Message #1354

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: Other 4D puzzles
Date: Wed, 26 Jan 2011 11:20:55 -0600

<https://mail.google.com/>Don’s latest engine does have all the polytopes
in it, 24-cell, 16-cell, 600-cell, and lots more, even the grand antiprism!
If you’ve never seen the last, I highly recommend checking out the wikipedia
page <http://en.wikipedia.org/wiki/Grand_antiprism> on it. It’s sort of a
strange hybrid between the 120-cell (with two rings of pentagonal antiprisms
playing the role of rings of dodecahedra) and 600-cell (it has 300
tetrahedra). However, even though his engine supports the shapes and
generally positioned slicing, he hasn’t actually sliced up any of the
polytopes with non-simplex vertex figures yet. I think there is probably a
good deal of hidden work there, and that it is a "90% done, 90% to go" kind
of situation. Generic implementation of twisting and slicing for the
stranger polytopes surely can’t be trivial.

Thanks Nan for the FTO links! I ordered one from the same site Melinda did,
and am already excited for it to arrive :D

Melinda and Matt, I agree the 120-cell is the most natural analogue to the
dodecahedron (agree there is no other 4D polytope the dodecahedron could
better be associated with, nor any other 3D shape that would deserve to be
associated with the 120-cell). The Magic120Cell page even says "4D
Megaminx." I guess my hesitation for the term had more to do with the
dramatic change in properties from 3D to 4D, contrasted with the cube or
simplex situations where properties change in more straightforward and
sequential ways. I hadn’t really put my finger on my discomfort, but now
realize part of it was a bias against the idea of an analogue being created
by repeating faces rather than stretching into the next dimension. I think
both your reactions and a little further reflection have removed that small
discomfort though, so I take it back :)

As an aside, there is a short paper titled "The Story of the
120-Cell<http://www.ams.org/notices/200101/fea-stillwell.pdf>"
which describes what may be the most insane of the connections between the
dodecahedron and the 120-cell. If you interpret the 4D vector locations of
the cell centers of the 120-cell as quaternions, the 3D rotations those
quaternions describe are none other than the symmetries of the dodecahedron
(and icosahedron). Likewise, the 24-cell encodes the symmetries of the
tetrahedron. This all really shocked me when I first read it, and was
almost enough to make me flirt with believing in intelligent design :)
Cheers,
Roice