Cubic pyramid

A cubic pyramid has nine edges, twenty vertices, and eighteen faces (which include twelve triangles and six squares). It has seven cells, six are square pyramids and one is a cube. By the calculation of Euler's characteristic for a four-dimensional polytope, the cubic pyramid is ${\displaystyle V-E+F-C=0}$; the letter ${\displaystyle V}$, ${\displaystyle E}$, ${\displaystyle F}$, and ${\displaystyle C}$ designates the number of vertices, edges, faces, and cells of a cubic pyramid.^[2]

Exactly eight regular cubic pyramids will fit together around a vertex in four-dimensional space (the apex of each pyramid). This construction yields a tesseract with eight cubical bounding cells, surrounding a central vertex with 16 edge-length long radii. The tesseract tessellates four-dimensional space as the tesseractic honeycomb.^{[citation needed]} The 4-dimensional content of a unit-edge-length tesseract is 1, so the content of the regular cubic pyramid is 1/8.^[3]

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

The dual four-dimensional polytope of a cubic pyramid is an octahedral pyramid, seen as an octahedral base, and eight regular tetrahedra meeting at an apex.

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.