Small ditrigonal dodecacronic hexecontahedron

Small ditrigonal dodecacronic hexecontahedron
	(3D model); (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Coxeter diagram	m5/3o3m5*a
Elements
Faces	60 darts
Edges	60+60
Vertices	20+12+12
Vertex figure	20 triangles, 12 pentagrams, 12 decagons
Measures (edge length 1)
Inradius
Dihedral angle
Central density	4
Number of external pieces	120
Related polytopes
Dual	Small ditrigonal dodecicosidodecahedron
Abstract & topological properties
Flag count	480
Euler characteristic	–16
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The small ditrigonal dodecacronic hexecontahedron is a uniform dual polyhedron. It consists of 60 darts.

It appears the same as the small dodecicosacron.

If its dual, the small ditrigonal dodecicosidodecahedron, has an edge length of 1, then the short edges of the darts will measure $3\frac{\sqrt{6\left(85+31\sqrt5\right)}}{22} ≈ 4.14937$ , and the long edges will be $3\frac{\sqrt{3\left(145+62\sqrt5\right)}}{19} ≈ 4.60584$ . The dart faces will have length $3\frac{\sqrt{10\left(3517-585\sqrt5\right)}}{418} ≈ 1.06668$ , and width $3\frac{3+\sqrt5}{2} ≈ 7.85410$ . The darts have two interior angles of $\arccos\left(\frac{5}{12}+\frac{\sqrt5}{4}\right) ≈ 12.66108°$ , one of $\arccos\left(-\frac{5}{12}-\frac{\sqrt5}{60}\right) ≈ 116.99640°$ , and one of $360°-\arccos\left(-\frac{1}{12}-\frac{19\sqrt5}{60}\right) ≈ 217.68145°$ .

Vertex coordinates[edit | edit source]

A small ditrigonal dodecacronic hexecontahedron with dual edge length 1 has vertex coordinates given by all even permutations of:

$\left(±3\frac{3+\sqrt5}{4},\,±3\frac{1+\sqrt5}{4},\,0\right),$
$\left(±3\frac{9\sqrt5-5}{76},\,±3\frac{15+11\sqrt5}{76},\,0\right),$
$\left(±3\frac{5+7\sqrt5}{44},\,±3\frac{15-\sqrt5}{44},\,0\right),$
$\left(±3\frac{10+\sqrt5}{38},\,±3\frac{10+\sqrt5}{38},\,±3\frac{10+\sqrt5}{38}\right).$