Message #3272

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] How big is a 120-cell?
Date: Sat, 12 Dec 2015 15:55:25 -0600

That’s an interesting question.

If you think of the 120-cell as living in S^3 (on the 3-sphere), then yes -
the longest length between portions of it are at antipodes. With a
normalized 3-sphere radius of 1, this distance would be 2 as a straight
line distance, or pi as a geodesic distance in the 3-sphere. Antipodal
points that are centers of cells or any other antipodal points would all be
the same distance from each other.

If you think of the 120-cell as a polytope living in R^4, then it’s a
little more complicated. Think about the dodecahedron. It has an
"inradius" through antipodal faces, a "midradius" through antipodal edges,
and a "circumradius" through antipodal vertices. The last are the furthest
from each other. The 120-cell would be have similarly, and so I gather you
are asking: What is the circumradius of the 120-cell, with a scaling so
that the edge length is 1? Note that the longest portion is *not* the
center of a cell to the center of the opposite cell.

Sounds like an interesting problem to calculate, but I was lazy and looked
it up.

http://mathworld.wolfram.com/120-Cell.html

That page says the vertices of a 120-cell with circumradius 2*sqrt(2) have
edge length 3 - sqrt(5). Therefore, the circumradius of a 120-cell with
edge length 1 have circumradius 2*sqrt(2)/(3-sqrt(5)), or approximately 3.7
<http://www.wolframalpha.com/input/?i=2*sqrt%282%29%2F%283-sqrt%285%29%29>.
The distance between antipodal vertices will be twice that amount.

Roice

On Sat, Dec 12, 2015 at 3:25 PM, llamaonacid@gmail.com [4D_Cubing] <
4D_Cubing@yahoogroups.com> wrote:

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> How big is the longest portion of a 120-cell using the measurement from
> the image below? Would the longest length be the center of a cell to the
> center of the opposite cell?
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