User:Dc.samizdat/Isoclinic rotations of the regular convex 4-polytopes

This is a draft of a work in progress. Missing sections are noted: ^{[tbd 1]}

Sequence of 6 regular convex 4-polytopes
Symmetry group	A4	B4		F4	H4
Name	5-cell Hyper-tetrahedron 5-point	16-cell Hyper-octahedron 8-point	8-cell Hyper-cube 16-point	24-cell Hyper-cuboctahedron 24-point	600-cell Hyper-icosahedron 120-point	120-cell Hyper-dodecahedron 600-point
Schläfli symbol	{3, 3, 3}	{3, 3, 4}	{4, 3, 3}	{3, 4, 3}	{3, 3, 5}	{5, 3, 3}
Coxeter mirrors
Mirror dihedrals	𝝅/3 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2	𝝅/3 𝝅/3 𝝅/4 𝝅/2 𝝅/2 𝝅/2	𝝅/4 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2	𝝅/3 𝝅/4 𝝅/3 𝝅/2 𝝅/2 𝝅/2	𝝅/3 𝝅/3 𝝅/5 𝝅/2 𝝅/2 𝝅/2	𝝅/5 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2
Graph
Vertices[a]	5 tetrahedral	8 octahedral	16 tetrahedral	24 cubical	120 icosahedral	600 tetrahedral
Edges	10 triangular	24 square	32 triangular	96 triangular	720 pentagonal	1200 triangular
Faces	10 triangles	32 triangles	24 squares	96 triangles	1200 triangles	720 pentagons
Cells	5 {3, 3}	16 {3, 3}	8 {4, 3}	24 {3, 4}	600 {3, 3}	120 {5, 3}
Tori	5 {3, 3}	8 {3, 3} x 2	4 {4, 3} x 2	6 {3, 4} x 4	30 {3, 3} x 20	10 {5, 3} x 12
Inscribed	120 in 120-cell	675 in 120-cell	2 16-cells	3 8-cells	25 24-cells	10 600-cells
Great polygons		2 squares x 3[b]	4 rectangles x 4	4 hexagons x 4	12 decagons x 6	100 irregular hexagons x 4
Petrie polygons	1 pentagon x 2	1 octagon x 3	2 octagons x 4	2 dodecagons x 4	4 30-gons x 6	20 30-gons x 4
Long radius	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$
Edge length[c]	${\displaystyle {\sqrt {\tfrac {5}{2}}}\approx 1.581}$	${\displaystyle {\sqrt {2}}\approx 1.414}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle {\tfrac {1}{\phi }}\approx 0.618}$	${\displaystyle {\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270}$
Short radius	${\displaystyle {\tfrac {1}{4}}}$	${\displaystyle {\tfrac {1}{2}}}$	${\displaystyle {\tfrac {1}{2}}}$	${\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}$	${\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}$	${\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}$
Area	${\displaystyle 10\left({\tfrac {5{\sqrt {3}}}{8}}\right)\approx 10.825}$	${\displaystyle 32\left({\sqrt {\tfrac {3}{4}}}\right)\approx 27.713}$	${\displaystyle 24}$	${\displaystyle 96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569}$	${\displaystyle 1200\left({\tfrac {\sqrt {3}}{4\phi ^{2}}}\right)\approx 198.48}$	${\displaystyle 720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{8\phi ^{4}}}\right)\approx 90.366}$
Volume	${\displaystyle 5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329}$	${\displaystyle 16\left({\tfrac {1}{3}}\right)\approx 5.333}$	${\displaystyle 8}$	${\displaystyle 24\left({\tfrac {\sqrt {2}}{3}}\right)\approx 11.314}$	${\displaystyle 600\left({\tfrac {\sqrt {2}}{12\phi ^{3}}}\right)\approx 16.693}$	${\displaystyle 120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}{\sqrt {8}}}}\right)\approx 18.118}$
4-Content	${\displaystyle {\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146}$	${\displaystyle {\tfrac {2}{3}}\approx 0.667}$	${\displaystyle 1}$	${\displaystyle 2}$	${\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863}$	${\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193}$

There is no natural way to arrange the van Ittersum row and column numbers so that they label the array cells they index with the rotational angles of the 24-cells. But we are not limited to labeling the cells with their own row and column labels. Here is a natural way to label the cells of the 5 x 5 array with those same 25 symbol pairs, but rearranged to show that each row and each column is an isoclinic rotation through five 24-cells:

We label the rows and columns with 10 distinct symbols as (Denney et al. 2020) do, so that each of Schoute’s partitions has a unique name, but we use the symbols 0_R through 4_R for the rows, and 0_L through 4_L for the columns. These symbols also number the vertices of a decagon: 0_R through 4_R clockwise on every second vertex (a pentagon), and 0_L through 4_L counter-clockwise on the antipodal pentagon.

Each cell label (𝝃_L, 𝝃_R) identifies a 600-cell vertex from the antipodal vertex pair or ray that is the intersection of the two non-orthogonal central decagons.^[d]

Each row and each column is an isoclinic (equal-angled double) pentagonal rotation, in which successive cells are rotated by one L unit and one R unit. Each 𝝃_L and 𝝃_R are the same angle, relative to any vertex in the same row or column. In each row 𝝃_L = i + j and in each column 𝝃_R = j - i, modulo 5.

(0_L, 0_R) is the north pole of the 600-cell. The cell labels (𝝃_L, 𝝃_R)) are the north poles of the 25 24-cells. Each 24-cell has a logical address (i, j) in the 5 x 5 array, its (column, row) numbers. It also has a physical address (𝝃_L, 𝝃_R) in the 600-cell, expressing the geodesic and angular distance of its north pole from vertex (0_L, 0_R).^[f] The set of (i, j) logical addresses and the set of (𝝃_L, 𝝃_R) physical addresses are the same set.

I found this labeling by noticing the (𝝃_L, 𝝃_R) pairs in the left column, the only ones where 𝝃_{L =} 𝝃_R so that row is trivially isoclinic and its 5 24-cells trivially lie at isoclinic inclinations from the basis orientation. There are 5 other 24-cells that lie (less trivially) at isoclinic inclinations from the basis orientation: the (𝝃_L, 𝝃_R) pairs in the top row. The rotation along the main diagonal is obviously a simple rotation (and must be among non-disjoint 24-cells as (Denney et al. 2020) have shown), as it has a fixed 𝝃_R = 0_R; so we call it the (R)ight central diagonal. The (L)eft central diagonal similarly has a fixed 𝝃_L = 4_L. There are four more right diagonal pentagons and four more left diagonal pentagons in the table, with the other fixed values of 𝝃_R and 𝝃_L.

Altogether the table directly represents 20 pentagonal rotations: 10 isoclinic (among disjoint 24-cells) and 10 simple (among non-disjoint 24-cells). No two of these 20 pentagons are orthogonal (they share one vertex). Four of these pentagons intersect at each vertex, e.g. at the north pole: the 0_L (left) column, the 0_R (top) row, the 0_R (main or right central) diagonal, and the 0_L (left central) diagonal. But six pentagons intersect at each vertex of the 600-cell (their 12 edges meeting at the center of an icosahedral vertex figure). Of the six pentagons that intersect at each vertex mentioned in the table, there are two with each of the three kinds of rotation: isoclinic double, simple, and unequal-angled double. Two of the six are isoclinic rotations within the table (a row and a column); two are simple rotations within the table (a right diagonal and a left diagonal); and two are unequal-angled double rotations that visit vertices outside the table. But they are all alike in being geodesic rotations along the edges of one of the 144 central pentagons. By symmetry there is nothing inherently different about the pentagons (all 144 are exactly alike). But isoclinic, simple and double rotations are very different operations with different properties and outcomes (as we see in this case, where the isoclinic rotations reach disjoint 24-cells, and the simple rotations reach 24-cells which share one pentagon). What we have in the 600-cell is 72 central planes which can be an invariant plane of rotation in either an isoclinic, simple, or double rotation between 24-cells: in three entirely different operations (taking each 24-cell to a different place), where each pentagon is the invariant plane in all of them. Their common invariant plane rotates exactly the same way in all three rotations, while everything outside the plane moves differently.

The table expresses the outcome of all three kinds of rotation, in terms of where the north pole 24-cell winds up after the rotation. For the double rotations, pick two rotation angles (a column label i and a row label j), and read the pair of rotation angles (𝝃_L, 𝝃_R) in the cell where they intersect. That is the new physical location (inclination) of the former north pole 24-cell. The isoclinic rotations, of course, are just double rotations where you picked two angles that are the same. The simple rotations are even simpler: you don't have to look at the table at all. The destination of the simple rotation is simply the (i, j) you picked.