A new crop circle formation in Wiltshire depicts the first 10 digits of pi.
posted on Jun 20, 2008 - View this thread

Polyhedral Maps is a website that explores unconventional methods of mapping the surface of the earth. The most famous of these unusual maps was Buckminster Fuller’s Dymaxion map, which used the net of an icosahedron. Da Vinci had experimented with this technique in his “Octant” map of 1514, which used Reuleaux triangles as map elements. This process is now being used by photographers and artists in manipulating panoramic images. A good example is Tom Lechner’s The Wild Highways of the Elongated Pentagonal Orthobicupola.
posted on Jun 1, 2008 - View this thread

It's art; it's geometry; it's green tech. It's the oloid.
posted on Mar 18, 2008 - View this thread

The connection between mathematics and music is often touted in awed, mysterious tones, but it is grounded in hard-headed science. For example, mathematical principles underlie the organization of Western music into 12-note scales. And even a beginning piano student encounters geometry in the "circle of fifths" when learning the fundamentals of music theory. ...according to Dmitri Tymoczko, a composer and music theorist at Princeton University, these well-known connections reveal only a few threads of the hefty rope that binds music and math.

The Geometry of Music
See also The Geometry of Musical Chords - Dmitri Tymoczko, Science 7 July 2006: Abstract
See also Dmitri Tymoczko, Composer and Music Theoristvia
posted on Mar 16, 2008 - View this thread

Vladimir Bulatov enjoys making polyhedra and abstract geometric sculptures.
posted on Feb 9, 2008 - View this thread

A Visual Dictionary of Famous Plane Curves is an outstanding resource for curves found in nature, man-made objects, and mathematics. Other websites that list exotically named curves also animate how they are created. One of the most unusually named curves, the “Witch of Agnesi”, has an unusual etymology. A number of these curves will be familiar to anyone who has used a Spirograph. Previously.
posted on Jan 19, 2008 - View this thread

The Gömböc is the first known convex, homogeneous shape having just one stable and one unstable point (i.e. altogether two points) of equilibrium. A little like some turtles' shells (or weebles), it's self-righting, but for purely geometric reasons.
posted on Dec 9, 2007 - View this thread

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 on Dec 1, 2007 - View this thread

The Great Pizza Orientation Test.
posted on Oct 24, 2007 - View this thread

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 on Oct 3, 2007 - View this thread

Möbius Transformations Revealed [yöutube alert] See also: Stereographic Projection Demo.
posted on Jun 26, 2007 - View this thread

Here are some beautifully rendered views of polytopes, and a few more. The rendering program, Jenn 3D, is free and downloadable, (OS X, Linux, Win) and includes some really dazzling fly-about and camera effects as well as tons of high-dimensional models to explore. There's also a mind-boggling possibility of playing Go on boards in projective space. Via the Math Paint blog, which leads to other interesting places...
posted on Jun 2, 2007 - View this thread

You have spacial skills. Apply them in Building Houses 2, on mathsnet.net. Or freestyle in Building Houses 1. Or at night! Oh and also there's like a hundred more puzzles over there too. Some java required.
posted on Apr 12, 2007 - View this thread

Did the roof of the Pantheon influence Copernicus? Are the planets of the solar system aligned in accordance with a nearly-forgotten hypothesis known (unfairly) as Bode's Law? A fascinating wide-ranging discussion on BLDGBLOG with Walter Murch, the visionary editor and sound designer for such films as The Conversation, Apocalypse Now, The English Patient, THX1138, and many others. [Murch's film work has previously been discussed here and here.]
posted on Apr 7, 2007 - View this thread

STEAM. Australian artist Donna Marcus uses kitchenware to make geodesic spheres to be placed in conspicuous locations.
posted on Mar 1, 2007 - View this thread

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 on Feb 13, 2007 - View this thread

The Institute for Figuring presents the Crocheted Hyperbolic Coral Reef Project and Hyperbolic Crocheted Cacti and Kelp (more at this flickr gallery). If you secretly spend your evenings crocheting mathematical models, help build the coral reef or send a photo of your other creations to The People's Hyperbolic Gallery. (via Wonderland)
posted on Sep 15, 2006 - View this thread

Order from chaos! Fill a cylindrical bucket with water and make it so the bottom can spin. At certain speeds, stable regular polygonal shapes will spontaneously form at the turbulent surface of the water. See the video. [2.6MB avi] [via last week's PRL]
posted on May 10, 2006 - View this thread

Under Foot and Between the Boards in the Laurential Library "Within the Laurentian Library, the enigmatic masterwork of Michelangelo, there exists a complex geometric pavement that is hidden from view, little known about and shrouded with mystery...Why had an immensely complicated pavement been constructed, only to be covered over?"
posted on Oct 23, 2005 - View this thread

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 on Sep 14, 2005 - View this thread

The Spidron is an interesting geometric construction that seems to lend itself to folding, dissection, and space-filling in two and three dimensions.
posted on Jul 17, 2005 - View this thread

The Geometry Center at the University of Minnesota, while now closed, maintains an awesome website with tons of math resources. I like sphere eversion, i.e. turning a sphere inside out. Link is to script of video, which explains things pretty well. Here is a clip [QT]. Also good: notes from a class on geometry and the imagination that John Conway and some friends gave awhile back. Old but good.
posted on Mar 1, 2005 - View this thread

Euclid in Colour. 'An unusual and attractive edition of Euclid was published in 1847 in England, edited by an otherwise unknown mathematician named Oliver Byrne. It covers the first 6 books of Euclid, which range through most of elementary plane geometry and the theory of proportions. What distinguishes Byrne's edition is that he attempts to present Euclid's proofs in terms of pictures, using as little text - and in particular as few labels - as possible. What makes the book especially striking is his use of colour ... '
posted on Aug 11, 2004 - View this thread

The House With Too Many Perpundiculars
posted on Jul 13, 2004 - View this thread

This giant Ukrainian Easter Egg (pysanka) was built in 1975 in Vegreville, Canada by (then) Univ. Utah Computer Science Professor Ronald Resch. Interesting egg factoids can be found here--including that it swivels like a weather vane. Vegreville has an annual festival. More images of egg here. The Vegreville Pysanka was the first physical structure completely designed with computer-aided geometric modeling software. There is a good description here of the complex geometry involved. It's based on a technique (PDF) he developed and patented for folding a flat material (i.e. sheet metal) into flexible surfaces. Ronald Resch has had an interesting career.
posted on Apr 8, 2004 - View this thread

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 on Oct 27, 2003 - View this thread

'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 on May 8, 2003 - View this thread

The Geometry Center: Center for the Computation and Visualization of Geometric Structures [more]
posted on Feb 2, 2003 - View this thread

After getting the inside story (ha?) on the inventor of everyone's favorite non-orientable surface, the Klein Bottle; and perhaps playing a few games inside of one, you can check out a few 3-dimensional immersions of klein bottles: in Lego, knitted fabric, paper, or glass.
posted on Oct 30, 2000 - View this thread