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/ End of document