Octahedral pyramid
Octahedral pyramid  

Rank  4 
Type  Segmentotope 
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  
Hypervolume  
Dichoral angles  Tet–3–tet: 120° 
Tet–3–oct: 60°  
Heights  Point atop oct: 
Trig atop gyro tet:  
Central density  1 
Related polytopes  
Army  Octpy 
Regiment  Octpy 
Dual  Cubic pyramid 
Conjugate  None 
Abstract & topological properties  
Euler characteristic  0 
Orientable  Yes 
Properties  
Symmetry  B_{3}×I, order 48 
Convex  Yes 
Nature  Tame 
The octahedral pyramid, or octpy, is a Blind polytope and CRF segmentochoron (designated K4.3 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[edit  edit source]
The vertices of an octahedral pyramid of edge length 1 are given by:
with all permutations of the first 3 coordinates of:
Representations[edit  edit source]
An octahedral pyramid has the following Coxeter diagrams:
 oo4oo3ox&#x (full symmetry)
 oo3ox3oo&#x (base is in A_{3} symmetry, tetratetrahedral pyramid)
 oxo3oox&#x (base is in A_{2} symmetry only, triangular antiprismatic pyramid)
Segmentochoron display[edit  edit source]

Point atop octahedron
External links[edit  edit source]
 Klitzing, Richard. "octpy".
 Wikipedia contributors. "Octahedral pyramid".
 Hi.gher.Space Wiki Contributors. "Octahedral pyramid".