Pentagrammic-order 600-cell honeycomb

Pentagrammic-order 600-cell honeycomb
(No image)
Type Hyperbolic regular honeycomb
Schläfli symbol {3,3,5,5/2}
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 5.pngCDel rat.pngCDel 2x.pngCDel node.png
4-faces Schlegel wireframe 600-cell vertex-centered.png {3,3,5}
Cells Tetrahedron.png {3,3}
Faces Regular polygon 3 annotated.svg {3}
Face figure Pentagram.svg {5/2}
Edge figure Great dodecahedron.png {5,5/2}
Vertex figure Ortho solid 007-uniform polychoron 35p-t0.png {3,5,5/2}
Dual Small stellated 120-cell honeycomb
Coxeter group H4, [5,3,3,3]
Properties Regular

In the geometry of hyperbolic 4-space, the pentagrammic-order 600-cell honeycomb is one of four regular star-honeycombs. With Schläfli symbol {3,3,5,5/2}, 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 alsoEdit


  • Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
  • Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)