Pentagonal-truncated tetrahedral duoalterprism

The pentagonal-truncated tetrahedral duoalterprism, or petuta, is a convex isogonal polyteron that consists of 10 truncated tetrahedral alterprisms, 6 digonal-pentagonal duoantiprisms, 12 pentagonal antiprismatic prisms, 8 triangular-pentagonal duoprisms, and 40 triangular cupofastegiums. 1 digonal-pentagonal duoantiprism, 1 triangular-pentagonal duoprism, 2 truncated tetrahedral cupoliprisms, 2 pentagonal antiprismatic prisms, and 4 trianguar cupofastegiums join at each vertex. It can be formed by tetrahedrally alternating the decagonal-small rhombicuboctahedral duoprism, so that all the small rhombicuboctahedra turn into truncated tetrahedra. However, it cannot be made scaliform.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\sqrt{\frac{20+2\sqrt5}{19}}$$ ≈ 1:1.13490. This occurs when it is a hull of 2 uniform pentagonal-truncated tetrahedral duoprisms.

Vertex coordinates
The vertices of a pentagonal-truncated tetrahedral duoalterprism, assuming that the edge length differences are minimized, centered at the origin, are given by: with all permutations and even changes of sign of the first three coordinates, and with all permutations and odd changes of sign of the first three coordinates.
 * $$\left(\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{3\sqrt2}{4},\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$\left(\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{3\sqrt2}{4},\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),$$
 * $$\left(\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{3\sqrt2}{4},\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right),$$
 * $$\left(\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{3\sqrt2}{4},\,0,\,-\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$\left(\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{3\sqrt2}{4},\,±\frac{1+\sqrt5}{4},\,-\sqrt{\frac{5-\sqrt5}{40}}\right),$$
 * $$\left(\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{3\sqrt2}{4},\,±\frac12,\,\sqrt{\frac{5+2\sqrt5}{20}}\right),$$