillers summarizes bounds and problems on convex hulls, unions, and intersections of spheres and unit spheres in high dimensions.
- Common misconception regarding a cube. From Paul Bourke's geometry page.
- Complex polytope. A diagram representing a complex polytope, from H. S. M. Coxeter's home page.
- 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.
- Conformal geometry. A project studying computability problems of Riemann surfaces, at the U. of Joensuu, Finland.
- Constructing Boy's surface out of paper and tape.
- Constructing a regular pentagon inscribed in a circle, by straightedge and compass. Scott Brodie. Also described by M. Gallant.
- Convex Archimedean polychoremata, 4-dimensional analogues of the semiregular solids, described by Coxeter-Dynkin diagrams representing their symmetry groups.
- John Conway in Zurich. A strip tease involving dense packing of self-inflating beach balls.
- 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 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. 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.
- Crumpled Menger Sponge. Paul D. Bourke.
- 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.)
- 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 puzzles collected by Johan Myrberger.
- 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 Simplexity
of the cube, R. B. Hughes and M. R. Anderson,
*Discrete Math.*158 (1996) 99-150; A lower bound for the simplexity of the cube via hyperbolic volumes, W. D. Smith,*Eur. J. Comb.*to appear; and Triangulating an n-dimensional cube, S. Finch, MathSoft. - Cuboctahedron, ink on paper, A. Glassner.
- 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.
- 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.
- Delaunay triangulation of projected points. Olivier Devillers asks how many different 2d Delaunay triangulations one gets when a 3d point set is projected in different ways onto a plane.
- 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.
- 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 hyperlattice. Tony Smith describes a tilted 4-dimensional hypercubic lattice with 8 links at each vertex: 4 in the future lightcone and 4 in the past lightcone.
- Dictionary of Combinatorics, Joe Fields, U. Illinois at Chicago.
- Digital
Diffraction, B. Hayes,
*Amer. Scientist*84(3), May-June 1996. What does the Fourier transform of a geometric figure such as a regular pentagon look like? The answer can reveal symmetries of interest to crystallographers. - 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?
- Direct opposite Reverse. David Sterner claims to have invented one of "the six simplest known solids to be mathematically defined" and uses its chromatic aberration in a 3d-photograph process for the seven-eyed.
- 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.
- Distinct point set with the same distance multiset. From K. S. Brown's Math Pages.
- 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. See also Stephen Werbeck's fractal iterations of a dodecahedron connecting through edges.
- Dodecahedron calendar, generated by a postscript program.
- Domegalomaniahedron. Clive Tooth makes polyhedra out of his deep and inscrutable singular name.
- 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.
- Dr. Matrix' programming challenge asks for a Windows Penrose tiler. This page also includes background material on tiling and aperiodicity as well as some of the theory of Penrose tilings.
- DUST software for visualization of Voronoi diagrams, Delaunay triangulations, minimum spanning trees, and matchings, U. Köln.
- Dutchman Designs. Pentomino and polyiamond patchwork quilt patterns.
- Dynamic formation of Poisson-Voronoi tiles. David Griffeath constructs Voronoi diagrams using cellular automata.
- An eight-point arrangement in which each perpendicular bisector passes through two other points. From Stan Wagon's PotW archive.
- Ellipse game, or whack-a-focus.
- 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
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.
- Erich's Combinatorial Geometry Page. Lots of information on covering grid points with circles, min max edge length triangulation, tree-planting problems, etc.
- 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.
- Escheresque wallpaper groups, Marcin Injarl Malinowski.
- Escherization.
How to find a periodic tile as close as possible to a given shape? Craig S.
Kaplan, U. Washington.
- Euclid's Elements. Online, in interesting colors, without all those annoying proofs. Also see D. Joyce's Java-animated version, Ralph Abraham's extensively illustrated edition, and this manuscript excerpt from a copy in the Bodleian library made in the year 888.
- Even pure mathematicians sometimes make large transpacific-bandwidth-wasting raytraces of buckyball jigsaw puzzles.
- 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 interactive fractal design software by John S. Stokes III.
- 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 this
dissection of a rectangle
with and without a hole.
- Famous curve applet index. Over fifty well-known plane curves, animated as Java applets.
- Fat triangulations. Mike Todd discusses methods for finding a linear transformation of a triangulation to optimize the shapes of the simplices.
- 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.
- Fibonacci spirals, Ned May.
- 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).
- Finding the wood by the trees. Marc van Kreveld studies strategies by which a blind man with a rope could map out a forest.
- First USA Computing Olympiad programming problems. Half of the four were geometrical: find a largest empty rectangle (any bets whether any of the solutions involved the SMAWK algorithm?), and enumerate polyominoes.
- 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.
- The five non-Platonic solids, "Sierpinskiized" versions of all the usual polyhedra.
- Five Platonic solids and a soccerball.
- 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.
- Flexagons. A 1962 technical report by Antony S. Conrad and Daniel K. Hartline.
- Flexible polyhedra. From Dave Rusin's known math pages.
- Foliations, partitions of topological spaces into lower dimensional subspaces, including the Hopf fibration partitioning a 3-sphere into circles.
- 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.
- The fractal art of Wolter Schraa. Includes some nice reptiles and sphere packings.
- 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 designs using pattern blocks, Jim Millar.
- The fractal gallery tour: Sierpinski tetrahedron
- Fractal geometry and complex bases. Publications and software by W. Gilbert.
- Fractal geometry summer workshop by Michael Frame and Benoit Mandelbrot.
- Fractal instances of the traveling salesman problem, P. Moscato, Buenos Aires.
- Fractal patterns formed by repeated inversion of circles: Inversion
graphics gallery, Xah Lee. Limit sets of
Kleinian groups, D. Wright, Oklahoma State. Inversive
circles, W. Gilbert, Waterloo.
- Fractal planet. Felix Golubov makes random triangulated polyhedra in Java by perturbing the vertices of a recursive subdivision.
- Fractal skewed web. Sierpinski tetrahedron by Mary Ann Conners.
- Fractal tetrahedron kite used by L. Hudgins to teach middle school students concepts of volume and surface area.
- Fractal tilings.
- The fractal translight newsletter. Roger Bagula mixes essays on random topics with Basic code for producing various Sierpinski-like fractal images.
- Fractals. The spanky fractal database at Canada's national meson research facility.
- Fractiles, multicolored magnetic rhombs with angles based on multiples of pi/7.
- Fractional Graph Theory, a rational approach to the theory of graphs, Edward R. Scheinerman and Daniel Ullman, Johns Hopkins. Explains why the fractional chromatic number of the plane is at most 7 and at least 32/9.
- French terms in computational geometry. Compiled by Otfried Schwarzkopf when his global circumnavigation passed through the French riviera.
- 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.)
- 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.
- 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.
- Gamelan educational geometry Java applet collection.
- Generating Convex Polyominoes at Random. W. Hochstättler , M. Loebl and C. Moll, U. Cologne.
- 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.
- Geombinatorics: Making Math Fun Again. A journal of open problems of combinatorial and discrete geometry and related areas.
- Geometria Java-based software for constructing and measuring polyhedra by transforming and slicing predefined starting blocks.
- Geometry problems involving circles and triangles, with proofs. Antonio Gutierrez.
- A geometry scavenger hunt!
- 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.
- Geometric
graph coloring problems from "Graph Coloring
Problems", a book by T. Jensen and B. Toft including a chapter on
geometric and combinatorial graphs.
- Geometric paper folding. David Huffman.
- Geometric probability constants. From MathSoft's favorite constants pages.
- Geometric topology preprint server.
- Geometrica, a Mathematica package for drawing and computing with geometrical forms.
- A Geometrical Picturebook of finite and combinatorial geometries, B. Polster, to be published by Springer.
- Geometrie mit dem Computer and Ka's Geometriepage. Teaching resources for geometry, in German, by Monika Schwarze.
- 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,
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.
- The geometry of the buckyball, Kim Allen, UC San Diego.
- Geometry corner with Martin Gardner. He describes some problems of cutting polygons into similar and congruent parts. From the MAT 007 I News.
- Geometry and Food. Janine Parker's schoolchildren make geometric models out of toothpicks and gumdrops.
- 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 jokes.
- The Geometry of the Mayan TimeStar, G. de Jong. Complexes of interlocking Platonic solids animated in Java.
- Geometry in Motion. Lots of Java animations by Daniel Scher.
- Geometry papers by Peter Woo. The Arbelos and circle inversion, ruler-only constructions, and triangle centers.
- Geometry poetry: Curves.
- Geometry turned on -- making geometry dynamic. A book on the use of interactive software in teaching.
- Geo-Sphere geodesic dome sculpture made from 3/4" steel tubes.
- Gerard's pentomino page.
- 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.
- The golden
spiral. This shape, constructed by inscribing circular arcs in a spiral tiling
of squares, resembles but is not quite the same as a logarithmic spiral. A
similar spiral is used as the Sybase Inc.
logo.
- Golden spiral jewelry image made by Keith Halewood to commemorate his Welsh heritage. This one combines three Archimedean spirals and doesn't have anything to do with the golden ratio. The meaning of the triple spiral symbol is explained by this note.
- Golygons, polyominoes with consecutive integer side lengths.
- 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.)
- Great math programs. Xah Lee reviews mathematical software, focusing on educational Macintosh applications. Includes sections on geometric visualization, fractals, cellular automata, and geometric puzzles.
- Greek mathematics and its modern heirs. Manuscripts of geometry texts by Euclid, Archimedes, and others, from the Vatican Library.
- Daniel Green's geometry page. Green makes models of regular sponges (infinite non-convex generalizations of Platonic solids) out of plastic "Polydron" pieces.
- Rona Gurkewitz' Modular Origami Polyhedra Systems Page. With many nice images from two modular origami books by Gurkewitz, Simon, and Arnstein.
- T. Hagerup's fancy Java traveling salesman 2-optimizer.
- 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.
- 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.
- Hecatohedra.
John Conway discusses the possible symmetry groups of hundred-sided polyhedra.
See also Anton's gallery.
- Hedronometry. Billy McConnell discusses equations relating the angles and face areas of tetrahedra.
- 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 and Xah's special curve index, and Eric Weisstein's treasure trove.
- 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.
- Helicopter and triceratops, covered with strips of triangles by the Stripe program.
- Hello polyomino! Arion Lei's polyomino page, with interactive Java demos and many links.
- 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.
- Heureka,
the Finnish science center uses Penrose tiles to pave the area in front of
its main entrance. (Unfortunately, the picture included here is not very good
-- see the
*Mathematical Intelligencer*18(4), Fall 1996, p. 65 for a better photo.) - Hexagon dissection and pentagon dissection. Five and six piece dissections into a square, made of walnut and cherry by Walter Hoppe. From Puzzle World.
- Hexagon tiling. The regular tiling by hexagons can be repeatedly subdivided and recombined into a tiling by hexagons 1/7 the size of the original, to form an interesting recursive structure. From Paul Bourke's geometry page.
- Hexiamond Home, Mark Paulhus, Calgary.
- Hexnet. The Hexnet
Corporation is a Hexagonal organization which promotes the use of Hexagons as
a replacement for other geometrical objects for many tasks.
- 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.
- Hinged
dissections of polyominoes
- Chuck Hoberman's Unfolding Structures.
- 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.
- 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(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 write "computational geometry" in Japanese (or Chinese).
- D. Huson's favorite hyperbolic tiling.
- Humongocubocuboctahedron and other constructions. Rod Rodrigues builds geometric models out of pressure treated lumber and deck reinforcing plates.
- Hyper hyper! Extra dimensions
- HypArr, Unix software for modeling and visualizing convex polyhedra and plane arrangements.
- 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 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 Geometry using Cabri
- Hypercube's Home Page. Speculations on the fourth dimension collected by Eric Saltsman.
- Hypercube game. Experience the fourth dimension with an interactive, stereoscopic java animation of the hypercube.
- 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. Kyoto University, Group for Hyperspace, English version. Graphic images of regular polytopes. See also their page of 4-polytope images (not linked to from their main page).
- 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.
- Iamond. Tilings of the plane by unions of equilateral triangles, found by Michael Dowle, a forthcoming book on polyiamonds by Ed Pegg Jr., and several more polyiamond links.
- 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.
- IFS attractors, a collection of fractal reptiles by Stewart Hinsley.
- Images of geometry. From the geometry center graphics archives. More images, from "Interactive Methods for Visualizable Geometry, A. Hanson, T. Munzner, and G. Francis.
- Infect. Eric Weeks generates interesting colorings of aperiodic tilings.
- Information on Pentomino Puzzles and Information on Polyominoes, from F. Ruskey's Combinatorial Object Server.
- 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).
- 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.
- 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.
- 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.
- 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...
- Isosceles pairs. Stan Wagon asks which triangles can be dissected into two isosceles triangles.
- 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.
- Jacqui's Polyomino Workshop. Activities associated with polyominoes, aimed at the level of primary (or elementary) school mathematics.
- Japanese
Temple Geometry. From
*Scientific American*, May 1998. See also this clickable temple geometry tablet map. - Japan Tessellation Design Society, Makoto Nakamura.
- 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. From Eric Weisstein's treasure trove of mathematics.
- Java applets on mathematics, Walter Fendt.
- Java gallery of geometric algorithms, Z. Zhao, Ohio State U.
- Interactive Delaunay triangulation and Voronoi diagrams:

VoroGlide, Icking, Klein, Köllner, Ma, Hagen.

D. Watson, CSIRO, Australia.

D. Abrahams-Gessel, Dartmouth U.

Baker et al., Brown U.

Frank Bossen, Lausanne.

Paul Chew, Cornell U.

Eric Olson, Berkeley.

Keith Voegele, Arizona State U.

Scandal, CMU (only for non-paranoid people running X-windows).

- 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.
- 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.
- Journey into the Menger sponge, Stephan Werbeck.
- Kabon
Triangles. How many disjoint triangles can you make out of
*n*line segments? From Eric Weisstein's treasure trove of mathematics. - Kadon Enterprises, makers of games and puzzles including polyominoes and Penrose tiles.
- The Kakeya-Besicovitch problem. Steven Finch 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 Eric Weisstein's page on the Kakeya Needle Problem.
- Kaleidoscope geometry, Ephraim Fithian.
- 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.
- 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.
- Kissing numbers. Eric Weisstein lists known bounds on the kissing numbers of spheres in dimensions up to 24.
- 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.
- Labyrinth
tiling. This aperiodic substitution tiling by equilateral and isosceles
triangles forms fractal space-filling labyrinths.
- Language Generator Tool and Die Lab. Tennis ball theorems, hourglass theorems, and cellular hierarchies. From a truly self-programmed individual.
- 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. - Layered
graph drawing.
- 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 sextic. Clive Tooth draws infinity symbols using lego linkages, and analyzes the resulting algebraic variety.
- Line fractal. Java animation allows user control of a fractal formed by repeated replacement of line segments by similar polygonal chains.
- Links2go: Polyhedra
- Logical art and the art of logic, pentomino art, philosophy, and DOS software, G. Albrecht-Buehler.
- 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.
- 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.
- 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 scalene triangle, Robert Simms.
- Maple polyhedron gallery.
- 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.
- Martin's pretty polyhedra. Simulation of particles repelling each other on the sphere produces nice triangulations of its surface.
- 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 or Art? German site on Menger-sponge-like fractals.
- Math Pages: Geometry
- Math, Penrose, Plato, and Pentagrams. Philip Marsh claims that Penrose's pentagonal tilings connect him to the numerology of the Pythagoreans.
- Math Quilts.
- Mathematica 3.0 Graphics Gallery: Polyhedra
- Mathematica Menger Sponge, Robert M. Dickau.
- 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, Kevin's 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.
- 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 sponge floating in space. Everyone and his brother makes ray-traced fractals with unlikely backgrounds nowadays, but Cliff Pickover was there first.
- Military Marge. An animated Escher-like tiling of the plane by images of a cartoon soldier. (Warning, mild nudity.)
- Mind Puzzles. Stefan Wolfrum investigates dissections of the 8*8 square into the 12 pentominoes together with a 2*2 square.
- 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.
- 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.
- 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.
- Modeling mollusc shells with logarithmic spirals, O. Hammer, Norsk Net. Tech. Also includes a list of logarithmic spiral links.
- 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.
- Modularity in art. Slavik Jablan explores connections between art, tiling, knotwork, and other mathematical topics.
- 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, CMU.
- Morin's Sphere Eversion. Robert Grzeszczuk, U. Chicago.
- 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.
- MuPAD Sierpinski Tetrahedron image and source code.
- Mutations and knots. Connections between knot theory and dissection of hyperbolic polyhedra.
- My face on a Voronoi Diagram.
- N-dimensional cubes, J. Bowen, Oxford.
- N-dimensional ray tracing, Pat Fleckenstein, RIT.
- 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.
- Netlib polyhedra. Coordinates for regular and Archimedean polyhedra, prisms, anti-prisms, and more.
- New directions in aperiodic tilings, L. Danzer, Aperiodic '94.
- Mark Newbold's Rhombic
Dodecahedron Page.
- 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 1.5*n*and 2*n*. Achim Flammenkamp discusses some new computational results including bounds on the number of symmetric solutions. - Non-Euclidean geometry with LOGO. A project at Cardiff, Wales, for using the LOGO programming language to help mathematics students visualise non-Euclidean geometry.
- 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.
- 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.
- Odd rectangles for
L
_{4n+2}. Phillippe Rosselet shows that any L-shaped (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. 156-159).
- Open problems:

From Jeff Erickson, Duke U.

From Jorge Urrutia, U. Ottawa.

From the 2nd MSI Worksh. on Computational Geometry. From SCG '98. - The Optiverse. An amazing 6-minute video on how to turn spheres inside out.
- Origami polyhedra. Jim Plank makes geometric constructions by folding paper squares.
- Origami mathematics, Tom Hull, Merrimack.
- Origami Menger Sponge built from Sonobe modules by K. & W. Burczyk.
- Origamic
tetrahedron. The image below depicts a way of making five folds in a 2-3-4
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.
- Ornamente, Parkette, visuelle Muster (pages on tilings and related lattice patterns, in German).
- 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 pennies in the plane, an illustrated proof of Kepler's conjecture in 2D by Bill Casselman.
- Packings in Grassmannian spaces, N. Sloane, AT&T. How to arrange lines, planes, and other low-dimensional spaces into higher-dimensional 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.
- Paper folding a 30-60-90 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.
- Paper models of polyhedra.
- Pappus on the Archimedean solids. Translation of an excerpt of a fourth century geometry text.
- 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 non-elegant.)
- 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 high-dimensional polytopes.
- Pennies in a tray, Ivars Peterson.
- Penrose mandala and five-way Borromean rings.
- Das Penrose Parkett. (In German.)
- Penrose Pavers in Penngrove. Pat Walp shows off a path of concrete Penrose rhombs he made for his garden.
- Penrose quilt on a snow bank, M.&S. Newbold. See also Lisbeth Clemens' Penrose quilt.
- Penrose-tiled swallow
- Penrose Tilings at Miami Univ.
- 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 tiles at Storey Hall, RMIT, Melbourne, Australia. See also Aperiodic Tiling, Penrose Tiling and the Generation of Architectural Forms, M. J. Ostwald, Nexus '98.
- Penrose tilings. This five-fold-symmetric tiling by rhombs or kites and darts is probably the most well known aperiodic tiling.
- Penrose tilings and the golden mean, K. Wiedman.
- Penrose-Wang tilings. Tony Smith describes some of the mathematics behind these aperiodic tilings, somehow leading to the concluding question "Can musical sequences also simulate the operation of any Turing machine?"
- Penrose's influence on Escher.
- Perplexing pentagons, Doris Schattschneider, from the Discovering Geometry Newsletter. A brief introduction to the problem of tiling the plane by pentagons.
- Perplexing poultry Penrose pieces from pentaplex. Also comes with alien space dogs.
- 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 and PentagOnline, Mike Korn's one-stop shop for five-sided polygons which tile the plane (but currently only vertex-transitive tilings).
- The pentagram and the golden ratio. Thomas Green, Contra Costa College.
- Pentamini. Italian site on pentominoes, by L. Zucca.
- The pentomino challenge. Aschig challenges all comers to write pentomino programs, and asks which square to omit from a chessboard so that the remaining 63 squares can be covered by 1*3 rectangles.
- Pentomino. Odette De Meulemeester is running a contest to see how much area can be enclosed by a contiguous boundary of squares formed by the twelve pentominos.
- Pentomino dissection of a square annulus. From Scott Kim's Inversions Gallery.
- Pentomino project-of-the-month from the Geometry Forum. List the pentominoes; fold them to form a cube; play a pentomino game. See also proteon's polyomino cube-unfoldings and Livio Zucca's polyomino-covered cube.
- Pento - A Program to Solve the Pentominoes Problem. Available in source or Linux binary.
- Pentomino dictionary, G. Esposito-Farèse. The twelve pentominoes resemble letters; what words do they spell? Also includes sections on "perecquian" configurations and a pentomino jigsaw puzzle.
- Pentomino HungarIQa. What happens to standard pentomino puzzles and games if you use poly-rhombs instead of poly-squares?
- Pentominoes, expository paper by R. Bhat and A. Fletcher.
- Penumbral shadows of polygons form projections of four-dimensional polytopes. From the Graphics Center's graphics archives.
- Perfect Square Dissection into unequal squares. From Eric Weisstein's treasure trove of mathematics.
- 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.
- 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 3-4-5 triangle.
- Pi and the Mandelbrot set.
- Pi squared by six rectangle dissected into unequal integer squares (or an approximation thereof) by Clive Tooth.
- Pictures of
minimal surfaces drawn with
*Mathematica*by Ute Fuchs. - Pictures of 3d and 4d regular solids, R. Koch, U. Oregon. Koch also provides some 4D regular solid visualization software in Java.
- Pinwheel and sphinx aperiodic
substitution tilings. Mathematica notebooks from M. Senechal, Smith College.
- Plan for pocket-machining Austria, M. Held, Salzburg.
- 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 solid fold-up patterns. Also showing inradii, circumradii, etc.
- Platonic solids and quaternion groups, J. Baez.
- Plexagons. Ron Evans proposes to use surfaces made out of pleated hexagons as modular construction units. Further details and computer models by Paul Bourke.
- 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 random-start hill-climbing 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.
- Polycell. George Olshevsky makes and sells polyhedra from colored cardstock.
- Polydron patented polychromatic plastic polygons.
- 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 n-gon counter.
- Polygon symbology.
- 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 three-dimensional convex polyhedron.
- Polyhedra. Bruce Fast is building a library of images of polyhedra. He describes some of the regular and semi-regular polyhedra, and lists names of many more including the Johnson solids (all convex polyhedra with regular faces).
- Polyhedra collection, V. Bulatov, Imperial College.
- The Polyhedra Page, Bruce Ross.
- 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. Ray-traced images by Tom Gettys, and a primer on constructing paper models.
- Polyhedron challenge: cuboctahedron.
- Polyhedron web scavenger hunt
- The Poly Pages. information on the various polyforms: poly-ominoes, -iamonds, -hexes, -cubes etc.
- Polyforms. From Miroslav Vicher's puzzle pages.
- Polyiamonds. This Geometry Forum problem of the week asks whether a six-point star can be dissected to form eight distinct hexiamonds.
- PolyMultiForms. L. Zucca uses pinwheel tilers to dissect an illustration of the Pythagorean theorem into few congruent triangles.
- Polyomino covers. Alexandre Owen Muniz investigates the minimum size of a polygon that can contain each of the n-ominoes.
- 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 n-ominoes 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.
- Polyominoes of orders 4 through 7. See also K. S. Brown's polyomino enumeration page.
- Polytopia CD-ROMs on tessellations, polyhedra, honycombs, and polytopes.
- Popsicle stick bombs, lashings and weavings in the plane, F. Saliola. See also this less mathematical treatment of popsicle stick bombs from Kid's Tracks.
- Portfolio: Polyhedra. Peter T. Wang makes geometric models out of strips of photocopy paper. See also his renderings of geometric models and Platonic solids page.
- Postscript geometry. Bill Casselman uses postscript to motivate a course in Euclidean geometry. See also his Coxeter group graph paper. and Phil Smith's Postscript Doodles page, especially the postscript spirograph. Beware, however, that postscript can not really represent such basic geometric primitives as circles, instead approximating them by splines.
- A pre-sliced 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.
- Pretty Penrose picture, J. Beale, Stanford.
- 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 14-omino. 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. Escher-like 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 incidence-preserving 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 soma-cube puzzle program
- Proofs of Euler's Formula. V-E+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.
- Pseudospherical surfaces. These surfaces are equally "saddle-shaped" 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 hand-crafted mechanical puzzles. Includes many geometric toys and puzzles.
- Puzzles. Discussions on the geometry.puzzles list, collected by topic at the Swarthmore Geometry Forum.
- 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.
- Quaquaversal
Tilings and Rotations. John Conway and Charles Radin describe a
three-dimensional generalization of the pinwheel tiling, the mathematics of
which is messier due to the noncommutativity of three-dimensional rotations.
Quasicrystals and aperiodic tilings, A. Zerhusen, U. Kentucky. Includes a nice description of how to make 3d aperiodic tiles from zometool pieces.

- Quasicrystals and color symmetry. Ron Lifshitz provides a light introduction to the symmetry of periodic and aperiodic crystals, and the complications introduced by including permutations of colors in a coloring as part of a symmetry operation. His publication list includes more technical material on the same subject.
- A
quasi-polynomial 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 n-face polytope are linked by
a chain of O(n) edges. This paper gives the weaker bound O(n
^{log d}). - Quasitiler image, E. Durand.
- Quicktime VR and mathematical visualization.
- Quincy Kim's World of Geometry. Illusions, space filling patterns, puzzles, fractals, and op art.
- 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 Escher-like 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 straight-line 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) 939-943, and rectilinear crossing constant, S. Finch, MathSoft.
- 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
- 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 right-angled tetrahedra.
- Realization Spaces of 4-polytopes are Universal, G. Ziegler and J. Richter-Gebert, 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.
- 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.)
- Regular 4d polytope foldouts. Java animations by Andrew Weimholt. Also includes some irregular polytops.
- Regular
polytopes in higher dimensions. Russell Towle uses Mathematica to slice
and dice simplices, hypercubes, and the other high-dimensional 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 project-of-the-month 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 Eric Weisstein's treasure trove.
- Reuleaux triangle entry from Kunkel's mathematics lessons.
- 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 48-gon, available with better color resolution
from his website.
- Riemann Surfaces and the Geometrization of 3-Manifolds, C. McMullen, Bull. AMS 27 (1992). This expository (but very technical) article outlines Thurston's technique for finding geometric structures in 3-dimensional topology.
- Right
Pentagonal Dodecahedron. Tessellating 3-space in hyperbolic geometry.
Robert Grzeszczuk, U. Chicago.
- Rigid
regular r-gons. Erich Friedman asks how many unit-length bars are needed
in a bar-and-joint linkage network to make a unit regular polygon rigid. What
if the polygon can have non-unit-length edges?
- 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 magnet-tipped 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.
- 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 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 Daniel 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 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 five-point 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 and coherent emotion.
- Sacred geometry, new discoveries linking the great pyramid to the human form. Charles Henry finds faces in raytraces of reflecting spheres.
- Sacred geometry discovery. This site also includes pictures of some bamboo polyhedral models.
- Saints Among Us. Anna Chupa makes kaleidoscopic photomontages based on the geometry of the Penrose tiling.
- Santa Rosa
Menger Cube made by Tom Falbo and helpers at Santa Rosa Junior College
from 8000 1"-cubed 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) 165-191). - The Schläfli Double Six. A lovely photo-essay of models of this configuration, in which twelve lines each meet five of thirty points. (This site also refers to related configurations involving 27 lines meeting either 45 or 135 points, but doesn't describe any mathematical details. For further descriptions of all of these, see Hilbert and Cohn-Vossen'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 unit-diameter rope that can be used to tie a given knot.
- Self-affine tiles, J. Lagarias and Y. Wang, DIMACS. Mathematics of a class of generalized reptiles.
- Self-affine tiles. Marina Khibnik computes the convex hulls and boundary dimensions of fractal tiles such as the twin dragon and fractal red cross.
- Semi-regular 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.
- 75-75-30 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.
- Shawn's mathematical gallery. Penrose tilings, Newton-iteration convergence-domain fractals, Schlegel diagrams of four-dimensional polyhedra, and more.
- Sierpinski carpet on the sphere. From Curtis McMullen's math gallery.
- Sierpinski gasket, green ocean. Rendered by Peter Wang.
- Sierpinski gaskets and variations rendered by D. H. Hepting.
- 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 on a flying carpet. Rendered by Jade Van Doren.
- Sierpinski triangle reptile based on a complex binary number system, R. W. Gosper.
- 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(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 LP-type algorithms [Matousek, Sharir, and Welzl, SCG 1992; Amenta, Bern, and Eppstein, SODA 1997]. - A simple dodecahedron tiling puzzle. Cover the dodecahedron's faces with pentagonal tetrominos.
- Simple polygonizations. Erik Demaine explores the question of how many different non-crossing traveling salesman tours an n-point 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 p-iterated cone over the product of m-1 and n-1 dimensional simplices.
- Six-regular toroid. Mike Paterson asks whether it is possible to make a torus-shaped 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.
- 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 sphere-packing and lattice theory publications.
- A small puzzle. Joe Fields asks whether a certain decomposition into L-shaped 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.
- SnapPea, powerful software for computing geometric properties of knot complements and other 3-manifolds.
- 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.
- Snub cube fountain at Caltech.
- Soap films and grid walks, Ivar Peterson. A discussion of Steiner tree problems in rectilinear geometry.
- Soap films on knots. Ken Brakke, Susquehanna.
- Soddy Spiral. R. W. Gosper calculates the positions of a sequence of circles, each tangent to the three previous ones.
- Sofa movers' problem. This well-known problem asks for the largest area of a two-dimensional region that can be moved through a hallway with a right-angled bend. Part of Mathsoft's collection of mathematical constants.
- Solid object which generates an anomalous picture. Kokichi Sugihara makes models of Escher-like illusions from folded paper. He has plenty more where this one came from, but maybe the others aren't on the web.
- Solution to the pentomino problem by pete@bignode.equinox.gen.nz, from the rec.puzzles archives.
- 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.
- Some planar tilings generated by the lattice projection method (of which the Penrose tiling is a special case) by Andrew Lewis, Queens U.
- Space Cubes plastic geometric modeling puzzle based on a rectangular Borromean link.
- 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.
- 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.
- Spiral
generator, web form for creating bitmap images of colored logarithmic
spirals.
- Spiral hexagonal circle
packings in the plane and figures. Beardon,
Dubejko, and Stephenson investigate the possible ways to pack circles in the
plane so that each circle is surrounded by six others.
- Spiral in a liquid crystal film.
- 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.
- 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.
- Spring into
action. Dynamic origami. Ben Trumbore, based on a model by Jeff Beynon
from Tomoko Fuse's book
*Spirals*. - Square Knots. This article by Brian Hayes for American Scientist examines how likely it is that a random lattice polygon is knotted.
- Squared square. Robert Harley provides a picture of a square, divided into unequal smaller squares; the resulting planar map is four-colored. 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 this Geometry Forum problem of the week.
- 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.
- 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.
- Stardust Polyhedron Puzzles. This U.K. company sells unfolded polyhedral puzzles and space-packing shapes (including a nice model of the Weaire-Phelan space-filling foam) on card-stock, to cut out and build yourself.
- 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 tangram-like shape-forming game based on a dissection of the square and studied by Archimedes.
- 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.
- Structors. Panagiotis Karagiorgis thinks he can get people to pay large sums of money for exclusive rights to use four-dimensional regular polytopes as building floor plans. But he does have some pretty pictures...
- Student of Hyperspace. Pictures of 6 regular polytopes, E. Swab.
- Subdivision kaleidoscope. Strange diatom-like 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 NP-completeness for geometric problems is a mismatch between the real-number 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, flexible polyhedra, and hyperbolic tiling.
- Sylvester's theorem. This states that any finite non-colinear 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 complex-number 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 torus-shaped polyhedra
- 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 bandwidth-intensive.
- The
Szilassi Polyhedron. This polyhedral torus, discovered by L.
Szilassi, has seven hexagonal faces, all adjacent to each other. It has an
axis of 180-degree 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. Here's
another picture with a Hungarian caption and some literature references.
See also Dave Rusin's page on polyhedral
tori with few vertices.
- 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 three-dimensional problems.
- Define: Tangent.
- The tea bag problem. How big a volume can you enclose by two square sheets of paper joined at the edges? See also Andrew Kepert's teabag problem page.
- 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.
- 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 Escher-like patterns of interlocking figure and really really long web page titles.
- Tessellations Tetrominos. This company sells sets of 60 foam rubber tetromino tiles.
- 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 were orphaned when
*maths with photographs*went offline. Does anyone know what places they are pictures of? (For another view of the cuboctohedron sculpture, see Rod's cuboctahedron page.) - This is your brain on Tetris. Are pentominos really "an ancient Roman puzzle"?
- Morwen Thistlethwait, sphere packing, computational topology, symmetric knots, and giant ray-traced 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
*n*-vertex thrackle has at most*n*edges. Stephan Wehner describes what is known about thrackles. - Three classical geek problems solved! Hauke Reddmann, Hamburg.
- Three-color the Penrose tiling? Mark Bickford asks if this tiling is always three-colorable. Ivars Peterson reports on a new proof by Tom Sibley and Stan Wagon that the rhomb version of the tiling is, but it's not clear whether this applies to the kites and darts version. 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.
- Three-dimensional models based on the works of M. C. Escher
- The
three dimensional polyominoes of minimal area, L. Alonso and R. Cert,
*Elect. J. Combinatorics*vol. 3. - 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
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.
- A tiling from ell. Stan Wagon asks which rectangles can be tiled with an ell-tromino.
- 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 one-dimensional 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 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, Brown U.
- Tiling stuff. J. L. King examines problems of determining whether a given rectangular brick can be tiled by certain smaller bricks.
- Tiling the unit square with rectangles. 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. According to fourth-hand rumor, Marc Paulhus can fit them into a square of side 1.000000001, to appear in J. Comb. Th. Erich Friedman shows that the 5/6 by 5/6 square can always be tiled with 1/(*k*+1) by 1/(*k*+1) squares. - Tiling with four cubes. Torsten Sillke summarizes results and conjectures on the problem of tiling 3-dimensional 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.
- Tilings of hyperbolic space.
- Tilings and visual symmetry, Xah Lee.
- Tobi Toys sell the Vector Flexor, a flexible cuboctahedron skeleton, and Fold-a-form, 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.
- Tom's Branch of Polytopia. An introduction to multi-dimensional regular solids.
- Touch-3d, commercial software for unfolding 3d models into flat printouts, to be folded back up again for quick prototyping and mock-ups.
- The topology and visualization of higher dimensions, Dollins, Hammel, Peck, and Williams, Brown Univ.
- Toroidal tile for tessellating three-space, C. Séquin, UC Berkeley.
- Totally Tessellated. Mosaics, tilings, Escher, and beyond.
- Toys from the Tech Museum Store.
- 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.
- Triangle centers.
- Triangle tiling. Geom. Ctr. exhibit at the Science Museum of Minnesota.
- Triangle to a square. David MacMillan asks geometry.puzzles about this dissection problem.
- Triangulated pig. M. Bern, Xerox.
- Triangulating
3-dimensional polygons. This is always possible (with exponentially many
Steiner points) if the polygon is unknotted, but NP-complete 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 n-vertex 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.
- Truncated
icosahedral symmetry. Explains why you might want to use a machined
aluminum buckyball as a gravity-wave detector...
- Truncated Nano-Octahedron. Ned Seeman makes polyhedra out of DNA molecules.
- Truncated Octahedra. Hop David has a nice picture of Coxeter's regular sponge {6,4|4}, 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.
- Turkey stuffing. A cube dissection puzzle from IBM research.
- Two-distance
sets. Timothy Murphy and others discuss how many points one can have in an
n-dimensional 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) 318-338) describe some related results. - 270-strut tensegrity sphere. Jim Leftwich makes polyhedra out of dowels and hairbands.
- Two-three-seven tiling of the hyperbolic plane with lines that connect to give a fiery appearance. From the Geometry Center archives.
- 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.
- Unbeatable Tetris. Java demonstration that this tetromino-packing game is a forced win for the side dealing the tetrominoes.
- 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.
- Unfold the polygon. Olivier Devillers asks, if one is given a simple polygon, treated as a linkage of rigid rods connected by hinges, can it be opened out into a convex polygon without crossing itself?
- Unfolding the tesseract. Peter Turney lists the 261 polycubes that can be folded in four dimensions to form the surface of a hypercube.
- 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
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. - An
uninscribable 4-regular 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 star-shaped 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 non-star-shaped 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 unit-diameter sets? Part of Mathsoft's collection of mathematical constants.
- Unreal project. Non-photorealistic 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.
- Untangling Un-Knots. Finding minimum-energy states of tangled ropes. Robert Grzeszczuk, U. Chicago.
- Variations of Uniform Polyhedra, Vince Matsko.
- 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. See also William Chow's
page on the Vegrevill easter egg.
- A Venn diagram
made from five congruent ellipses. From F. Ruskey's Combinatorial Object
Server.
- Helena Verrill's Fun Page
- Virtual Image, makers of CD-ROMS of ray-traced mathematical animation.
- Vision test.
Can you spot the hidden glide reflection symmetry lurking in (the infinite
continuation of) this pattern?
- Visual Mathematics, journal and exhibitions relating art and math.
- Visualising fractals in 3D. Sierpinski tetrahedron in Stonehenge, and a Menger sponge.
- Visualization of the Carrillo-Lipman Polytope. Geometry arising from the simultaneous comparison of multiple DNA or protein sequences.
- 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 semi-regular 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 equal-volume chunks.
- 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. Higher-dimensional
generalizations (in which the Voronoi cells form zonotopes) are much harder --
one can find a basis of short vectors using the well-known LLL algorithm, but
this doesn't necessarily find the vectors corresponding to Voronoi
adjacencies. (In fact, according to Schattschneider'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.) - The Voronoi Experience (Jason Smith, Oberlin). A couple pretty pictures of Voronoi diagrams but little actual content.
- Wallpaper groups. An illustrated guide to the 17 planar symmetry patterns. See also Xah Lee's wallpaper group page.
- Walt's toy box. Walt Venables collects geometric toys, and uses them to help design geodesic domes.
- Wei and Stan's Puzzle Selections, Key Press.
- Fr. Magnus Wenninger, OSB, mathematician, builder of polyhedra.
- 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 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?
- 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.
- Which heptiamonds tile the plane? Part of Kurt's tiling project.
- Whimsical rendering of a 4-cube. 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 two-dimensional 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.
- 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.
- A word
problem. Group theoretic mathematics for determining whether a polygon
formed out of hexagons can be dissected into three-hexagon triangles, or
whether a polygon formed out of squares can be dissected into
restricted-orientation triominoes.
- 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.
- Wrinkle Java applet creates reaction-diffusion patterns with a choice of symmetries.
- 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, and the postscript spirograph machine on Phil Smith's Postscript Doodles page. 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.
- Yet another ray-traced Sierpinski tetrahedron with a fractal background
- 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.)
- Z
^{2}section of a Penrose tiling. Robbie Robinson explains his work on the dynamical theory of tilings. - Zometool. The 31-zone 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 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 pseudo-line arrangements.

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.

Semi-automatically filtered from a
common source file. Last update: 01 Jan 2001, 23:05:46 PST.