The Geometry Junkyard

Sierpinski cookies

Thu, 10 Apr 2008 00:00:00 GMT

Sierpinski cookies. Actually more like Menger cookies, but whatever.

Non periodic tiling of the plane

Thu, 10 Apr 2008 00:00:00 GMT

Non periodic tiling of the plane. Including Penrose tiles, Pinhweel tiling, and more. Paul Bourke.

Sierpinski gaskets and Menger sponges

Thu, 10 Apr 2008 00:00:00 GMT

Sierpinski gaskets and Menger sponges, Paul Bourke. Including stacks of coke cans, radio antennas, crumpled sponges, and more.

Platonic solids and Euler's formula

Sat, 01 Mar 2008 00:00:00 GMT

Platonic solids and Euler's formula. Vishal Lama shows how the formula can be used to show that the familiar five Platonic solids are the only ones possible.

Greg's favorite math party trick

Sat, 12 Jan 2008 00:00:00 GMT

Greg's favorite math party trick. A nice visual proof of van Aubel's theorem, that equal perpendicular line segments connect the opposite centers of squares exterior to the sides of any quadrilateral. See also Wikipedia, MathWorld, Geometry from the land of the Incas, interactive Java applet.

Platonic tesselations of Riemann surfaces

Fri, 21 Dec 2007 00:00:00 GMT

Platonic tesselations of Riemann surfaces, Gerard Westendorp.

The trouble with five

Fri, 21 Dec 2007 00:00:00 GMT

The trouble with five. Craig Kaplan explains why five-fold symmetry doesn't work in regular plane tilings, but does work for the Penrose tiling.

Gömböc

Sun, 09 Dec 2007 00:00:00 GMT

Gömböc, a convex body in 3d with a single stable and a single unstable point of equilibrium. Placed on a flat surface, it always rights itself; it may not be a coincidence that some tortoise shells are similarly shaped. See also Wikipedia, Metafilter, New York Times.

Moebius transformations revealed

Fri, 16 Nov 2007 00:00:00 GMT

Moebius transformations revealed. Video by Douglas N. Arnold and Jonathan Rogness explaining 2d Moebius transformations in terms of the motions of a 3d sphere. See also MathTrek.

Modular pie-cosahedron

Sat, 06 Oct 2007 00:00:00 GMT

Modular pie-cosahedron. Turkey Tek makes geometric models out of pecan pie.

Hebesphenomegacorona

Wed, 03 Oct 2007 00:00:00 GMT

Hebesphenomegacorona ~~onna stick~~ in space! Space Station Science picture of the day. In case you don't remember what a hebesphenomegacorona is, it's one of the Johnson solids: convex polyhedra with regular-polygon faces.

In plane sight

Wed, 26 Sep 2007 00:00:00 GMT

In plane sight. Equilateral triangle visibility problem from Andy Drucker. See also here.

Escher's buildings in origami

Thu, 19 Jul 2007 00:00:00 GMT

Escher's buildings in origami.

Poncelet's porism

Mon, 16 Jul 2007 00:00:00 GMT

Poncelet's porism, the theorem that if a polygon is simultaneously inscribed in one circle and circumscribed in another, then there exists an infinite family of such polygons, one touching each point of each circle. From the secret blogging seminar.

A golden sales pitch

Fri, 29 Jun 2007 00:00:00 GMT

A golden sales pitch. Julie Rehmeyer dissects the myth of the golden ratio in classical art and describes some new uses for it in commerce.

Trisecting an angle with origami

Thu, 31 May 2007 00:00:00 GMT

Trisecting an angle with origami. Julie Rehmeyer, MathTrek.

Self-righting shapes

Thu, 05 Apr 2007 00:00:00 GMT

Self-righting shapes. Figures with only one stable and one unstable equilibrium, when placed on a level surface. Surprisingly, they look much like certain kinds of turtles. Julie J. Rehmeyer in MathTrek.

The Origami Lab

Mon, 26 Feb 2007 00:00:00 GMT

The Origami Lab. New Yorker article on Robert Lang's origami mathematics.

Ancient Islamic Penrose Tiles

Thu, 22 Feb 2007 00:00:00 GMT

Ancient Islamic Penrose Tiles. Peter Lu uncovers evidence that the architects of a 500-year-old Iranian shrine used Penrose tiling to lay out the decorative patterns on its archways. From Ivars Peterson's MathTrek.

Qubits

Tue, 23 Jan 2007 00:00:00 GMT

Qubits, modular geometric building blocks by architect Mark Burginger, inspired by Fuller's geodesic domes.

Geometric Arts

Sun, 21 Jan 2007 00:00:00 GMT

Geometric Arts. Knots, fractals, tesselations, and op art. Formerly Quincy Kim's World of Geometry.

The Szilassi Polyhedron

Thu, 18 Jan 2007 00:00:00 GMT

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. See also Dave Rusin's page on polyhedral tori with few vertices and Ivars' Peterson's MathTrek article.

Solution of Conway-Radin-Sadun problem

Mon, 15 Jan 2007 00:00:00 GMT

Solution of Conway-Radin-Sadun problem. Dissections of combinations of regular dodecahedra, regular icosahedra, and related polyhedra into rhombs that tile space. By Dehn's solution to Hilbert's third problem this is impossible for individual dodecahedra and icosahedra, but Conway, Radin, and Sadun showed that certain combinations could work. Now Izidor Hafner shows how.

Bending a soccer ball mathematically

Sun, 14 Jan 2007 00:00:00 GMT

Bending a soccer ball mathematically. Michael Trott animates morphs between a torus and a double-covered sphere, to illustrate their topological equivalence, together with several related animations.

Explore the 120-cell!

Sun, 14 Jan 2007 00:00:00 GMT

Explore the 120-cell! Free Windows+OpenGL+.Net software.

Exploring hyperspace with the geometric product

Sun, 14 Jan 2007 00:00:00 GMT

Exploring hyperspace with the geometric product. Thomas S. Briggs explains some four-dimensional shapes.

Sierpinski pentatope

Sun, 14 Jan 2007 00:00:00 GMT

Sierpinski pentatope video by Chris Edward Dupilka. A four-dimensional analogue of the Sierpinski triangle.

Art of the Tetrahedron

Thu, 11 Jan 2007 00:00:00 GMT

Art of the Tetrahedron. And by "Art" he means "Arthur". Arthur Silverman's geometric sculpture, from Ivars Peterson's MathTrek.

Laying Track

Thu, 04 Jan 2007 00:00:00 GMT

Laying Track. The combinatorics and topology of Brio train layouts. From Ivars Peterson's MathTrek.