Cubic pyramid
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.[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.
References
- ↑ Klitzing, Richard. "3D convex uniform polyhedra o3o4x - cube". https://bendwavy.org/klitzing/dimensions/polyhedra.htm. sqrt(3)/2 = 0.866025
- ↑ Quadling, Douglas (2007). "Further Forays into Dimensions". The Mathematical Gazette 91 (522): 462-468. doi:10.1017/S0025557200182105.
- ↑ 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. doi:10.1007/s11075-022-01278-y.
- ↑ Coxeter, H.S.M. (1973). Regular Polytopes (Third ed.). New York: Dover Publications. pp. 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. https://web.archive.org/web/20070204075028/members.aol.com/Polycell/glossary.html#Pyramid.
- Klitzing, Richard. "4D Segmentotopes". https://bendwavy.org/klitzing/dimensions/../explain/segmentochora.htm. Klitzing, Richard. "Segmentotope cubpy, K-4.26". https://bendwavy.org/klitzing/dimensions/..//incmats/cubpy.htm.
- Richard Klitzing, Axial-Symmetrical Edge Facetings of Uniform Polyhedra
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