# Small ditrigonal dodecacronic hexecontahedron

Small ditrigonal dodecacronic hexecontahedron
Rank3
TypeUniform dual
Notation
Coxeter diagramm5/3o3m5*a
Elements
Faces60 darts
Edges60+60
Vertices12+12+20
Vertex figure20 triangles, 12 pentagrams, 12 decagons
Measures (edge length 1)
Inradius${\displaystyle 3{\frac {\sqrt {305\left(9+2{\sqrt {5}}\right)}}{122}}\approx 1.57627}$
Dihedral angle${\displaystyle \arccos \left(-{\frac {44+3{\sqrt {5}}}{61}}\right)\approx 146.23066^{\circ }}$
Central density4
Number of external pieces120
Related polytopes
DualSmall ditrigonal dodecicosidodecahedron
ConjugateGreat ditrigonal dodecacronic hexecontahedron
Convex coreNon-Catalan deltoidal hexecontahedron
Abstract & topological properties
Flag count480
Euler characteristic–16
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

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 ${\displaystyle 3{\frac {\sqrt {6\left(85+31{\sqrt {5}}\right)}}{22}}\approx 4.14937}$, and the long edges will be ${\displaystyle 3{\frac {\sqrt {3\left(145+62{\sqrt {5}}\right)}}{19}}\approx 4.60584}$. ​The dart faces will have length ${\displaystyle 3{\frac {\sqrt {10\left(3517-585{\sqrt {5}}\right)}}{418}}\approx 1.06668}$, and width ${\displaystyle 3{\frac {3+{\sqrt {5}}}{2}}\approx 7.85410}$. ​The darts have two interior angles of ${\displaystyle \arccos \left({\frac {5+3{\sqrt {5}}}{12}}\right)\approx 12.66108^{\circ }}$, one of ${\displaystyle \arccos \left(-{\frac {25+{\sqrt {5}}}{60}}\right)\approx 116.99640^{\circ }}$, and one of ${\displaystyle 360^{\circ }-\arccos \left(-{\frac {5+19{\sqrt {5}}}{60}}\right)\approx 217.68145^{\circ }}$.

## Vertex coordinates

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

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