Topology and Geometry Software by Jeff Weeks.
posted by Eideteker on Apr 22, 2009 - 5 comments

Mathematicians create videos that help in visualizing four-dimensional objects. Science News writes about it: seeing in four dimensions.
posted by Surfin' Bird on Aug 24, 2008 - 26 comments

Interactive mathematics miscellany and puzzles, including 75 proofs of the Pythagorean Theorem, an interactive column using Java applets, and eye-opening demonstrations. (Actually, much more.)
posted by parudox on Dec 1, 2007 - 11 comments

The Johnson Solids are a set of 92 semi-regular polyhedra, all of which are uniquely named and numbered. Except for the familiar square pyramid they all have exotic names like the Hebesphenomegacorona. A Hebesphenomegacorona in space. Number 26, the Gyrobifastigium, is unique in that if copies of itself are properly stacked together they will leave no gaps, thus making it the only space filling Johnson Solid.
posted by Tube on Oct 3, 2007 - 28 comments

What if Euclid had been Japanese? There are traditionally stated and proved theorems about origami. And MetaFilter has previously explored modular origami (as well as the boring old artistic kind), which has a geometric foundation. However, origami itself is a powerful mathematical framework that allows one to, for instance, solve the famously insoluable problem of trisecting an angle. More generally: Traditional geometry solves quadratic equations, origami solves cubic ones. (Many more mathematical items about and using origami can be found in the excellent mathematics teachers' book: Project Origami: Activities for Exploring Mathematics, most of which are unfortunately not findable online).
posted by DU on Feb 13, 2007 - 9 comments

Jim Loy's Mathematics Page is (among other things) a collection of interesting theorems (like Napoleon's Triangle theorem), thoughtful discussions of both simple and complex math, and geometric constructions (my personal favorite); the latter of which contains surprisingly-complex discussions on the trisection of angles, or the drawing of regular pentagons.

Similarly enthralling are the pages on Billiards (and the physics of), Astronomy (and the savants of), and Physics (and the Phlogiston Theory of), all of which are rife with illustrations and diagrams. See the homepage for much more.

If you like your geometric constructions big, try Zef Damen's Crop Circle Reconstructions.
posted by odinsdream on Sep 14, 2005 - 8 comments

The House With Too Many Perpundiculars
posted by DevilsAdvocate on Jul 13, 2004 - 8 comments

Astonishing geometric art using only folded paper plates, from Bradford Hansen-Smith at wholemovement. View the gallery of fantastic polyhedral creations, and learn how to do it yourself. (For more fun with paper plates, see also Paper Plate Education: Serving the Universe on a Paper Plate.)
posted by taz on Oct 27, 2003 - 7 comments

'The Poincare Conjecture' Solved? "Dr Grigori Perelman, of the Steklov Institute of Mathematics of the Russian Academy of Sciences, St Petersburg, claims to have proved the Poincare Conjecture, one of the most famous problems in mathematics. The Poincare Conjecture, an idea about three-dimensional objects, has haunted mathematicians for nearly a century. If it has been solved, the consequences will reverberate throughout geometry and physics."

Also of note is that Perelman's solution is only a benign side effect of his efforts toward defining all three-dimensional surfaces mathematically, which if successful would allow humanity to "produce a catalogue of all possible three-dimensional shapes in the Universe, meaning that [mankind] could ultimately describe the actual shape of the cosmos itself."
posted by eyebeam on May 8, 2003 - 13 comments