Hexagonal-hexagonal prismantiprismoid

The hexagonal-hexagonal prismantiprismoid or hihipap, also known as the edge-snub hexagonal-hexagonal duoprism or 6-6 prismantiprismoid, is a convex isogonal polychoron that consists of 6 hexagonal antiprisms, 6 hexagonal prisms, 12 ditrigonal trapezoprisms, and 36 wedges. 1 hexagonal antiprism, 1 hexagonal prism, 2 ditrigonal trapezoprisms, and 3 wedges join at each vertex. It can be obtained through the process of alternating one class of edges of the dodecagonal duoprism so that one ring of dodecagons become ditrigons. However, it cannot be made uniform, as it generally has 4 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:$$\frac{7+3\sqrt3+\sqrt{384+174\sqrt3}}{22}$$ ≈ 1:1.74436.

Vertex coordinates
The vertices of a hexagonal-hexagonal prismantiprismoid based on a dodecagonal duoprism of edge length 1, centered at the origin, are given by:


 * $$\left(0,\,±\frac{\sqrt2+\sqrt6}{2},\,±\frac12,\,\frac{2+\sqrt3}{2}\right),$$
 * $$\left(0,\,±\frac{\sqrt2+\sqrt6}{2},\,±\frac{2+\sqrt3}{2},\,-\frac12\right),$$
 * $$\left(0,\,±\frac{\sqrt2+\sqrt6}{2},\,±\frac{1+\sqrt3}{2},\,-\frac{1+\sqrt3}{2}\right),$$
 * $$\left(±\frac{\sqrt2+\sqrt6}{2},\,0,\,±\frac12,\,-\frac{2+\sqrt3}{2}\right),$$
 * $$\left(±\frac{\sqrt2+\sqrt6}{2},\,0,\,±\frac{2+\sqrt3}{2},\,\frac12\right),$$
 * $$\left(±\frac{\sqrt2+\sqrt6}{2},\,0,\,±\frac{1+\sqrt3}{2},\,\frac{1+\sqrt3}{2}\right),$$
 * $$\left(±\frac{3\sqrt2+\sqrt6}{4},\,±\frac{\sqrt2+\sqrt6}{4},\,±\frac12,\,\frac{2+\sqrt3}{2}\right),$$
 * $$\left(±\frac{3\sqrt2+\sqrt6}{4},\,±\frac{\sqrt2+\sqrt6}{4},\,±\frac{2+\sqrt3}{2},\,-\frac12\right),$$
 * $$\left(±\frac{3\sqrt2+\sqrt6}{4},\,±\frac{\sqrt2+\sqrt6}{4},\,±\frac{1+\sqrt3}{2},\,-\frac{1+\sqrt3}{2}\right),$$
 * $$\left(±\frac{\sqrt2+\sqrt6}{4},\,±\frac{3\sqrt2+\sqrt6}{4},\,±\frac12,\,-\frac{2+\sqrt3}{2}\right),$$
 * $$\left(±\frac{\sqrt2+\sqrt6}{4},\,±\frac{3\sqrt2+\sqrt6}{4},\,±\frac{2+\sqrt3}{2},\,\frac12\right),$$
 * $$\left(±\frac{\sqrt2+\sqrt6}{4},\,±\frac{3\sqrt2+\sqrt6}{4},\,±\frac{1+\sqrt3}{2},\,\frac{1+\sqrt3}{2}\right).$$