Pentagrammic-order 600-cell honeycomb explained
In the
geometry of
hyperbolic 4-space, the
pentagrammic-order 600-cell honeycomb is one of four
regular star-
honeycombs. With
Schläfli symbol, it has five
600-cells around each face in a
pentagrammic arrangement. It is dual to the
small stellated 120-cell honeycomb. It can be considered the higher-dimensional analogue of the 4-dimensional
icosahedral 120-cell and the 3-dimensional
great dodecahedron. It is related to the
order-5 icosahedral 120-cell honeycomb and
great 120-cell honeycomb: the
icosahedral 120-cells and
great 120-cells in each honeycomb are replaced by the
600-cells that are their convex hulls, thus forming the pentagrammic-order 600-cell honeycomb.
This honeycomb can also be constructed by taking the order-5 5-cell honeycomb and replacing clusters of 600 5-cells meeting at a vertex with 600-cells. Each 5-cell belongs to five such clusters, and thus the pentagrammic-order 600-cell honeycomb has density 5.
See also
References
- Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
- Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II, III, IV, V, p212-213)
