User:Dc.samizdat/The progression of the convex polytopes

Author
David Brooks Christie

Abstract
The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more content in the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering. Irregular uniform convex 4-polytopes may be included in this series, as intermediate forms between the regular elements. The progression of unit-radius convex 4-polytopes frames and illuminates the transitional operations by which each polytope in the series is constructed from its predecessor (or inversely, from its successor).

Introduction
The uniform convex n-polytopes (for any n) are ordered the same way by size (their n-content in the same radius) and by relative complexity (number of vertices). This suggests an alternative numerical naming scheme for uniform convex polytopes in which, for example, the tesseract (or 8-cell) is the 16-point 4-polytope: third in the ascending sequence of the six regular 4-polytopes that runs from 5-point 4-polytope (5-cell) to 600-point 4-polytope (120-cell). It also suggests that we study the sequence by comparing polytopes of unit radius (rather than unit edge length), and the operations by which each may be constructed from an instance of its predecessor (or its successor).

Inscribings
The most obvious relationship among the convex regular 4-polytopes of the same radius is that they nest, one within the other. Two 8-point 16-cells are inscribed in the 16-point tesseract, occupying opposite vertices. Three 16-point tesseracts are inscribed in the 24-point 24-cell, with each vertex shared by two tesseracts. One might suspect that 5 24-point 24-cells could be inscribed in the 120-point 600-cell, occupying disjoint sets of vertices, but in fact they are not inscribed that way; instead 25 24-cells are inscribed in the 120-point 600-cell, with each vertex shared by five 24-cells. Ten 120-point 600-cells are inscribed in the 600-point 120-cell, with each vertex shared by two 600-cells. Thus each convex regular 4-polytope is inscribed in all its successors, with the curious exception of the smallest one: the 5-point 5-cell cannot be inscribed in the 8-point 16-cell or any of its other successors except the largest one: 120 5-cells are inscribed in the 600-point 120-cell.

Rotations
The inscribings show how the entire series of polytopes can be generated by rotations in equatorial planes.

In 3 dimensions we can construct an octahedron centered at the origin of the cartesian coordinate system, with its vertices at the permuted coordinates (±1, 0, 0) forming three orthogonal squares which bisect the polytope. Analogously in 4 dimensional Euclidean space (in which the 4 coordinate axes cross six orthogonal planes through the origin, in the six ways we can pick a plane by choosing two of them), we can place six orthogonal squares at the permuted coordinates (±1, 0, 0, 0), the vertices of an 8-point 16-cell.

In 3 dimensions each orthogonal square can generate the other two by 90 degree rotations around each of its two axes (its diagonals). In 4 dimensions, each square can generate these two others in a single simple rotation (since simple rotations in effect rotate around both diagonals at once, by tilting a hyperplane around a fixed plane, rather than by tilting a plane around a fixed line), but it takes a double rotation around two orthogonal planes at once to reach the other three squares. Nonetheless all six squares are rotationally equidistant and completely orthogonal from each other, because the double rotation is a single 'diagonal' transition through 90 degrees, not two simple rotations.

Having generated the 8-point 16-cell by rotations around its equatorial square polygons, we can generate all its successors by double rotations of the 4-polytopes themselves (in each of two orthogonal planes of rotation).

Two 8-point 16-cells are inscribed in the 16-point tesseract, occupying opposite vertices. Given one of the 16-cells, the other one can be generated from it by an isoclinic double rotation through 90° = $𝜋⁄2$, producing the 16-point tesseract.

Three 16-point tesseracts are inscribed in the 24-point 24-cell, with each vertex shared by two tesseracts. Given one of the tesseracts, the other two can be generated from it by isoclinic double rotations through 60° = $𝜋⁄3$, producing the 24-cell.

Twenty-five 24-cells are inscribed in the 120-point 600-cell, with each vertex shared by five 24-cells. Given one of the 24-cells, the other twenty-four can be generated from it by double rotations through multiples of 36° = $𝜋⁄5$, producing the 600-cell.

Ten 120-point 600-cells are inscribed in the 600-point 120-cell, with each vertex shared by two 600-cells. Given one of the 600-cells, the other nine can be generated from it by isoclinic double rotations through multiples of .., producing the 120-cell.