# Octahedral pyramid

Octahedral pyramid
Rank4
TypeSegmentotope
Notation
Bowers style acronymOctpy
Coxeter diagramoo4oo3ox&#x
Elements
Cells8 tetrahedra, 1 octahedron
Faces8+12 triangles
Edges6+12
Vertices1+6
Vertex figures1 octahedron, edge length 1
6 square pyramids, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Hypervolume${\displaystyle {\frac {1}{12}}\approx 0.083333}$
Dichoral anglesTet–3–tet: 120°
Tet–3–oct: 60°
HeightsPoint atop oct: ${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Trig atop gyro tet: ${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Central density1
Related polytopes
ArmyOctpy
RegimentOctpy
DualCubic pyramid
ConjugateNone
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryB3×I, order 48
ConvexYes
NatureTame

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:

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

with all permutations of the first 3 coordinates of:

• ${\displaystyle \left(0,\,0,\,\pm {\frac {\sqrt {2}}{2}},\,0\right).}$

## 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)