bgcolor=#e7dcc3 colspan=3 | Octahedral pyramid | ||
---|---|---|---|
align=center colspan=3 | Schlegel diagram | ||
Type | Polyhedral pyramid | ||
Schläfli symbol | ∨ ∨ r ∨ s ∨ [{4} + { }] ∨ [{ } + { } + { }] | ||
Cells | 9 | ||
Faces | 20 | ||
Edges | 18 | ||
Vertices | 7 | ||
Dual | Cubic pyramid | ||
Symmetry group | B3, [4,3,1], order 48 [3,3,1], order 24 [2<sup>+</sup>,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.
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.
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)
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)
bgcolor=#e7dcc3 colspan=3 | Square-pyramidal pyramid | ||
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align=center colspan=3 | |||
Type | Polyhedral pyramid | ||
Schläfli symbol | ∨ [ ∨ {4}] [∨] ∨ = ∨ ∨ [{ } × { }] ∨ [{ } + { }] | ||
Cells | 6 | ||
Faces | 12 1 | ||
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 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)