**New:**
Tyler
is a simple applet that lets you explore planar tilings using regular
polygons.
Please visit the Tyler Art Gallery to see
the incredible variety of beautiful forms that can be easily created.
With
the Tyler applet you can create polygons of various sizes and attach
them
to edges of other polygons. Click the image above or the following link
to try the Tyler applet yourself. The
image
is from is from Kepler's Harmonice Mundi volume 2 and is easily
reconstructed
using Tyler. For a mathematical description of planar tilings see Jim
McNeill's excellent description.

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.

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.

Don Hatch has done a beautiful treatment on hyperbolic tessellations. 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 only 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 of hyperbolic spaces.

**Other great geometry sites**

- The Geometry Junkyard - a huge site and great resource
- George Olshevsky's Uniform Polytopes in Four Dimensions
- Vladimir Bulatov'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
- Jim McNeill's Toroids.
- Spiral Tilings by Paul Gailiunas
- Impossible Illusions

**Geometric construction kits**

- Polydron home page. Excellent for building plastic geometric models.
- Zometool edge construction sets. Favorite of artist George Hart.
- Polymorph home page. Similar to Polydron but up to six faces may share an edge.
- Astro-logix edge construction sets