The Geometry Page

★ wanayoo — archive 1999 http://www.slip.net/~danielgr/geometry/geometry.htm

Geometry Page

Symmetry has always been attractive to mathematicians, and the most symmetric of all figures are the regular polyhedra, or Platonic solids. A regular polyhedron is defined as a finite polyhedron composed of a single type of regular polygon such that each element (vertex, edge and face) is surrounded identically. In three dimensions there are exactly five such polyhedra which don't intersect themselves, and four more that do. There are many other interesting such figures, many of which are defined by relaxing one or more of the conditions defining regular polyhedra. For instance, the figure above is composed of only regular triangular faces, but it has three types of edges and three types of vertices. (The three types of vertices are surrounded by 4, 6 and 10 triangles.) Click on the following link for more information on deltahedra.

I have done substantial work exploring an interesting and often overlooked class of polyhedra which satisfy most or all the criteria defining regular polyhedra except that they are not finite. In other words, it would take an infinite number of polygons to complete such a figure which would then fill all of space with a latticework. Of course an infinite model cannot be completely constructed, but large enough sections can be built to show their geometry and prove their existence. The image above (courtesy of Steve Dutch) shows a portion of one of the simplest such models. Many more elaborate and beautiful figures exist. Click the following link for a fuller description of infinite polyhedra along with images and interactive 3D models of many of them.

Another very interesting and overlooked area is that of flexible polyhedra. If polyhedra are built out of perfectly thin, perfectly stiff faces but which are free to hinge where faces meet, then almost all polyhedra are rigid. The image above is of a rare example of a polyhedron that actually can flex. Click the following link for a description of flexible polyhedra plus ways of interacting with 3D computer models of them.

A new section on hyperbolic tessellations by Don Hatch has just been added. The image above shows 2D space tessellated by regular seven-sided polygons (the white lines). That can't be done on a flat 2D space but it's no problem on the appropriately curved space. The reason that the polygons above don't appear perfectly symmetric and get infinitely small at the edges is because that curved space has been stretched to fit a flat screen. Follow the link above for more information and lots of images of other beautiful tessellations.

Finaly, here are a few links to great geometry sites:
The Geometry Junkyard - a huge site and great resource
George Olshevsky's Uniform Polytopes in Four Dimensions
Vladimir Bulitov's Polyhedra Collection
George Hart's Virtual Polyhedra
Jim McNeill's Polyhedra
Rona Gurkewitz' Modular Origami Polyhedra Systems
Steven Dutch's Symmetry, Crystals and Polyhedra
The personal home page for the famous polyhedra model builder Fr. Magnus Wenninger
A great Uniform Polyhedra site by Dr. R. Mäder
Rolf Asmund's Polyhedra site

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