Cubic pyramid

In 4-dimensional geometry, the cubic pyramid is bounded by one cube on the base and 6 square pyramid cells which meet at the apex. Since a cube has a circumradius divided by edge length less than one,[1] the square pyramids can be made with regular faces by computing the appropriate height.

Cubic pyramid

Schlegel diagram
Type Polyhedral pyramid
Schläfli symbols ( ) ∨ {4,3}
( ) ∨ [{4} × { }]
( ) ∨ [{ } × { } × { }]
Cells 7 1 {4,3}
6 ( ) ∨ {4}
Faces 18 12 {3}
6 {4}
Edges 20
Vertices 9
Dual Octahedral pyramid
Symmetry group B3, [4,3,1], order 48
[4,2,1], order 16
[2,2,1], order 8
Properties convex, regular-faced

Images

3D projection while rotating

Exactly 8 regular cubic pyramids will fit together around a vertex in four-dimensional space (the apex of each pyramid). This construction yields a tesseract with 8 cubical bounding cells, surrounding a central vertex with 16 edge-length long radii. The tesseract tessellates 4-dimensional space as the tesseractic honeycomb. The 4-dimensional content of a unit-edge-length tesseract is 1, so the content of the regular octahedral pyramid is 1/8.

The regular 24-cell has cubic pyramids around every vertex. Placing 8 cubic pyramids on the cubic bounding cells of a tesseract is Gosset's construction[2] of the 24-cell. Thus the 24-cell is constructed from exactly 16 cubic pyramids. The 24-cell tessellates 4-dimensional space as the 24-cell honeycomb.

The dual to the cubic pyramid is an octahedral pyramid, seen as an octahedral base, and 8 regular tetrahedra meeting at an apex.

A cubic pyramid of height zero can be seen as a cube divided into 6 square pyramids along with the center point. These square pyramid-filled cubes can tessellate three-dimensional space as a dual of the truncated cubic honeycomb, called a hexakis cubic honeycomb, or pyramidille.

References
  1. Klitzing, Richard. "3D convex uniform polyhedra o3o4x - cube". sqrt(3)/2 = 0.866025
  2. Coxeter, H.S.M. (1973). Regular Polytopes (Third ed.). New York: Dover. p. 150.
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