Recent Additions to the Junkyard

• 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.

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

• 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.

• Escher's buildings in origami.

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

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

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

• 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.

• 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.

• 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.

• 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. Equilateral triangle visibility problem from Andy Drucker. See also here.

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

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

• 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.

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

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

• 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.

• Platonic tesselations of Riemann surfaces, Gerard Westendorp.

• 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.

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

• 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.

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

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

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

• 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.

• 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.
  

• Trisecting an angle with origami. Julie Rehmeyer, MathTrek.

• 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.

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.
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