Cube atop icosahedron

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Cube atop icosahedron
Rank4
TypeSegmentotope
Notation
Bowers style acronymCubaike
Coxeter diagramxo4os3os&#x
Elements
Cells8 tetrahedra, 12 square pyramids, 6 triangular prisms, 1 cube, 1 icosahedron
Faces8+12+12+24 triangles, 6+12 squares
Edges6+12+24+24
Vertices8+12
Vertex figures8 chiral triangular antipodiums, edge lengths 1 and 2 (each 3 base and 3 side edges)
 12 digonal-pentagonal wedges, edge lengths 1 and 2
Measures (edge length 1)
Circumradius1
Hypervolume
Dichoral anglesSquippy–3–tet:
 Cube–4–trip:
 Ike–3–tet:
 Ike–3–squippy:
 Squippy–3–trip:
 Squippy–4–trip:
Height
Central density1
Related polytopes
DualOctahedral-dodecahedral tegmoid
ConjugateCube atop great icosahedron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryB3/2×I, order 24
ConvexYes
NatureTame

The cube atop icosahedron, or cubaike, is a CRF segmentochoron (designated K-4.21 on Richard Klitzing's list). It consists of a cube and an icosahedron located on parallel hyperplanes, connected by 6 triangular prisms (attached to the cube's faces), 12 square pyramids (attached to the icosahedron and the remaining square faces of the triangular prisms), and 8 tetrahedra that fill in the remaining gaps.

The drastically different symmetries of the two "bases" set the cube atop icosahedron apart from other segmentochora. In fact the common symmetry here is only pyritohedral.

This segmentochoron may at first not seem to be related to any other polychora, but it can be thought of as a subset of the vertices of the hexacosichoron, when seen vertex-first. The top cube forms some of the vertices of the dodecahedral layer, while the bottom icosahedron is the larger icosahedral layer from the hexacosichoron.

Gallery[edit | edit source]

Vertex coordinates[edit | edit source]

Coordinates for the vertices of a cube atop icosahedron of edge length 1 are given by:

and all cyclic permutations in the first 3 coordinates of

The first set of coordinates defines the cube, and the second set defines the icosahedron.

Representations[edit | edit source]

A cube atop icosahedron can be represented by the following Coxeter diagrams:

  • os3os4xo&#x
  • x(xfo) x(fox) x(oxf)&#x

External links[edit | edit source]