Octahedron atop small rhombicuboctahedron

Octahedron atop small rhombicuboctahedron
(OFF file)
Rank	4
Type	Segmentotope
Notation
Bowers style acronym	Octasirco
Coxeter diagram	ox4oo3xx&#x
Elements
Cells	6 square pyramids, 8+12 triangular prisms, 1 octahedron, 1 small rhombicuboctahedron
Faces	8+8+24 triangles, 6+12+24 squares
Edges	12+24+24+24
Vertices	6+24
Vertex figures	6 square antiprisms, edge lengths 1 (base) and √2 (sides)
	12 isosceles trapezoidal pyramids, base edge lengths 1, √2, √2, √2, side edge lengths 1, 1, √2, √2
Measures (edge length 1)
Circumradius	${\sqrt {2+{\sqrt {2}}}}\approx 1.84776$
Hypervolume	${\frac {21+16{\sqrt {2}}}{24}}\approx 1.81781$
Dichoral angles	Trip–4–trip: $\arccos \left(-{\frac {2{\sqrt {2}}}{3}}\right)\approx 160.52878^{\circ }$
	Oct–3–trip: 150°
	Squippy–3–trip: 150°
	Sirco–4–squippy: 45°
	Sirco–4–trip: $\arccos \left({\frac {\sqrt {6}}{3}}\right)\approx 35.26439^{\circ }$
	Sirco–3–trip: 30°
Height	${\frac {1}{2}}=0.5$
Central density	1
Related polytopes
Army	Octasirco
Regiment	Octasirco
Dual	Cubic-deltoidal icositetrahedral tegmoid
Conjugate	Octahedron atop quasirhombicuboctahedron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B3×I, order 48
Convex	Yes
Nature	Tame

Octahedron atop small rhombicuboctahedron, or octasirco, is a CRF segmentochoron (designated K-4.107 on Richard Klitzing's list). As the name suggests, it consists of an octahedron and a small rhombicuboctahedron as bases, connected by 8+12 triangular prisms, and 6 square pyramids.

It is also sometimes referred to as an octahedral cupola, as one generalization of the definition of a cupola is to have a polytope atop an expanded version.

It can be obtained as a segment of the small prismatotetracontoctachoron, which can be constructed by attaching 8 of these segmentochora to the small rhombicuboctahedral cells of the small rhombated tesseract.

Segmentochoron display

• 

  Octahedron atop small rhombicuboctahedron

Vertex coordinates

The vertices of an octahedron atop small rhombicuboctahedron segmentochoron of edge length 1 are given by:

• $\left(\pm {\frac {\sqrt {2}}{2}},\,0,\,0,\,{\frac {1}{2}}\right)$ and all permutations of first three coordinates
• $\left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0\right)$ and all permutations of first three coordinates

External links

• Klitzing, Richard. "octasirco".
• Wikipedia contributors. "Octahedral cupola".
• Quickfur. "Octahedron atop Rhombicuboctahedron".