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
Geometric construction kits
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