It has been two weeks since SUUSI and it's a long wait until 2006. One of the things I did at this camp or conference was that I gave a workshop on how to weave polyhedra out of various construction paper strips. This led me to think about the polyhedra and reminded me of a neat site for them. This is George Hart's site, at www.georgehart.com in which he has VRML files of many polyhedra models. He even has a page on which you can design and construct your own VRML polyhedra, at
http://www.georgehart.com/virtual-polyhedra/conway_notation.html
This introduces a code by John H. Conway, author of "Numbers and Games" and of numerous neat mathematical things. Conway introduces about 12 operators or so, and 8 objects on which to operate. The operators include a for "ambo", which means take the in-between model (e.g., the cuboctahedron for the cube/octahedron pair) , t for "truncate", j for "join" (make rhombuses out of all the edges) and so forth. I found that there was only one basic object letter (except for prisms, pyramids, and antiprisms) and 6 basic operations. The letter is T for Tetrahedron. The operations are a (ambo), t (truncate), s (snub), d (dual), r (mirror-reverse), and p (propeller - not Conway's invention). I am going to consider only the first 4. I note that Cube, for instance, is daT, and Icosahedron (or I) is actually sT, the snub tetrahedron. Here are some of the other examples:
aaT Cuboctahedron (or aC)
asT Icosadodecahedron (or aD)
taT Truncated octahedron (or tO)
daaaT Trapezoidal Icosatetrahedron (or 24-hedron) (or deC)
ssaT The snub of the snub cube (Hart points out that there are four versions of this)
and so forth. These symbols form a logic of sorts with syntax rules; for example, dd = identity, ad = a, and sd = s. Not all of these can be formed with regular polygons; for example, datsT is Hart's favorite solid, the rhombic enneacontahedron (90-hedron). atsT is its dual, and this turns out to have pentagons, triangles and hexagons. However, these polygons cannot be regular, as atsT (the truncatedicosahedronpentakisdodecahedron or ambo-truncated icosahedron) is not one of the 13 regular Archimedean polyhedra. Presumably Hart has a formula for producing an image of these polyhedra and figuring out where the points on them go, and this might not be easy; for example, the pentakis dodecahedron is made up of somewhat irregular triangles. Where are the midpoints of the faces?
Also, the site does not work right. It is supposed to pop up a VRML window showing the polyhedron defined by the entered Conway notation. Instead, when you enter the notation in the blank, and click "generate", you get a window with VRML code in it. You can copy this by dragscrolling through the document, and putting it in word and making sure each comment is on a line of its own, then saving it as a text document of type .vrml. But one should not have to go through this. George Hart should fix it so you get the image immediately.
But it is one of the best sites on the Internet, in my opinion, and you can enter any of the codes in this blog into that site and after copying to Word and fixing the comments, you can get a VRML page showing the polyhedron whose code you selected.
