Double antiprismoid

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A double antiprismoid is an isogonal polytope formed from the alternation of a ditetragoltriate, a powertope formed by a polytope to the power of a ditetragon, if and and only if the base polytope is alternable. An n-gonal double antiprismoid is composed of two orthogonal rings of n n-gonal antiprisms each, with the rings aligned so that the edges belonging to the antiprisms' triangles on one ring are perpendicular to the other, creating tetragonal disphenoids, with sphenoids filling in the remaining gaps. As such, they are also the convex hull of two duoantiprisms (made of similar but not congruent bases which are alternated polytopes) and are generally nonuniform. The simplest non-trivial double antiprismoid is the digonal double antiprismoid, while the only uniform double antiprismoids are the grand antiprism (n=5) and its conjugate, the pentagrammatic double antiprismoid (n=5/3). The dual of a double antiprismoid is a double antitegmoid. They are also swirlchora based on an antiprism.

In four dimensions, the vertex figure of a double antiprismoid is a variant of the Johnson solid sphenocorona with 2 trapezoidal and 12 triangular faces, except for the digonal double antiprismoid, which has a hexakis digonal-hexagonal gyrowedge vertex figure, as the top edges of the trapezoids collapse into a vertex.

The n-gonal double antiprismoid could be generalized into an n-m double antiprismoid with rings of n- and m-gonal antiprisms respectively, with two vertex types.

Special cases[edit | edit source]

In four dimensions, an n-gonal double antiprismoid can have the least possible edge length difference if the ratio of the n-gons is equal to 1:(1+cos(π/n)+(1+cos(π/n))^2-4cos(2π/n))/2. There is also always a variant that uses uniform antiprisms only.