Duoexpandoprism

A duoexpandoprism is an isogonal polytope formed from the convex hull of two orthogonal duoprisms where one base is an expanded version of the other, forming two orthogonal rings of 2n n-gonal prisms (with alternating heights). In fact, they generally can be described as tegum sums xo-n-xx ox-n-xx&#zy, where ${\frac {y}{x}}={\frac {1}{{\sqrt {2}}\sin(\pi /m)}}$. (As such they clearly are just Stott expansions of the bi-'n'-gonal duotegum xo-n-oo ox-n-oo&#zy.)

They are closely related to the duotruncatoprisms, differing only in the prisms and the number of rings. The simplest possible duoexpandoprism is the digonal-square prismantiprismoid, which can be thought of as a digonal duoexpandoprism, while the simplest unique duoexpandoprism is the triangular duoexpandoprism, which is the convex hull of two orthogonal triangular-ditrigonal duoprisms. The dual of a duoexpandoprism is a duoexpandotegum.

In four dimensions, the related n-gonal double truncatoprismantiprismoid is topologically identical to the 2n-gonal duoexpandoprism, but with only half the symmetry. This variant alternates prisms and trapezoprisms.

The 4D duoexpandoprisms have 2 rings of 2n n-gonal prisms, with the space between filled by a layer of rectangular trapezoprisms, tetragonal disphenoids, and wedges. The vertex figure is a highly distorted variant of the octahedron with only mirror symmetry. If n is equal to 2, then the prisms become rectangles and the vertex figure becomes a mirror-symmetric notch. Analogs in higher dimensions also exist.

If one of the bases is a hypercube, then the resulting polytope will be the expanded hypercube, which can be made uniform. Other duoexpandoprisms are nonuniform.

Duoexpandoprisms can be blended with the base duoprisms to produce analogs of the four-dimensional inverted quasiprismatodishexadecachoron.

Special cases

In four dimensions, an n-gonal duoexpandoprism can have the least possible edge length difference, assuming that the prism heights are identical and the n-gon length is 1, if the n-gonal prism height is equal to (1+tan(π/2n)√3+4cos(π/n))/4. This ensures that the lateral edges have length 1.

External links

• Klitzing, Richard. "n-dep".