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.
Polyhedra composed entirely of equlateral triangles.
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. Notice that disallowing self-intersections is not part of the definition of regular polyhedra, but people generally find those less appealing. The self intersecting faces are due to using star polygons which themselves contain self-intersecting edges. My point is that there are no mathemetical reasons for excluding self intersections, and that the rules that restrict and therefore define types of polyhedra are purely aesthetic.
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 Archimedean solids are the result of allowing two types of faces and edges. And the deltahedron shown in the previous section is composed of only regular triangular faces, but it has three types of edges and three types of vertices.
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 lovely example. Many more elaborate and beautiful figures exist. Click the image 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.
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.
Win $100 by being the first to find an example of this type of polyhedron or prove that none exist. Click image for details.
Ed Baumann has generated unit edge length versions of near-miss Johnson solids. He sacrafices polygon regularity in favor of equal edge lengths and planar polygons. These near misses were one of the things that fell out of the original search for a 554.
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