Octahedral pyramid explained

bgcolor=#e7dcc3 colspan=3Octahedral pyramid
align=center colspan=3
Schlegel diagram
TypePolyhedral pyramid
Schläfli symbol
∨ r
∨ s
∨ [{4} + { }]
∨ [{ } + { } + { }]
Cells9
Faces20
Edges18
Vertices7
DualCubic pyramid
Symmetry groupB3, [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
Propertiesconvex, 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.

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.

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

bgcolor=#e7dcc3 colspan=3Square-pyramidal pyramid
align=center colspan=3
TypePolyhedral pyramid
Schläfli symbol ∨ [ ∨ {4}]
[∨] ∨ = ∨
∨ [{ } × { }]
∨ [{ } + { }]
Cells6
Faces12
1
Edges13
Vertices6
DualSelf-dual
Symmetry group[4,1,1], order 8
[4,2,1], order 16
[2,2,1], order 8
Propertiesconvex, 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 [ ∨ ] ∨ = ∨, joining an edge to a perpendicular square.

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)

External links

Notes and References

  1. 1/sqrt(2) = 0.707107