Cubic pyramid

Cubic pyramid
Rank4
TypeSegmentotope
Notation
Bowers style acronymCubpy
Coxeter diagramox4oo3oo&#x
Tapertopic notation[111]1
Elements
Cells6 square pyramids, 1 cube
Faces12 triangles, 6 squares
Edges8+12
Vertices1+8
Vertex figures1 cube, edge length 1
8 triangular pyramids, edge lengths 2 (base) and 1 (sides)
Measures (edge length 1)
Hypervolume${\displaystyle {\frac {1}{8}}=0.125}$
Dichoral anglesSquippy–3–squippy: 120°
Squippy–4–cube: 45°
HeightsPoint atop cube: ${\displaystyle {\frac {1}{2}}=0.5}$
Square atop squippy: ${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Central density1
Related polytopes
ArmyCubpy
RegimentCubpy
DualOctahedral pyramid
ConjugateNone
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryB3×I, order 48
ConvexYes
NatureTame

The cubical pyramid, or cubpy, is a CRF segmentochoron (designated K-4.26 on Richard Klitzing's list). It has 6 square pyramids and 1 cube as cells. As the name suggests, it is a pyramid based on the cube.

The cubic pyramid occurs as a vertex-first cap of the regular icositetrachoron.

A regular tesseract can be exactly decomposed into 8 CRF cubic pyramids.

Vertex coordinates

The vertices of a cubic pyramid of edge length 1 are given by:

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

Representations

A cubic pyramid has the following Coxeter diagrams:

• ox4oo3oo&#x (full symmetry)
• ox ox4oo&#x (BC2×A1 base, square prismatic pyramid)
• ox ox ox&#x (A1×A1×A1 base, cuboid pyramid)
• oxx4ooo&#x (base square atop square)
• oxx oxx&#x (*similar with rectangles)