Bipyramid

A square bipyramid, constructed from a square.

A bipyramid is a polytope constructed by joining two congruent pyramids at their base. Usually the term refers to a 3D bipyramid constructed from a base polygon, but any n-polytope may be the base for a bipyramid, which has rank n + 1. To disambiguate them, 3D bipyramids may be called polygonal bipyramids. In the hi.gher.space community, the neologism tegum is frequently used for bipyramids, coined by Wendy Krieger after the Latin term tegere ("to cover") in allusion to how the bipyramid wraps around the base polytope and a line segment.

The facets of a bipyramids comprise two pyramids for every facet of the base. The bipyramid $\operatorname {Bpy} ({\mathcal {P}})$ in general does not actually contain the base polytope 𝓟, but does contain 𝓟 as a pseudofacet, i.e. all proper elements of 𝓟 can be found in $\operatorname {Bpy} ({\mathcal {P}})$.

The bipyramid operator is a special case of the direct sum. Particularly, a bipyramid built from a polytope 𝓟 is the same as the direct sum of 𝓟 and a line segment.

The dual of a bipyramid is a prism based on the dual polytope.

Any orbiform CRF polytope with a circumradius of less than 1 has a CRF bipyramid in the next dimension. The bipyramid itself is not guaranteed to be circumscribable.

The regular orthoplex of each dimension is the bipyramid of the orthoplex of the previous dimension.

Volume

The hypervolume of a bipyramid in n  dimensions can be calculated with the formula:

• V  = A h  / n ,

where A  is the hypervolume of the base polytope, and h  is the bipyramid’s height, the distance between apices.

External links

• Wikipedia contributors. "Bipyramid".
• Weisstein, Eric W. "Dipyramid" at MathWorld.