| 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
|
| Net
|
|
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.
Related polytopes and honeycombs
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 cubic 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.
The cubic pyramid can be folded from a three-dimensional net in the form of a non-convex tetrakis hexahedron, obtained by gluing square pyramids onto the faces of a cube, and folded along the squares where the pyramids meet the cube.
