Cubic pyramid
| Cubic pyramid | |
|---|---|
| Type | Polyhedral pyramid |
| Schläfli symbol | ( ) ∨ {4,3} ( ) ∨ [{4} × { }] ( ) ∨ [{ } × { } × { }] |
| Cells | 1 cube 6 square pyramids |
| Faces | 12 triangles 6 squares |
| 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 |
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,[1] the square pyramids can be made with regular faces by computing the appropriate height.
Construction and properties
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 ; the letter , , , and 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. 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.
References
- ^ Klitzing, Richard. "3D convex uniform polyhedra o3o4x - cube". sqrt(3)/2 = 0.866025
- ^ Quadling, Douglas (2007). "Further Forays into Dimensions". The Mathematical Gazette. 91 (522): 462–468. doi:10.1017/S0025557200182105. JSTOR 40378419.
- ^ Petrov, Miroslav S.; Todorov, Todor D.; Walters, Gage S.; Willams, David M.; Witherden, Freddie D. (2022). "Enabling four-dimensional conformal hybrid meshing with cubic pyramids". Numerical Algorithms. 91: 671–709. arXiv:2101.05884. doi:10.1007/s11075-022-01278-y.
- ^ Coxeter, H.S.M. (1973). Regular Polytopes (Third ed.). New York: Dover Publications. p. 150.
Further reading
- Zamboj, Michal (2018). "Sections and Shadows of Four-Dimensional Objects". Nexus Network Journal. 20: 475–487. doi:10.1007/s00004-018-0384-x.
External links
- 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