View source for Octahedral pyramid

{{short description|4-D polyhedral pyramid}}
{{Infobox 4-polytope
| Name = Octahedral pyramid
| Image_File = Octahedral pyramid.png
| Image_Caption = [[Schlegel diagram]]
| Type = [[Polyhedral pyramid]]
| Schläfli = {{nowrap|( ) ∨ {3,4} }}<BR>{{nowrap|( ) ∨ r{3,3} }}<BR>{{nowrap|( ) ∨ s{2,6} }}<BR>{{nowrap|( ) ∨ [{4} + { }] }}<BR>{{nowrap|( ) ∨ [{ } + { } + { }] }}
| CD =
| Cell_List = 1 [[octahedron|{3,4}]] [[Image:octahedron.png|30px]]<BR>8 [[tetrahedron|( ) ∨ {3}]] [[Image:Tetrahedron.png|30px]]
| Face_List = 20 [[triangle|{3}]]
| Edge_Count = 18
| Vertex_Count = 7
| Vertex_Figure =
| Petrie_Polygon =
| Coxeter_Group =
| Symmetry_Group = B<sub>3</sub>, [4,3,1], order 48<BR>[3,3,1], order 24<BR>[2<sup>+</sup>,6,1], order 12<BR>[4,2,1], order 16<BR>[2,2,1], order 8
| Dual = [[Cubic pyramid]]
| Property_List = [[Convex polytope|convex]], regular-cells, [[Blind polytope]]
| Index =
}}

In 4-dimensional [[geometry]], the '''octahedral pyramid''' is bounded by one [[octahedron]] on the base and 8 [[triangular pyramid]] [[cell (mathematics)|cells]] which meet at the apex. Since an octahedron has a circumradius divided by edge length less than one,<ref>{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|x3o4o - oct}} 1/sqrt(2) = 0.707107</ref> the triangular pyramids can be made with regular faces (as regular [[tetrahedron]]s) 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]], {{CDD|node_1|3|node_1|3|node|3|node|4|node}}.
:[[File:Truncated pentacross.png|160px]]

The graph of the octahedral pyramid is the only possible minimal counterexample to [[Negami's conjecture]], that the connected graphs with [[planar cover]]s are themselves projective-planar.<ref>{{citation
 | last = Hliněný | first = Petr
 | doi = 10.1007/s00373-010-0934-9
 | issue = 4
 | journal = [[Graphs and Combinatorics]]
 | mr = 2669457
 | pages = 525–536
 | title = 20 years of Negami's planar cover conjecture
 | url = http://www.fi.muni.cz/~hlineny/papers/plcover20-gc.pdf
 | volume = 26
 | year = 2010| citeseerx = 10.1.1.605.4932
 | s2cid = 121645
 }}</ref>

Example 4-dimensional coordinates, 6 points in first 3 coordinates for cube and 4th dimension for the apex. 
<math display=block>\begin{array}{lllr}
  (\pm 1, & 0,     & 0;    & 0) \\
  (0,     & \pm 1, & 0;    & 0) \\
  (0,     & 0,     & \pm1; & 0) \\
  (0,     & 0,     & 0;    & \ 1)
\end{array}</math>

== Other polytopes ==

=== Cubic pyramid ===
The dual to the octahedral pyramid is a [[cubic pyramid]], seen as a cubic base and 6 [[square pyramid]]s meeting at an [[apex (geometry)|apex]].
:[[File:Cubic pyramid.png|160px]]

Example 4-dimensional coordinates, 8 points in first 3 coordinates for cube and 4th dimension for the apex.
<math display=block>\begin{array}{lllr}
  (\pm 1, & \pm 1, & \pm 1; & 0) \\
  (0,     & 0,     & 0;     & 1)
\end{array}</math>
{{-}}

=== Square-pyramidal pyramid ===

{{Infobox 4-polytope
| Name = Square-pyramidal pyramid
| Image_File = Square pyramid pyramid.png
| Image_Caption =
| Type = [[Polyhedral pyramid]]
| Schläfli = {{nowrap|( ) ∨ [( ) ∨ {4}] }}<BR>{{nowrap|1=[( )∨( )] ∨ {4} = { } ∨ {4} }}<BR>{{nowrap|{ } ∨ [{ } × { }]<BR>{ } ∨ [{ } + { }] }}
| CD =
| Cell_List = 2 [[square pyramid|(&nbsp;)∨{4}]] [[Image:square pyramid.png|30px]]<BR>4 [[tetrahedron|(&nbsp;)∨{3}]] [[Image:Tetrahedron.png|30px]]
| Face_List = 12 [[triangle|{3}]]<BR>1 [[square|{4}]]
| Edge_Count = 13
| Vertex_Count = 6
| Vertex_Figure = 
| Petrie_Polygon =
| Coxeter_Group =
| Symmetry_Group = [4,1,1], order 8<BR>[4,2,1], order 16<BR>[2,2,1], order 8
| Dual = Self-dual
| Property_List = [[Convex polytope|convex]], regular-faced
| Index =
}}
[[File:Square pyramid pyramid edgecenter.png|150px|thumb|Edge-centered view]]


The '''square-pyramidal pyramid''', {{nowrap|( ) ∨ [( ) ∨ {4}]}}, is a bisected octahedral pyramid. It has a [[square pyramid]] base, and 4 [[tetrahedron]]s 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 {{nowrap|1=[( ) ∨ ( )] ∨ {4} = { } ∨ {4},}} joining an edge to a perpendicular square.<ref> {{KlitzingPolytopes|..//incmats/squasc.htm|Segmentotope|squasc, K-4.4}}</ref>

The ''square-pyramidal pyramid'' can be distorted into a ''rectangular-pyramidal pyramid'', {{nowrap|{ } ∨ [{ } × { }] }} or a ''rhombic-pyramidal pyramid'', {{nowrap|{ } ∨ [{ } + { }],}} or other lower symmetry forms.

The ''square-pyramidal pyramid'' exists as a vertex figure in uniform polytopes of the form {{CDD|node|p|node_1|q|node_1|r|node|4|node}}, including the [[bitruncated 5-orthoplex]] and [[bitruncated tesseractic honeycomb]].
:[[File:Bitruncated pentacross verf.png|120px]][[File:Bitruncated tesseractic honeycomb verf.png|120px]]

Example 4-dimensional coordinates, 2 coordinates for square, and axial points for pyramidal points.
<math display=block>\begin{array}{lllr}
  (\pm 1, & \pm 1; & 0; & 0) \\
  (0,     & 0;     & 1; & 0) \\
  (0,     & 0;     & 0; & \ \ 1)
\end{array}</math>

== References==
{{reflist}}

==External links==
*{{GlossaryForHyperspace |anchor=Pyramid |title=Pyramid}}
* {{KlitzingPolytopes|../explain/segmentochora.htm|4D|Segmentotopes}} 
** {{KlitzingPolytopes|..//incmats/octpy.htm|Segmentotope|octpy, K-4.3}}
* Richard Klitzing, [http://bendwavy.org/klitzing/pdf/edge-facetings_color.pdf Axial-Symmetrical Edge Facetings of Uniform Polyhedra]

[[Category:4-polytopes]]
[[Category:Pyramids (geometry)]]
{{Polychora-stub}}