Octahedron atop triangular cupola

Octahedron atop triangular cupola
Rank	4
Type	Segmentotope
Notation
Bowers style acronym	Octatricu
Coxeter diagram	xoxx3oxox&#xr
Elements
Cells	2+3 octahedra 2 triangular cupolas 6 square pyramids
Faces	1+2+3+6+6+6+12 triangles 6 squares 1 hexagon
Edges	3+3+6+6+12+12 3-fold 3 4-fold
Vertices	3+6+6
Vertex figures	3 cubes, edge length 1
	6 wedges, edge lengths √2 (two base edges) and 1 (remaining edges)
	6 skewed square pyramids, 1 (base and one side edge), √2 (two side edges), √3 (one side edge)
Measures (edge length 1)
Circumradius	$1$
Hypervolume	${\frac {2}{3}}\approx 0.66667$
Height	${\frac {\sqrt {2}}{2}}\approx 0.70711$
Central density	1
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A2×A1×I, order 12
Convex	Yes
Nature	Tame

Octahedron atop triangular cupola is a CRF segmentochoron (designated K-4.30 on Richard Klitzing's list). As the name suggests, it consists of an octahedron and a triangular cupola as bases, connected by 4 further octahedra, 1 further triangular cupola, and 6 square pyramids. It also has a higher symmetry orientation with 3 layers of vertices: a triangle (joining two octahedra) on one side, a hexagon (joining two triangular cupolas) on the other side, and 6 vertices in between in the shape of a non-uniform triangular prism.

An octahedron atop triangular cupola segmentochoron can be cut into two triangular antiwedges. Three octahedron atop triangular cupola segmentochora can be joined around a shared hexagonal face to form the regular icositetrachoron.

Vertex coordinates

The vertices of an octahedron atop triangular cupola segmentochoron of edge length 1 are given by:

• $\left(\pm {\frac {\sqrt {2}}{2}},\,0,\,0,\,{\frac {\sqrt {2}}{2}}\right)$ and all permutations of first three coordinates,
• $\pm \left({\frac {\sqrt {2}}{2}},\,-{\frac {\sqrt {2}}{2}},\,0,\,0\right)$ and all permutations of first three coordinates,
• $\left({\frac {\sqrt {2}}{2}},\,{\frac {\sqrt {2}}{2}},\,0,\,0\right)$ and all permutations of first three coordinates.

External links

• Klitzing, Richard. "octatricu".