Octahedral pyramid

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

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-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

The 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.

(±1, 0, 0; 0)
( 0,±1, 0; 0)
( 0, 0,±1; 0)
( 0, 0, 0; 1)

Other polytopes

Cubic pyramid

The 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.

(±1,±1,±1; 0)
( 0, 0, 0; 1)

Square-pyramidal pyramid
Square-pyramidal pyramid
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.

Example 4-dimensional coordinates, 2 coordinates for square, and axial points for pyramidal points.

(±1,±1; 0; 0)
( 0, 0; 1; 0)
( 0, 0; 0; 1)

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, CiteSeerX 10.1.1.605.4932, doi:10.1007/s00373-010-0934-9, MR 2669457, S2CID 121645
  3. Klitzing, Richard. "Segmentotope squasc, K-4.4".


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