Excavated expanded cuboctahedron

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Excavated expanded cuboctahedron
Rank3
TypeQuasi-convex Stewart toroid
Elements
Faces8 triangles, 6+24+48 squares
Edges24+24+24+24+24+48
Vertices6+8+24+24
Measures (edge length 1)
Volume
Surface area
Dihedral angles4-4 (rhombic prism lacing, narrow):
 4-4 (rhombic prism lacing, wide):
 3-4:
 4-4 (6-fold square to 24-fold square):
 4-4 (trip lacings): 60°
Related polytopes
Convex hullExpanded cuboctahedron
Abstract & topological properties
Flag count672
Euler characteristic–20
OrientableYes
Genus11
Properties
SymmetryB3, order 48
Flag orbits14
ConvexNo
NatureTame
History
Discovered byAlex Doskey
First discovered2004

The excavated expanded cuboctahedron is a quasi-convex Stewart toroid. It can be obtained by excavating 12 unit-edge-length rhombic prisms from an expanded cuboctahedron (blending their rhombus faces such that the prisms lie inside the shape), then excavating a rhombic dodecahedron from the center (blending it with the remaining rhombus faces of the prisms and opening up that space). This removes all of the rhombic faces, so the toroid is regular-faced. Its convex hull is an equilateral expanded cuboctahedron, with 30 squares (divided into two sets of 6 and 24), 12 rhombi, and 8 triangles.

It can also be obtained by outer-blending six square pyramids, eight tetrahedra, and twenty-four triangular prisms together.

Vertex coordinates[edit | edit source]

An excavated expanded cuboctahedron of edge length 1 has vertex coordinates given by all permutations of

  • ,
  • ,
  • ,
  • .

Related polyhedra[edit | edit source]

If the square pyramids and tetrahedra are partially-expanded into their respective cupolae (square and triangular), an excavated truncated rhombicuboctahedron will be formed.

If the triangular prisms are removed (and the pyramids brought together), a dissection of the cuboctahedron will be formed. It is a part of the tetrahedral-octahedral honeycomb.

External links[edit | edit source]