# Square pyramidal prism

Square pyramidal prism
Rank4
TypeSegmentotope
Notation
Bowers style acronymSquippyp
Coxeter diagramxx ox4oo&#x
Tapertopic notation[11]11
Elements
Cells2 square pyramids, 4 triangular prisms, 1 cube
Faces8 triangles, 2+4+4 squares
Edges1+4+8+8
Vertices2+8
Vertex figures2 square pyramids, edge lengths 1 (base) and 2 (legs)
8 Sphenoids, edge lengths 1 (2) and 2 (4)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.86603}$
Hypervolume${\displaystyle {\frac {\sqrt {2}}{6}}\approx 0.23570}$
Dichoral anglesTrip–4–trip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Squippy–3–trip: 90°
Squippy–4–cube: 90°
trip–4–cube: ${\displaystyle \arccos \left({\frac {\sqrt {3}}{3}}\right)\approx 54.73561^{\circ }}$
HeightsSquippy atop squippy: 1
Square atop trip: ${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Dyad atop cube: ${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Central density1
Related polytopes
DualSquare pyramidal tegum
ConjugateNone
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryB2×A1×I, order 16
ConvexYes
NatureTame

The square pyramidal prism, or squippyp, is a CRF segmentochoron (designated K-4.12 on Richard Klitzing's list). It consists of 2 square pyramids, 1 cube, and 4 triangular prisms.

As the name suggests, it is a prism based on the square pyramid. As such, it is a segmentochoron between two square pyramids. It can also be viewed as a segmentochoron between a cube and a dyad, or between a triangular prism and a square.

Two square pyramidal prisms can be joined at their cubes to form an octahedral prism. By rotating one of the square pyramidal prisms before joining, one can instead form a dyadic gyrotegmipucofastegium.

## Vertex coordinates

Coordinates of the vertices of a square pyramidal prism of edge length 1 centered at the origin are given by:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(0,\,0,\,{\frac {\sqrt {2}}{2}},\,\pm {\frac {1}{2}}\right).}$

## Representations

A square pyramidal prism has the following Coxeter diagrams:

• xx ox4oo&#x (full symmetry)
• xx ox ox&#x (A1×A1×A1 symmetry, rectangular pyramidal prism)
• oxx xxx&#x (A1×A1 axial only, trapezoidal pyramidal prism)
• oxxo4oooo&#xr (BC2 axial only, square pyramid atop square pyramid)