Wikipedia:Reference desk/Archives/Mathematics/2020 February 26

= February 26 =

Compound of 3 16-cells?
In my head, a compound of 3 16-cells would have the same verticies as a 24-cell, but I dont see anything about that in Polytope compound. Should it be there?Naraht (talk) 14:14, 26 February 2020 (UTC)
 * This? It's there under "4-polytope compounds". Double sharp (talk) 14:24, 26 February 2020 (UTC)
 * Ah. Misunderstood that line of the 4-polytope compounds as being a compound of both 3 16-cells *and* 3-hypercubes.Naraht (talk) 16:40, 26 February 2020 (UTC)

Hypervolume comparison
If the hypervolume inside the compound of 3 16-cells (inside being inside any of them, not just inside the core) is 1, what is the volume of the 24-cell that makes up the convex hull and what is the hypervolume of the core.Naraht (talk) 16:43, 26 February 2020 (UTC)
 * According to my calculations, the intersection of the 3 16-cells is another 24-cell smaller than the original 24-cell by a factor of 1/√2. The hypervolume would then be 1/4 that of the original. --RDBury (talk) 19:31, 26 February 2020 (UTC)
 * Sorry, I misread the question and thought you were looking for the ratio of the volumes of the core to the original 24-cell. But rereading it, it looks like you want the volume of the union of the 3 16-cells. It's not mentioned in the article for some reason, but the volume of a 24-cell with side s is 2s4. (Again according to my calculations; I couldn't find a reference but the computation, while messy, isn't difficult.) So if the original 24-cell has side 1 then its volume is 2, and the core has side 1/√2 and volume 1/2. Each 16-cell has side √2 and the volume of a 16-cell with side s is s4/6, in this case that makes 2/3. The volume of the union is then 3⋅(2/3) - 2⋅(the volume of the intersection) = 2 - 1 = 1. Note that the intersection of any two of the 16-cells is equal to the intersection of all three. So the ratios of three volumes are 2:1:1/2. It really seems like the hypervolumes of the 16 and 24-cells (and the other regular polychora) should be in the articles, but they seem to be strong on combinatorics and short on metrics. --RDBury (talk) 21:47, 26 February 2020 (UTC)
 * How do you know that the entire intersection of two of the 16 cells is also inside the third?Naraht (talk) 03:27, 2 March 2020 (UTC)
 * Also, hypervolume isn't a parameter at Template:Infobox polychoron.Naraht (talk) 03:42, 2 March 2020 (UTC)
 * On the first point, the 24-cell's 24 cells can be grouped into 3 groups of 8, with the hyperplanes in each group bounding a hypercube. This is easier to describe using coordinates, so take the vertices of a 24-cell to be all permutations of (±1, ±1, 0, 0). The planes defining the faces are then ±xi=1 and ±x1±x2±x3±x4=2. The first set forms a group by itself and the second set splits into two groups according to whether there are an even or odd number of minus signs. The first set of 8 bounds the hypercube with vertices (±1, ±1, ±1, ±1) and the other two sets of 8 form two other hypercubes; all three cubes together form a compound whose vertices are a larger 24-cell. The last two sets of 8 together bound a 16-cell with vertices permutations of (±2, 0, 0, 0), and by the symmetry of the 24-cell, any two groups of 8 bound a 16-cell. There are three possibilities for two groups out of three which gives you the three 16-cells in a compound whose vertices form the larger 24-cell. Take 2 of the 16-cells and suppose one is bounded by the planes in groups A and B, and the second is bounded by the planes in group A and C. Then the intersection is bounded by the planes in all three groups, so it's the 24-cell in the center. Note that this is possible because eight cells of one 16-cell are defined by the same planes which define 8 cells of the other 16-cell. (The cells themselves aren't the same but they occupy the same 3-dimensional space. Each cell is a tetrahedron and together they form a complex of 2 tetrahedra in a cube. Their intersection is an octahedron which is one of the cells of the 24-cell.)
 * On the second point, it shouldn't be that hard to update the template; it's finding a citable source that's the sticking point for me. Regular Polytopes (book) might be a place to start. --RDBury (talk) 11:10, 2 March 2020 (UTC)
 * Can't find my copy. *whimper* :(18:07, 4 March 2020 (UTC)