Cubic pyramid

Cubic pyramid

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 B3, [4,3,1], order 48
[4,2,1], order 16
[2,2,1], order 8
Properties convex, regular-faced

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.

Images


3D projection while rotating

The regular 24-cell has cubic pyramids around every vertex.

The dual to the cubic pyramid is a 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.

References

  1. ↑ Klitzing, Richard. "3D convex uniform polyhedra o3o4x - cube". sqrt(3)/2 = 0.866025
  • 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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