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

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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 do, so that each of Schoute’s partitions has a unique name, but we use the symbols 0R through 4R for the rows, and 0L through 4L for the columns. These symbols also number the vertices of a decagon: 0R through 4R clockwise on every second vertex (a pentagon), and 0L through 4L counter-clockwise on the antipodal pentagon.

Each cell label (𝝃L, 𝝃R) identifies a 600-cell vertex from the antipodal vertex pair or ray that is the intersection of the two non-orthogonal central decagons.

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 𝝃L and 𝝃R are the same angle, relative to any vertex in the same row or column. In each row 𝝃L = i + j and in each column 𝝃R = j - i, modulo 5.

(0L, 0R) is the north pole of the 600-cell. The cell labels (𝝃L, 𝝃R)) 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 (𝝃L, 𝝃R) in the 600-cell, expressing the geodesic and angular distance of its north pole from vertex (0L, 0R). The set of (i, j) logical addresses and the set of (𝝃L, 𝝃R) physical addresses are the same set.

I found this labeling by noticing the (𝝃L, 𝝃R) pairs in the left column, the only ones where 𝝃undefined 𝝃R 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 (𝝃L, 𝝃R) 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 have shown), as it has a fixed 𝝃R = 0R; so we call it the (R)ight central diagonal. The (L)eft central diagonal similarly has a fixed 𝝃L = 4L. There are four more right diagonal pentagons and four more left diagonal pentagons in the table, with the other fixed values of 𝝃R and 𝝃L.

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 0L (left) column, the 0R (top) row, the 0R (main or right central) diagonal, and the 0L (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 (𝝃L, 𝝃R) 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.