Triangular-pentagonal duoprism

{{Infobox polytope The triangular-pentagonal duoprism or trapedip, also known as the 3-5 duoprism, is a uniform duoprism that consists of 3 pentagonal prisms and 5 triangular prisms, with two of each at each vertex.
 * type=Uniform
 * dim = 4
 * img= auto
 * off = auto
 * obsa = Trapedip
 * symmetry = A2×H2, order 60
 * coxeter = x3o x5o
 * army=Trapedip
 * reg=Trapedip
 * cells = 5 triangular prisms, 3 pentagonal prisms
 * faces = 5 triangles, 15 squares, 3 pentagons
 * edges = 15+15
 * vertices = 15
 * verf = Digonal disphenoid, edge lengths 1 (base 1), (1+$\sqrt{5}$)/2 (base 2), and $\sqrt{2}$ (sides)
 * circum = $$\sqrt{\frac{25+3\sqrt5}{30}} ≈ 1.02808$$
 * height = $$\frac{\sqrt3}{2 ≈ 0.86603$$
 * hypervolume = $$\frac{\sqrt{75+30\sqrt5}}{16} ≈ 0.74499$$
 * dich = Trip–3–trip: 108°
 * dich2 = Trip–4–pip: 90°
 * dich3 = Pip–5–pip: 60°
 * conjugate=Triangular-pentagrammic duoprism
 * dual=Triangular-pentagonal duotegum
 * conv = Yes
 * orientable=Yes
 * nat=Tame}}

It is also a CRF segmentochoron, as pentagon atop pentagonal prism. It is designated K-4.34 on Richard Klitzing's list.

Vertex coordinates
Coordinates for the vertices of a triangular-pentagonal duoprism with edge length 1 are given by:


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

Representations
A triangular-pentagonal duoprism has the following Coxeter diagrams:


 * x3o x5o (full symmetry)
 * ox xx5oo&#x (pentagon atop pentagon prism)
 * ofx xxx3oooo&#xt (triangle-first)