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The geometry of \(H_4\) polytopes. (English) Zbl 1450.52009

The paper studies the 600-cell, one of the four-dimensional analogues of the Platonic solids. Several facts about the 600-cell are collected together and proven. Some of these facts are previously known, with proofs; others are previously known but without proof. In these latter cases, the paper completes the picture by providing a proof.
Of central importance in the paper is the fact that the 600-cell contains 25 24-cells; these 24-cells may be arranged in a \(5 \times 5\) array such that each row and column gives a partition of the 600-cell into 24-cells. These ten partitions of the 600-cell into five 24-cells are the only such partitions, as Schoute claimed, without proof, over one hundred years ago [P. H. Schoute, Mehrdimensionale Geometrie. II. Teil. Die Polytope. Leipzig: G. J. Göschen (1905; JFM 36.0600.03)]. The paper provides a proof of this fact in Section 3.
In Section 2, the authors explain how the vertices of the 600-cell may be labelled. The first method is by coordinates in 4-dimensional space, where the 120 vertices of the 600-cell are labelled by the 120 icosians. The second method is by labelling each vertex pair using five duads. These five duads arise from the five different 24-cells that each vertex pair lies in, with the duads giving the coordinates of the 24-cells in the \(5 \times 5\) array. A similar analysis is then conducted for the 120-cell, the dual polytope of the 600-cell.
As noted above, Section 3 proves the result that the 600-cell admits ten different partitions into five 24-cells. This section further addresses the other regular four-dimensional polytopes that can be found using the vertices of the 600-cell. Namely, there are 75 16-cells and 75 8-cells, where each 16-cell and each 8-cell lies in just one 24-cell.
Section 4 deals with a different aspect of the 600-cell’s geometry: the planar polygons that can be found in its vertices. In particular, pentagons, hexagons, and decagons are studied, along with how these polygons interact with the 24-cells found in the 600-cell.
In Section 5, a new perspective on the geometry of the 600-cell is initiated. This uses the \(E_{8}\) root lattice, or rather, the space \(E_{8}/2E_{8}\) over \(\mathbb{F}_{2}\). Section 5 records facts about the geometry of this space, while Section 6 shows how one may embed the 600-cell into the \(E_{8}\) lattice in two different ways. These two different embeddings give an endomorphism of the \(E_{8}\) lattice which transforms the \(8\)-space \(E_{8}/2E_{8}\) over \(\mathbb{F}_{2}\) into a \(4\)-space over \(\mathbb{F}_{4}\). Finally, in Section 7, the authors show how the geometry of this \(4\)-space relates to the geometry of the 600-cell. For instance, the 85 projective points of this 4-space correspond to the 60 vertex pairs of the 600-cell, along with the 25 24-cells it contains.

MSC:

52B11 \(n\)-dimensional polytopes
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)

Citations:

JFM 36.0600.03
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Full Text: DOI arXiv

References:

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