The Geometry Junkyard: All Topics

All Topics

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

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.
Almost research-related maths pictures. A. Kepert approximates superellipsoids by polyhedra.
Angle trisection, from the geometry forum archives.
Anna's pentomino page. Anna Gardberg makes pentominoes out of sculpey and agate.
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.
Antiprism (Archimedean). Some self-explanatory pictures from Eric Weisstein's treasure trove of mathematics.
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.
Aperiodic tiling in Eⁿ. C. Goodman-Strauss generates tilings via "the illusion of beveling".
Aperiodic Wang tilings. K. Culik [Discrete Math. 160 (1996) 245-251] and J. Kari [Discrete Math. 160 (1996) 259-264] have recently discovered matching rules for as few as 13 square tiles that force the tiles to cover the plane aperiodically. Their constructions are based on finite automata that perform real number multiplication.
Archimedean solids: John Conway describes some interesting maps among the Archimedean polytopes. Eric Weisstein lists properties and pictures of the Archimedean solids.
Are most manifolds hyperbolic? From Dave Rusin's known math pages.
Area of hyperbolic triangles. From the Geometry Center's Java gallery of interactive geometry
Area of the Mandelbrot set. One can upper bound this area by filling the area around the set by disks, or lower bound it by counting pixels; strangely, Stan Isaacs notes, these two methods do not seem to give the same answer.
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.
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.
Aunt Annie's Craft Page / Geometric Playthings. Paper snowflake symmetry, flexagons, regular polyhedra, and the tangram square dissection puzzle.
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.
Henry Baker's hypertext version of HAKMEM includes a dissection of square and hexagon, depicted below.
Basic Research -- Combinatorial Geometry. A. Bachem, U. Koeln.
Books on polyhedra and polytopes. Collected by Tony Davie, St. Andrews U.
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.
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?
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.
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.
Buildings. These abstract structures are a way of interpreting group-theoretic questions as a form of combinatorial geometry.
Can't we make it non-Euclidean?
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).
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. For a recent result in the same area, see Another six-coloring of the plane, I. Hoffman and A. Soifer, Disc. Math. 150 (1996) 427-429.
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 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 and Dave Watson compare formulas for computing the coordinates of a circle's center from three boundary points.
Colinear points on knots. Greg Kuperberg shows that a non-trivial knot or link in R³ necessarily has four colinear points.
Coloring line arrangements. The graphs formed by overlaying a collection of lines require three, four, or five colors, depending on whether one allows three or more lines to meet at a point, and whether the lines are considered to wrap around through infinity.
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 Numbers and Geometry. Contents from a book by Liang-shiu Hahn, published by the MAA.
Complex polytope. A diagram representing a complex polytope, from H. S. M. Coxeter's home page.
Conformal geometry. A project studying computability problems of Riemann surfaces, at the U. of Joensuu, Finland.
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.
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.
Crumpling paper: states of an inextensible sheet.
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, Disc. Math. 158 (1996) 99-150.
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.
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. This lecture includes a mention of the possible application of regular triangulations to the famous problem of whether, if one pushes a collection of disks closer together, the covered area goes down. (Marshall Bern and Amit Sahai have recently made major progress on this problem using these methods.)
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.
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.
Diamond hyperlattice. Tony Smith of Georgia Tech. 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.
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 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.
Dissections: Plane & Fancy, Greg Frederickson's forthcoming dissection book.
Distinct point set with the same distance multiset. From K. S. Brown's Math Pages.
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.)
Do buckyballs fill hyperbolic space?
Dodecafoam. A fractal froth of polyhedra fills space.
Dodecahedron T-shirts from the makers of Mathematica.
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.
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.
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. 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.
Equivalents of the parallel postulate. David Wilson quotes a book by George Martin, listing 26 axioms equivalent to Euclid's parallel postulate.
Escher Fish. Silvio Levy's tesselation of the Poincare model of the hyperbolic plane by fish in M.C. Escher's style. From the Geometry Center archives.
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.
Exploring area and perimeter through pentominoes.
Fagnano's theorem. This involves differences of lengths in an ellipse. Joe Keane asks why it is unusual.
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.
Fat triangulations. Mike Todd discusses methods for finding a linear transformation of a triangulation to optimize the shapes of the simplices.
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).
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.
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.
Flexible polyhedra. From Dave Rusin's known math pages.
The Four Color Theorem. A new proof by Robertson, Sanders, Seymour, and Thomas.
Four dice hypercube visualization.
The Fourth Dimension.
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.
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 geometry and complex bases. Publications and software by W. Gilbert.
Fractal instances of the traveling salesman problem, P. Moscato, Buenos Aires.
Fractal patterns formed by repeated inversion of circles: Fractals Generated by M�bius Transforms, Mark Krosky, Cornell. Limit sets of Kleinian groups, D. Wright, Oklahoma State. Inversive circles, W. Gilbert, Waterloo.
Fractal tilings.
Fractals. The spanky fractal database at Canada's national meson research facility.
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.
Greg Frederickson's home page includes a 17-piece dissection of five decagons into one.
French terms in computational geometry. Compiled by Otfried Schwarzkopf while visiting INRIA.
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.
Generating Convex Polyominoes at Random. W. Hochst�ttler , M. Loebl and C. Moll, U. Cologne.
Generating Fractals from Voronoi Diagrams, Ken Shirriff, Berkeley and Sun. Unfortunately without illustrations.
Generators of crystallographic space groups. From Dave Rusin's known math pages.
Geombinatorics: Making Math Fun Again. A journal of open problems of combinatorial and discrete geometry and related areas.
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 topology and knot theory. From Dave Rusin's known math pages.
Geometrinity, geometric sculpture by Denny North.
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. Tim Kurtz' schoolchildren make geometric models out of toothpicks and marshmallows.
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.
Geometry poetry: Curves.
Geometry web page project: symmetry, Capital High School (Wash. State?)
Gerard's pentomino page.
The Graph of the Truncated Icosahedron and the Last Letter of Galois, B. Kostant, Not. AMS, Sep. 1995. Group theoretic mathematics of buckyballs. Recently reviewed by J. Baez.
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.
T. Hagerup's fancy Java traveling salesman 2-optimizer.
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.
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.
Hecatohedra. John Conway discusses the possible symmetry groups of hundred-sided polyhedra.
Hedronometry. Billy McConnell discusses equations relating the angles and face areas of tetrahedra.
Heilbronn triangle constants. How can you place n points in a square so that all triangles formed by triples of points have large area?
Hello polyomino! Arion Lei's polyomino page, with interactive Java demos and many links.
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.
Hexnet. The Hexnet Corporation is a Hexagonal organization which promotes the use of Hexagons as a replacement for other geometrical objects for many tasks.
Hilbert's 3rd Problem and Dehn Invariants. How to tell whether two polyhedra can be dissected into each other.
How many intersection points can you form from an n-line arrangement? It must be a number between n-1 and n(n-1)/2, but not all of those values are possible.
How many points can one find in three-dimensional space so that all triangles are equilateral or isosceles? One eight-point solution is formed by placing three points on the axis of a regular pentagon. This problem seems related to the fact that any planar point set forms O(n^7/3) isosceles triangles; in three dimensions, Theta(n³) are possible (by generalizing the pentagon solution above). From Stan Wagon's PotW archive.
How to construct a golden rectangle, K. Wiedman.
How to write "computational geometry" in Japanese (or Chinese).
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 Tesselations, 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.
Hyperspheres. Eric Weisstein calculates volumes and surface areas of hyperspheres, which curiously reach a maximum for dimensions around 5.257 and 7.257 respectively.
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 Erd�s) 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.
Interim report on Peek, software for visualizing high-dimensional polytopes.
Interlocking fractal trees. Treelike shapes grown by coloring and recoloring a hexagonal tiling cover the plane.
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.
Introduction to fractal geometry. Course offered by A. Kirillov at U. Penn.
Islamic geometric art.
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.
Japanese Triangulation Theorem. The sum of inradii in a triangulation of a convex polygon doesn't depend on which triangulation you choose! From Eric Weisstein's treasure trove of mathematics.
Java interactive Delaunay triangulation and Voronoi diagrams: D. Abrahams-Gessel, Dartmouth U. Baker et al., Brown U. Frank Bossen, Lausanne. Paul Chew, Cornell U. Eric Olson, Berkeley. Harald Spanring, Salzburg. And a WWW interaction without Java: Keith Voegele, Arizona State U.
Java lamp, S. M. Christensen.
Java Penrose Tiler, Geert-Jan van Opdorp.
Java pentomino puzzle solver, D. Eck, Hobart and William Smith Colleges.
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.
Kabon Triangles. How many disjoint triangles can you make out of n line segments? From Eric Weisstein's treasure trove of mathematics.
The Kakeya-Besicovitch problem. Paul Wellin describes this famous problem of rotating a needle in a planar set of minimal area. As it turns out the area can be made arbitrarily close to zero. See also Eric Weisstein's page on the Kakeya Needle Problem.
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 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. The Geometry Center has a nice picture of this construction.
Richard Kenyon's Gallery of tilings by squares and equilateral triangles of varying sizes.
Kepler-Poinsot Solids, concave polyhedra with star-shaped faces. From Eric Weisstein's treasure trove of mathematics.
Kepler's plague (vertex figures of regular polytopes and regular tilings).
Kissing numbers. Eric Weisstein lists known bounds on the kissing numbers of spheres in dimensions up to 24.
Knot pictures. Energy-minimized smooth and polygonal knots, from the ming knot evolver, Y. Wu, U. Iowa. See also the UMass Gang library of energy minimizing knots and links.
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.
Labyrinth tiling. This aperiodic substitution tiling by equilateral and isosceles triangles forms fractal space-filling labyrinths.
Erez Lirov has implemented several geometric algorithms in Java: three fractals and a TSP approximation heuristic.
Logical art and the art of logic, pentomino art, philosophy, and DOS software, G. Albrecht-Buehler.
A look at tilings, particularly emphasizing symmetry groups, strips, and spirals.
Making your own set of Penrose rhombs, N. Casey.
Manifolds from regular solids. Brent Everitt lists the finite volume orientable hyperbolic and spherical 3-manifolds obtained by identifying the faces of regular solids.
Maple polyhedron gallery.
The Margulis Napkin Problem. Jim Propp asks for a proof that the perimeter of a flat origami figure must be at most that of the original starting square.
Matching rules and substitution tilings. These two phrases denote different methods of generating aperiodic tilings. C. Goodman-Strauss shows that any tiling generated by substitution can also be generated by matching rules.
Math Pages: Geometry
Mathematica 3.0 Graphics Gallery: Polyhedra
The mathematics of polyominoes, K. Gong. Counts of k-ominoes, Kevin's Macintosh polyomino software, and more links.
Mathenautics. Visualization of 3-manifold geometry at UIUC.
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.
Mind Puzzles. Stefan Wolfrum investigates dissections of the 8*8 square into the 12 pentominoes together with a 2*2 square.
Minimax elastic bending energy of sphere eversions. Rob Kusner, U. Mass. Amherst.
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.
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.
Movies by Impulse. Computational geometry applied to the simulation of bowling allies and poolhalls.
Mutations and knots. Connections between knot theory and dissection of hyperbolic polyhedra.
My face on a Voronoi Diagram.
The Mystery of the Linked Triangles. Mathematical analysis of a sculpture by J. Robinson, consisting of three hollow triangles linked together like the Borromean rings. See also the linked triangle picture in the Geometry Center's graphics archive.
N-dimensional cubes, J. Bowen, Oxford.
Natural neighbors. Dave Watson supplies instances where shapes from nature are (almost) Voronoi polygons. He also has a page of related references.
Netlib polyhedra. Coordinates for regular and Archimedean polyhedra, prisms, anti-prisms, and more.
New directions in aperiodic tilings, L. Danzer, Aperiodic '94.
The no-three-in-line problem. How many points can be placed in an n*n grid with no three on a common line? The solution is known to be between 1.5n and 2n. Achim Flammenkamp discusses some new computational results including bounds on the number of symmetric solutions.
T. Nordstrand's gallery of surfaces.
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 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).
Open problems: From Jeff Erickson at Duke. From the 2nd MSI Worksh. on Computational Geometry.
Origami polyhedra. Jim Plank makes geometric constructions by folding paper squares.
Origami: a study in symmetry. M. Johnson and B. Beug, Capital H.S.
Packing polyominoes into rectangles. B. Purohit of Kings College applies neural networks to the problem.
Packings in Grassmannian spaces, N. Sloane, AT&T. How to arrange lines, planes, and other low-dimensional spaces into higher-dimensional spaces.
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.
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.)
The pavilion of polyhedreality. George Hart makes geometric constructions from coffee stirrers and dacron thread. Includes many pointers to related web pages.
Penrose quilt on a snow bank, M.&S. Newbold.
Penrose tiles and worse. This article from Dave Rusin's known math pages discusses the difficulty of correctly placing tiles in a Penrose tiling, as well as describing other tilings such as the pinwheel.
Penrose Tiles entry from E. Weisstein's treasure trove.
Penrose tilings. This 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 tiling puzzle, Kadon Enterprises.
Penrose-Wang tilings. Tony Smith of Georgia Tech. 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?"
Perplexing poultry Penrose pieces from pentaplex.
Pentagons that tile the plane, Bob Jenkins.
The pentamino challenge [sic]. 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 project-of-the-month from the Geometry Forum. List the pentominoes; fold them to form a cube; play a pentomino game.
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.
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 21, 24, 26, 26, 38, and 69 unequal squares. Many further solutions are possible; for instance combining solutions with m and n squares yields one with m+n-1 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.
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.
Plane Geometry. From Dave Rusin's known math pages.
Planet J Paperfolding. The mathematics of origami.
Pl�cker 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.
Polygon symbology.
Polygonal and polyhedral geometry. Dave Rusin, Northern Illinois U.
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.
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.
Polyiamonds. This Geometry Forum problem of the week asks whether a six-point star can be dissected to form eight distinct hexiamonds.
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.
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 of orders 4 through 7. See also K. S. Brown's polyomino enumeration 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.
Pretty Penrose picture, J. Beale, Stanford.
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.
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.
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.
Puzzles. Discussions on the geometry.puzzles list, collected by topic at the Swarthmore Geometry Forum.
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.
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.
Rabbit style object on geometrical solid. Complete and detailed instructions for this origami construction, in 3 easy steps and one difficult step.
A Ramsey-type problem on right-angled triangles. Any 3-coloring of 3-space has a monochromatic copy of any right triangle. M. Bona and G. Toth, Courant Inst. See also their paper in Disc. Math. 150 (1996) 61-67.
Random domino tiling of an Aztec diamond and other MIT 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 -- 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, as well as Mathsoft's page on geometric probability constants.
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).
Rec.puzzles archive: dissection problems.
Rec.puzzles archive: coloring problems.
The reflection of light rays in a cup of coffee or the curves obtained with b^n mod p, S. Plouffe, Simon Fraser U.
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.)
Reptile project-of-the-month from the Geometry Forum. Form tilings by dividing polygons into copies of themselves.
Reuleaux triangles: This demo of Sketchpad includes a Reuleaux triangle rolling between two parallel lines. The Reuleaux! WWW CAD/CAM service is designed to showcase the Reuleaux triangle, which forms the rotating element in a Wankel Rotary Engine. Ignore the line about this and the circle being the only constant-width shapes. Geometry forum discussion on the Reuleux triangle and its ability to drill out (most of) a square hole. Demo of Fields&Operators includes the path followed by a Reuleaux triangle rotating in a square. A magic geometric constant optimized by the Reuleaux triangle. Reuleaux triangle entry from Eric Weisstein's treasure trove.
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. Tesselating 3-space in hyperbolic geometry. Robert Grzeszczuk, U. Chicago. See also the Geometry Center's images of hyperbolic space tiled with dodecahedra.
Right triangles. I. Luttrell and J. Healy, Capital H.S.
Rigidity of polyhedra and geometric games. S. Schteingold describes how to make a rigid icosahedron out of only 16 plastic "Clixi" triangles, and contemplates more general questions of the rigidity of partial polytopes.
Robinson Friedenthal polyhedral explorations. Geometric sculpture.
Sascha Rogmann's hyperbolic geometry page
Rolling polyhedra. Dave Boll investigates Hamiltonian paths on (duals of) regular polyhedra.
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.
Rubino. This font, designed by Noel Rubin and available through ImageClub Graphics, "is based on Albrect Durer's mathematic interpretations of the typographic form". After dithering for a few months, I got a copy which I am using for the Geometry in Action banner. It's also available in a sans serif. The Junkyard banner is in another ImageClub font, Rockabilly.
The RUG FTP origami archive contains several papers on mathematical origami.
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.
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 Schafli 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".)
Self-affine tiles, J. Lagarias and Y. Wang, DIMACS. Mathematics of a class of generalized reptiles.
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.
Shape metrics. Larry Boxer and David Fry provide many bibliographic references on functions measuring how similar two geometric shapes are.
Sierpinski carpet on the sphere. From Curtis McMullen's math gallery.
Sierpinski Tetrahedron. I have seen ki tes made in this shape; if viewed from the right direction it projects one-to-one onto a square. Here's another view, another, and yet another, a paper model constructed by schoolchildren, instructions for making one out of straws, plus some history of tetrahedral kites. John Hart has still more: the five non-Platonic solids, "Sierpinskiized" versions of all the usual polyhedra.
Sighting point. John McKay asks, given a set of co-planar points, how to find a point to view them all from in a way that maximizes the minimum viewing angle between any two points. Somehow this is related to monodromy groups. I don't know whether he ever got a useful response. This is clearly polynomial time: the decision problem can be solved by finding the intersection of O(n²) shapes, each the union of two disks, so doing this naively and applying parametric search gives O(n⁴ polylog), but it might be interesting to push the time bound further. A closely related problem of smoothing a triangular mesh by moving points one at a time to optimize the angles of incident triangles can be solved in linear time by LP-type algorithms [Matousek, Sharir, and Welzl, SCG 1992].
Similarity tiling. A plane tiling using warped Koch snowflakes of various sizes.
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 R³ have exactly two lines crossing them all, and asks how this generalizes to higher dimensions.
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.
SMAPO library of polytopes encoding the solutions to optimization problems such as the TSP.
Smoothing regular and stellated polyhedra by spline surfaces. J. Chen, Purdue.
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.
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.
Solution to the pentomino problem by pete@bignode.equinox.gen.nz, from the rec.puzzles archives.
Some generalizations of the pinwheel tiling, L. Sadun, U. Texas.
Some images made by Konrad Polthier.
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 geometry. From Dave Rusin's known math pages. Includes a survey on placing points evenly on a sphere.
Spherical Julia set with dodecahedral symmetry discovered by McMullen and Doyle in their work on quintic equations and rendered by Don Mitchell.
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 tilings. These similarity tilings are formed by applying the exponential function to a lattice in the complex number plane.
Squared square. Robert Harley provides a picture of a square, divided into unequal smaller squares; the resulting planar map is four-colored. 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 MathPro open problem list and the geometry.puzzles archive.
Splitting the hair. Matthew Merzbacher discusses how many times one can subdivide a line segment by following certain rules.
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.
Student of Hyperspace. Pictures of 6 regular polytopes, E. Swab.
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.
Sylvester's theorem. This states that any finite non-colinear point set has a line containing only two points. Michael Larsen, Tim Chow, and Noam Elkies discuss two proofs and a complex-number generalization.
SymmeToy, windows shareware for creating paint patterns, symmetry roses, tesselated art and symmetrically decorated 3D polyhedron models.
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.
The symmetry of origami. R. Morse and T. Mosqueda, Capital H.S.
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. Its combinatorial dual, the Császár polyhedron, forms a torus with seven vertices. Here's another picture with a Hungarian caption and some literature references. See also Dave Rusin's page on polyhedral tori with few vertices.
Define: Tangent.
Tensegrity zoology. A catalog of stable structures formed out of springs, somehow forming a quantum theory of what used to be described as time.
Tesselations, a company which makes Puzzellations puzzles, posters, prints, and kaleidoscopes inspired in part by Escher, Penrose, and Mendelbrot.
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".
This is your brain on Tetris. Are pentominos really "an ancient Roman puzzle"?
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.
Three-dimensional geometry. From Dave Rusin's known math pages.
The three dimensional polyominoes of minimal area, L. Alonso and R. Cert, Elect. J. Combinatorics vol. 3.
3D strange attractors and similar objects, Tim Stilson, Stanford.
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.
Tiles at Wadham College, Oxford.
Tiling the integers with one prototile. A one-dimensional tiling problem on the boundary between geometry and number theory, with connections to factorization of finite cyclic groups.
The Tiling Problem. Karen Van Houten of U. Idaho describes Wang tiles as an example of an undecidable problem for her intro to CS class.
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 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. Open problem from the MathPro list.
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 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.
Toroidal tile for tesselating three-space, C. S�quin, UC Berkeley.
Toys from the Tech Museum Store: Rogers Connection (magnetic balls and metal tubes), Tensegritoy, Polygonzo.
Triangle tiling. Geom. Ctr. exhibit at the Science Museum of Minnesota.
Triangle to a square. David MacMillan asks geometry.puzzles about this dissection problem.
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 Octahedron. Ned Seeman makes polyhedra out of DNA molecules.
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 n²/2 + O(n).
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 zonotope provides a lower bound for the complexity of the set of centroids of points with approximate weights.
Unbeatable Tetris. Java demonstration that this tetromino-packing game is a forced win for the side dealing the tetrominoes.
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. T. Hahn supplies Platonic polyhedron patterns to be folded into paper models. Dave Rusin's known math pages include another article by J. O'Rourke on the same problem. Brian Hayes describes Eisenberg and Nishioka's HyperGami program for unfolding polyhedra in this article from the American Scientist.
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?
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.
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. According to R. J. Gardner, this problem has recently been put to rest by Rychlik.
Unit area triangle. Is there a constant A such that any plane region of area A contains the vertices of a unit area triangle? Open problem from the MathPro list.
Untangling Un-Knots. Finding minimum-energy states of tangled ropes. Robert Grzeszczuk, U. Chicago.
A Venn diagram made from five congruent ellipses. From F. Ruskey's Combinatorial Object Server.
Visualization of the Carrillo-Lipman Polytope. Geometry arising from the simultaneous comparison of multiple DNA or protein sequences.
Volumes in synergetics. Volumes of various regular and semi-regular polyhedra, scaled according to inscribed tetrahedra.
Volumes of ideal hyperbolic hypercubes.
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 are much harder -- the best solutions are based on the famous LLL algorithm, which finds a collection of short (but not shortest) vectors in any lattice.
The Voronoi Experience (Jason Smith, Oberlin). A couple pretty pictures of Voronoi diagrams but little actual content.
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.
Why "snub cube"? John Conway provides a lesson on polyhedron nomenclature and etymology. From the geometry.research archives.
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.
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.
Z² 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".
Zonohedron Beta. A flexible polyhedron model made by Bathsheba Grossman out of aluminum, stainless steel, and brass (bronze optional).
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.

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: 18 Oct 1996, 16:13:20 PDT.