From: "Clive Tooth"
Subject: Re: 120-cell mentioned on pathetic 4-d page
Date: Mon, 13 Sep 1999 09:45:25 +0100
Newsgroups: sci.math
Keywords: visualizing the 16-cell (solid in R^4)
David Goodenough wrote in message ...
>This falls into the "I ought to know better than to post in this
>thread" class, but here goes.
>
>I have often thought of trying to use time as the fourth dimension in
>thought experiments. This would be the analogy of viewing three
>dimensional objects as a series of 2 d slices as a plane moves through
>the solid over a period of time.
>
>That way, a tetrahedron could be viewed as a point that grows to a
>triangle, and a cube would be a square that exists for a length of
>time.
>
>Using the same model in three dimensions, the pentatope becomes a
>point that expands to a tetrahedron, and the tesseract becomes a cube
>that exists for a length of time.
>
>Problem is, I lose it *COMPLETELY* trying to visualise the other 4
>guys. :-) I suspect that the problem is the equivalent of the fact
>that in the "2 d plane model" the structure of a dodecahedron or an
>icosahedron is close to impossible to see (think about it), and an
>octahedron looks more square than triangular.
Visualizing the 16-cell (the 4-dimensional cross polytope) being pushed
vertex-first through 3-space is not too bad. Recall that it can be
constructed by taking an octahedron and joining its vertices to two points
on a line through the center of the octahedron and perpendicular to the
3-space containing the octahedron. So as the 16-cell is pushed through
3-space we see a growing, then shrinking, octahedron.
Pushing the 16-cell tetrahedral-face-first through 3-space is slightly more
difficult to visualize. It starts as a tetrahedron... push it in a little
way: all six edges get shaved off, giving an object with 12 vertices...
--
Clive Tooth
http://www.pisquaredoversix.force9.co.uk/
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