Octahedral pyramid

Octahedral pyramid
Rank	4
Type	Segmentotope
Space	Spherical
Notation
Bowers style acronym	Octpy
Coxeter diagram	oo4oo3ox&#x
Elements
Cells	8 tetrahedra, 1 octahedron
Faces	8+12 triangles
Edges	6+12
Vertices	1+6
Vertex figures	1 octahedron, edge length 1
	6 square pyramids, edge length 1
Measures (edge length 1)
Circumradius	$\frac{\sqrt2}{2} \approx 0.70711$
Hypervolume	$\frac{1}{12} \approx 0.083333$
Dichoral angles	Tet–3–tet: 120°
	Tet–3–oct: 60°
Heights	Point atop oct: $\frac{\sqrt2}{2} \approx 0.70711$
	Trig atop gyro tet: $\frac{\sqrt2}{2} \approx 0.70711$
Central density	1
Related polytopes
Army	Octpy
Regiment	Octpy
Dual	Cubic pyramid
Conjugate	None
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B3×I, order 48
Convex	Yes
Nature	Tame

The octahedral pyramid, or octpy, is a Blind polytope and CRF segmentochoron (designated K-4.3 on Richard Klitzing's list). It has 8 regular tetrahedra and 1 regular octahedron as cells. As the name suggests, it is a pyramid based on the octahedron.

Two octahedral pyramids can be attached at their bases to form a regular hexadecachoron. An octahedral pyramid can be further cut in half to produce two square scalenes.

It is part of an infinite family of Blind polytopes known as the orthoplecial pyramids, which generalize the square pyramid to higher dimensions.

Apart from being a point atop octahedron, it has an alternate segmentochoron representation as a triangle atop gyro tetrahedron seen as a triangular pyramid.

Vertex coordinates

The vertices of an octahedral pyramid of edge length 1 are given by:

• $\left(0,\,0,\,0,\,\frac{\sqrt2}{2}\right),$

with all permutations of the first 3 coordinates of:

• $\left(0,\,0,\,±\frac{\sqrt2}{2},\,0\right).$

Representations

An octahedral pyramid has the following Coxeter diagrams:

• oo4oo3ox&#x (full symmetry)
• oo3ox3oo&#x (base is in A₃ symmetry, tetratetrahedral pyramid)
• oxo3oox&#x (base is in A₂ symmetry only, triangular antiprismatic pyramid)

Segmentochoron display

• 

  Point atop octahedron

External links

• Klitzing, Richard. "octpy".

• Wikipedia Contributors. "Octahedral pyramid".
• Hi.gher.Space Wiki Contributors. "Octahedral pyramid".