Cubic pyramid

Net

In four-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, the square pyramids can be made with regular faces by computing the appropriate height.

Construction and properties

A cubic pyramid has nine vertices, twenty edges, 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 $V-E+F-C=0$; the letter $V$, $E$, $F$, and $C$ designates the number of vertices, edges, faces, and cells of a cubic pyramid.Template:R

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.Template:Cn The 4-dimensional content of a unit-edge-length tesseract is 1, so the content of the regular cubic pyramid is 1/8.Template:R

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.Template:R The 24-cell tessellates 4-dimensional space as the 24-cell honeycomb.

Octahedral pyramid, the dual of a cubic pyramid

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.

Further reading

• Template:Cite journal

External links

• Template:GlossaryForHyperspace
• Template:KlitzingPolytopes Template:KlitzingPolytopes
• Richard Klitzing, Axial-Symmetrical Edge Facetings of Uniform Polyhedra