Augmented truncated dodecahedron

{{Infobox polytope The augmented truncated dodecahedron, or autid, is one of the 92 Johnson solids (J{{sub|68}}). It consists of 5+5+5+5+5 triangles, 5 squares, 1 pentagon, and 1+5+5 decagons. It can be constructed by attaching a pentagonal cupola to one of the decagonal faces of the truncated dodecahedron..
 * type=CRF
 * img=Augmented truncated dodecahedron 2.png
 * 3d=J68 augmented truncated dodecahedron.stl
 * dim = 3
 * obsa = Autid
 * faces = 5+5+5+5+5 triangles, 5 squares, 1 pentagon, 1+5+5 decagons
 * edges = 5+5+5+5+5+5+5+5+5+10+10+10+10+10+10
 * vertices = 5+5+5+5+5+10+10+10+10
 * verf = 5 trapezoids, edge length 1, $\sqrt{2}$, (1+$\sqrt{5}$)/2, $\sqrt{2}$
 * verf2 = 10 irregular tetragons, edge length 1, $\sqrt{2}$, 1, $\sqrt{(5+√5)/2}$
 * verf3 = 50 isosceles triangles, edge lengths 1, $\sqrt{2+√2}$, $\sqrt{2+√2}$
 * army=Autid
 * reg=Autid
 * symmetry = H2×I, order 10
 * volume = (505+243$\sqrt{5}$)/12 ≈ 87.37361
 * dih = 3–4 join: acos(–$\sqrt{(23+23√5)/30}$) ≈ 174.34011º
 * dih2 = 3–4 cupolaic: acos(–($\sqrt{3}$+$\sqrt{15}$)/6) ≈ 159.09484º
 * dih3 = 3–10 join: acos(–$\sqrt{(65–2√5)/75}$) ≈ 153.94242º
 * dih4 = 4–5: acos(–$\sqrt{(5+√5)/10) ≈ 148.28253º
 * dih5 = 3–10 tid: acos(–√(5+2√5)/15) ≈ 142.62263º
 * dih6 = 10–10: acos(–√5/5) ≈ 116.56505º
 * smm = Yes
 * dual = Rhombirhombistellated triakis icosahedron
 * conjugate = Augmented quasitruncated great stellated dodecahedron
 * conv=Yes
 * orientable=Yes
 * nat=Tame}$

Vertex coordinates
An augmented truncated dodecahedron of edge length 1 has vertices given by all even permutations of: Plus the following additional vertices:
 * (0, ±1/2, ±(5+3$\sqrt{5}$)/4),
 * (±1/2, ±(3+$\sqrt{5}$)/4, ±(3+$\sqrt{5}$)/2),
 * (±(3+$\sqrt{5}$)/4, ±(1+$\sqrt{5}$)/2, ±(2+$\sqrt{5}$)/2),
 * ((15+13$\sqrt{5}$)/30, ±1/2, 3(5+$\sqrt{5}$)/10),
 * ((25+13$\sqrt{5}$)/20, ±(1+$\sqrt{5}$)/4, (25+$\sqrt{5}$)/20),
 * ((10+9$\sqrt{5}$)/10, 0, (15+$\sqrt{5}$)/20).