Antiditetragoltriate

An antiditetragoltriate is an isogonal polytope formed as the convex hull of two orthogonal duoprisms (made of similar but not congruent bases), but antialigned so that the lateral facets are pyramids of the base prism's lateral facets, with simplexes filling the gaps. They are related to the ditetragoltriates in the same way three-dimensional prisms are related to antiprisms. The simplest non-trivial antiditetragoltriate is the triangular antiditetragoltriate. The dual of an antiditetragoltriate is an antitetrambitriate.

In 4 dimensions, an antiditetragoltriate has 2 rings of prismatic cells, with rectangular pyramids attached to the sides of the prisms, with tetragonal disphenoids filling the gaps. The vertex figure of an antiditetragoltriate in four dimensions is a variant of the Johnson solid biaugmented triangular prism.

Unlike the ditetragoltriates, optimization of antiditetragoltriates cannot be done in any meaningful way. A proof is shown below:

Take the coordinates of a square antiditetragoltriate as follows: where x > $\sqrt{2}$/2.
 * (±1/2, ±1/2, ±x, ±x),
 * (0, ±$\sqrt{2}$x, 0, ±$\sqrt{2}$/2),
 * (0, ±$\sqrt{2}$x, ±$\sqrt{2}$/2, 0),
 * (±$\sqrt{2}$x, 0, 0, ±$\sqrt{2}$/2),
 * (±$\sqrt{2}$x, 0, ±$\sqrt{2}$/2, 0),

The distances are therefore:
 * d1 = 1
 * d2 = 2x
 * d3 = $\sqrt{4x^2-2√2x+1}$

Using the ratio method, the lowest value is at x = $\sqrt{2}$/2 (intended ratio 1:$\sqrt{2}$), which instead gives a rectified tesseract. Therefore, any square antiditetragoltriate necessarily has a variant that approaches the minimal value but never reaches it.

Some antiditetragoltriates also exist in higher dimensions, but only when the base has two alternate orientations, with a simple example being the 6D tetrahedral antiditetragoltriate.