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-faced

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, the triangular pyramids can made with regular faces (as regular tetrahedrons) by computing the appropriate height.

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.

The octahedral pyramid is the vertex figure for a truncated 5-orthoplex, .

Other polytopes

The dual to the octahedral pyramid is a cubic pyramid, seen as an cubic base, and 6 square pyramids meeting at an apex.

Square-pyramidal pyramid

Square-pyramidal pyramid
Schlegel diagrams
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.

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.

External links

• 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