Triangular-pentagonal duoantiprism

The triangular-pentagonal duoantiprism or trapedap, also known as the 3-5 duoantiprism, is a convex isogonal polychoron that consists of 6 pentagonal antiprisms, 10 triangular antiprisms, and 30 digonal disphenoids. 2 pentagonal antiprisms, 2 triangular antiprisms, and 4 digonal disphenoids join at each vertex. It can be obtained through the process of alternating the hexagonal-decagonal duoprism. However, it cannot be made uniform, as it generally has 3 edge lengths, which can be minimized to no fewer than 2 different sizes.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\sqrt{\frac{75+9\sqrt5}{58}}$$ ≈ 1:1.28066.

Vertex coordinates
The vertices of a triangular-pentagonal duoantiprism based on triangles and pentagons of edge length 1, centered at the origin, are given by:


 * $$±\left(0,\,\frac{\sqrt3}{3},\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$±\left(0,\,\frac{\sqrt3}{3},\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),$$
 * $$±\left(0,\,\frac{\sqrt3}{3},\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right),$$
 * $$±\left(±\frac12,\,-\frac{\sqrt3}{6},\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$±\left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),$$
 * $$±\left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right),$$