Message #3274

From: Roman Tershak <t_roma@yahoo.com>
Subject: Re: [MC4D] How big is a 120-cell?
Date: Sun, 13 Dec 2015 11:49:09 +0000

Hi!
To give you a visual perception of this figure I created two gif animations (attached) from a program that I wrote a couple of years ago. This program has this 120-cell monster called ‘hecatonicosachoron’. A real name for a real monster :-). Although, mostly it is called just 120-cell. The figure is projected plainly on 3d and then uses cross-eye ‘stereoscopic’ view. The 4th dimension is presented by color, the more reddish the color is the more negative 4th coordinate it represents, blue color is the oposite. Since there are a lot of lines all this projection is rather messy, so there is the second animation ‘hecatonicosachoron-half.gif’ that contains half of 120 cell, negative part is effectively cut off.
Hope you will enjoy!Roman

On Saturday, December 12, 2015 11:55 PM, "Roice Nelson roice3@gmail.com [4D_Cubing]" <4D_Cubing@yahoogroups.com> wrote:

  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.  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:

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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