| bgcolor=#e7dcc3 colspan=3 | Cubic pyramid | ||
|---|---|---|---|
| align=center colspan=3 | Schlegel diagram | ||
| Type | Polyhedral pyramid | ||
| Schläfli symbols | ∨ ∨ [{4} × { }] ∨ [{ } × { } × { }] | ||
| Cells | 7 | ||
| Faces | 18 | 12 6 | |
| 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 | |||
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.