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.
| Octahedral pyramid | |
|---|---|
 | |
| Type | Polyhedral pyramid |
| Schläfli symbol | ( ) ∨ {3,4} ( ) ∨ r{3,3} ( ) ∨ s{2,6} ( ) ∨ [{4} + { }] ( ) ∨ [{ } + { } + { }] |
| Cells | 1 {3,4}  8 ( ) ∨ {3}  |
| Faces | 20 {3} |
| Edges | 18 |
| Vertices | 7 |
| 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 |
| Dual | Cubic pyramid |
| Properties | convex, regular-cells, Blind polytope |
Having all regular cells, it is a Blind polytope. Two copies can be augmented to make an octahedral bipyramid which is also a Blind polytope.
Occurrences of the octahedral pyramid
editThe regular 16-cell has octahedral pyramids around every vertex, with the octahedron passing through the center of the 16-cell. Therefore placing two regular octahedral pyramids base to base constructs a 16-cell. The 16-cell tessellates 4-dimensional space as the 16-cell honeycomb.
Exactly 24 regular octahedral pyramids will fit together around a vertex in four-dimensional space (the apex of each pyramid). This construction yields a 24-cell with octahedral bounding cells, surrounding a central vertex with 24 edge-length long radii. The 4-dimensional content of a unit-edge-length 24-cell is 2, so the content of the regular octahedral pyramid is 1/12. The 24-cell tessellates 4-dimensional space as the 24-cell honeycomb.
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]
Example 4-dimensional coordinates, 6 points in first 3 coordinates for cube and 4th dimension for the apex.
Other polytopes
editCubic pyramid
editThe dual to the octahedral pyramid is a cubic pyramid, seen as a cubic base and 6 square pyramids meeting at an apex.
Example 4-dimensional coordinates, 8 points in first 3 coordinates for cube and 4th dimension for the apex.
Square-pyramidal pyramid
edit| Square-pyramidal pyramid | |
|---|---|
 | |
| Type | Polyhedral pyramid |
| Schläfli symbol | ( ) ∨ [( ) ∨ {4}] [( )∨( )] ∨ {4} = { } ∨ {4} { } ∨ [{ } × { }] { } ∨ [{ } + { }] |
| Cells | 2 ( )∨{4}  4 ( )∨{3} |
| Faces | 12 {3} 1 {4} |
| Edges | 13 |
| Vertices | 6 |
| Symmetry group | [4,1,1], order 8 [4,2,1], order 16 [2,2,1], order 8 |
| Dual | Self-dual |
| 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.
Example 4-dimensional coordinates, 2 coordinates for square, and axial points for pyramidal points.
References
edit- ^ Klitzing, Richard. "3D convex uniform polyhedra x3o4o - oct". 1/sqrt(2) = 0.707107
- ^ Hliněný, Petr (2010), "20 years of Negami's planar cover conjecture" (PDF), Graphs and Combinatorics, 26 (4): 525–536, CiteSeerX 10.1.1.605.4932, doi:10.1007/s00373-010-0934-9, MR 2669457, S2CID 121645
- ^ Klitzing, Richard. "Segmentotope squasc, K-4.4".
External links
edit- 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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