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.
| Cubic pyramid | |
|---|---|
 | |
| Type | Polyhedral pyramid |
| Schläfli symbol | ( ) ∨ {4,3} ( ) ∨ [{4} × { }] ( ) ∨ [{ } × { } × { }] |
| Cells | 1 {4,3}  6 ( ) ∨ {4}  |
| Faces | 12 {3} 6 {4} |
| Edges | 20 |
| Vertices | 9 |
| Coxeter group | B3 |
| Symmetry group | [4,3,1], order 48 [4,2,1], order 16 [2,2,1], order 8 |
| Dual | Octahedral pyramid |
| Properties | convex, regular-faced |
Images
edit 3D projection while rotating |
Related polytopes and honeycombs
editExactly 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.
References
edit- ^ Klitzing, Richard. "3D convex uniform polyhedra o3o4x - cube". sqrt(3)/2 = 0.866025
- ^ Coxeter, H.S.M. (1973). Regular Polytopes (Third ed.). New York: Dover. p. 150.
External links
edit- 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
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