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. The vertex figure of an antiditetragoltriate in four dimensions is a 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 = $\sqrt{4x^2-2√2x+1}$

Taking the sum of the absolute values of the differences of all distances, we get: and when plotting the appropriate result, there is no local minimum for any permissible x, and hence there is no way to find the appropriate value of x that gives the least difference.
 * f(x) = |1-2x|+|1-$\sqrt{4x^2-2√2x+1}$|+|2x-$\sqrt{4x^2-2√2x+1}$|,