Perkel's graph and the 57-cell

H.S.M. Coxeter found the 57-cell in 1982, a few years after Branko Grünbaum had found the similar 11-cell, or hendekachoron, in 1976. But Coxeter didn't notice, so he independently rediscovered the 11-cell in 1984.

Both are abstract, locally projective polytopes, that is, their faces and vertex figures are tessellations of projective planes. They are similar to the hemi-platonic polyhedra one can obtain by identifying opposite features of platonic polyhedra.


The 57-cell is a regular 4-polytope, meaning its largest-dimensional toroidal cells have 4 dimensions. It consists of 57 hemi-dodecahedra, the same number of vertices, making up 171 pentagons and again, the same number of edges!

It is self-dual, its symmetry is L_2(19), also known as PSL(2,F_19), and of order 57•60=3420.


The 57-cell's skeleton is the Perkel graph, which is the unique distance-regular graph with intersection array {6, 5, 2; 1, 1, 3}. It has the Petersen graph as subgraph, 57 times, forming a single orbit!

This graph is also distance-transitive, which means that whenever we have two pairs of vertices with the same distances between their vertices, then there's an automorphism between the two pairs.


This is true for any skeleton of a platonic graph, but in fact there are only 12 3-valent distance-transitive graphs, which implies we can't keep building ever higher-dimensional 3-valent vertex-transitive polytopes where this holds.

The definition of distance-transitive graphs was given only in 1971 by Norman Linstead Biggs and D. H. Smith, preceding the discovery of the 57-cell by over a decade.


That there are graphs that are distance-regular but not distance-transitive was discovered just two years earlier in 1969 by a group lead by Georgy Adelson-Velsky. Two nice examples are the 3-valent Tutte 12-cage and the 6-valent Shrikhande graph.

Anyways, Manley Perkel was looking for bounds on the valence of polygonal graphs with odd girth when he came up with his example in 1979.


To find a realization of the 57-cell one has to go up to dimension 56! But even that is not faithful: its 3-cells are skew, which means they aren't contained in flat R³.

It seems, Coxeter hadn't heard about any of this. He certainly wouldn't have needed to. Typically, he would systematically classify all the things, before widening his perspective slightly so he could continue his quest: cartographing as much of geometrical cosmos as he could!


For some more details about the Perkel graph see here:

Carlo H. Séquin and James F. Hamlin attempted to visualize this wildly sel-intersecting and warped 57-cell for SIGGRAPH '07, giving a nice construction and insights about its embeddings in R³:

They did a similar thing for the 11-cell.


For full detail look at Coxeters paper "Ten toroids and fifty-seven hemidodecahedra"



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