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-cells, Blind polytope

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.

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.

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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.