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Cubic pyramid | ||
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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 | B_{3}, [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.

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 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.

- Olshevsky, George. "Pyramid".
*Glossary for Hyperspace*. Archived from the original on 4 February 2007. - Klitzing, Richard. "4D Segmentotopes". Klitzing, Richard. "Segmentotope cubpy, K-4.26".
- Richard Klitzing, Axial-Symmetrical Edge Facetings of Uniform Polyhedra