# Small ditrigonal dodecacronic hexecontahedron

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

Small ditrigonal dodecacronic hexecontahedron Rank3
TypeUniform dual
SpaceSpherical
Notation
Coxeter diagramm5/3o3m5*a
Elements
Faces60 darts
Edges60+60
Vertices20+12+12
Vertex figure20 triangles, 12 pentagrams, 12 decagons
Measures (edge length 1)
Inradius$3\frac{\sqrt{305\left(9+2\sqrt5\right)}}{122} ≈ 1.57627$ Dihedral angle$\arccos\left(-\frac{44+3\sqrt5}{61}\right) ≈ 146.23066°$ Central density4
Number of external pieces120
Related polytopes
DualSmall ditrigonal dodecicosidodecahedron
Abstract & topological properties
Flag count480
Euler characteristic–16
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

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

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

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