A prismantiprismoid is an isogonal polytope with a structure composed of one ring of alternating prisms and antiprisms and are generally nonuniform. The simplest non-trivial prismantiprismoid is the digonal-square prismantiprismoid. The dual of a prismantiprismoid is a tegmotrapezohedroid. All prismantiprismoids are edge-snub polytopes. They are also a special class of the duoprismatic prismantiprismatoswirlprisms, having only two polygonal rotations for each ring.
Unlike other duoprismatic families, the n-m prismantiprismoid is not equivalent to the m-n prismantiprismoid, meaning that the square-hexagonal prismantiprismoid and the hexagonal-square prismantiprismoid are not topologically the same polychoron, as the former has 6 rings of alternating square prisms and square antiprisms, while the latter has 4 rings of alternating hexagonal prisms and hexagonal antiprisms. Another additional restriction is that n must be an alternated polytope (such as a snub cube), while m must be an alternable polytope (such as a hexagonal prism).
4D prismantiprismoids generally have one ring of alternating prisms and antiprisms, and orthogonal ring of trapezoprisms, and a layer of wedges connecting these rings. If the first polygon is a digon, the antiprisms become tetragonal disphenoids, and the prisms become rectangular faces only. The vertex figure of the general prismantiprismoid is a 6-vertex heptahedron that can be obtained by augmenting one of the side faces of an isosceles trapezoidal pyramid, excepting the digonal case again, when one edge of the vertex figure collapses, the trapezoid becomes a triangle, and the overall shape becomes a notch.
Special cases[edit | edit source]
In four dimensions, an n-m prismantiprismoid can have the least possible edge length difference, assuming that the edge lengths belonging to the n-gon and the longest edge are 1, if the n-gonal prism height is equal to (sec(π/2n)cos(π/m)√-1)/(2+4cos(2π/m)). This ensures that the isosceles trapezoids have three equal edges.
[edit | edit source]
- Klitzing, Richard. "(n,2m)-pap".
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