Octahedral pyramid

Octahedral pyramid

Schlegel diagram
Type Polyhedral pyramid
Schläfli symbol ( ) ∨ {3,4}
( ) ∨ r{3,3}
( ) ∨ s{2,6}
( ) ∨ [{4} + { }]
( ) ∨ [{ } + { } + { }]
Cells 9 1 {3,4}
8 ( ) ∨ {3}
Faces 20 {3}
Edges 18
Vertices 7
Dual Cubic pyramid
Symmetry group B3, [4,3,1], order 48
[3,3,1], order 24
[2+,6,1], order 12
[4,2,1], order 16
[2,2,1], order 8
Properties convex, regular-faced

In 4-dimensional geometry, the octahedral pyramid is bounded by one octahedron on the base and 8 triangular pyramid cells which meet at the apex. Since an octahedron has a circumradius divided by edge length less than one,[1] the triangular pyramids can be made with regular faces (as regular tetrahedrons) by computing the appropriate height.

Occurrences of the octahedral pyramid

The regular 16-cell has octahedral pyramids around every vertex, with the octahedron passing through the center of the 16-cell.

The octahedral pyramid is the vertex figure for a truncated 5-orthoplex, .

The graph of the octahedral pyramid is the only possible minimal counterexample to Negami's conjecture, that the connected graphs with planar covers are themselves projective-planar.[2]

Other polytopes

The dual to the octahedral pyramid is a cubic pyramid, seen as an cubic base, and 6 square pyramids meeting at an apex.

Square-pyramidal pyramid

Square-pyramidal pyramid

Schlegel diagrams
Type Polyhedral pyramid
Schläfli symbol ( ) ∨ [( ) ∨ {4}]
[( )∨( )] ∨ {4} = { } ∨ {4}
{ } ∨ [{ } × { }]
{ } ∨ [{ } + { }]
Cells 6 2 { } ∨ {4}
4 { } ∨ {3}
Faces 12 {3}
1 {4}
Edges 13
Vertices 6
Dual Self-dual
Symmetry group [4,1,1], order 8
[4,2,1], order 16
[2,2,1], order 8
Properties convex, regular-faced

The square-pyramidal pyramid, ( ) ∨ [( ) ∨ {4}], is a bisected octahedral pyramid. It has a square pyramid base, and 4 tetrahedrons along with another one more square pyramid meeting at the apex. It can also be seen in an edge-centered projection as a square bipyramid with four tetrahedra wrapped around the common edge. If the height of the two apexes are the same, it can be given a higher symmetry name [( ) ∨ ( )] ∨ {4} = { } ∨ {4}, joining an edge to a perpendicular square.[3]

The square-pyramidal pyramid can be distorted into a rectangular-pyramidal pyramid, { } ∨ [{ } × { }] or a rhombic-pyramidal pyramid, { } ∨ [{ } + { }], or other lower symmetry forms.

The square-pyramidal pyramid exists as a vertex figure in uniform polytopes of the form , including the bitruncated 5-orthoplex and bitruncated tesseractic honeycomb.

References

  1. ↑ Klitzing, Richard. "3D convex uniform polyhedra x3o4o - oct". 1/sqrt(2) = 0.707107
  2. ↑ Hliněný, Petr (2010), "20 years of Negami's planar cover conjecture" (PDF), Graphs and Combinatorics, 26 (4): 525–536, doi:10.1007/s00373-010-0934-9, MR 2669457
  3. ↑ Klitzing, Richard. "Segmentotope squasc, K-4.4".
  • Olshevsky, George. "Pyramid". Glossary for Hyperspace. Archived from the original on 4 February 2007.
  • Klitzing, Richard. "4D Segmentotopes".
    • Klitzing, Richard. "Segmentotope octpy, K-4.3".
  • Richard Klitzing, Axial-Symmetrical Edge Facetings of Uniform Polyhedra
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