Great ditrigonal dodecacronic hexecontahedron

Great ditrigonal dodecacronic hexecontahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Notation
Coxeter diagram	m5/3m3o5*a
Elements
Faces	60 kites
Edges	60+60
Vertices	12+12+20
Vertex figure	20 triangles, 12 pentagons, 12 decagrams
Measures (edge length 1)
Inradius	$3{\frac {\sqrt {305(9-2{\sqrt {5}})}}{122}}\approx 0.91381$
Dihedral angle	$\arccos \left(-{\frac {44-3{\sqrt {5}}}{61}}\right)\approx 127.68652^{\circ }$
Central density	4
Number of external pieces	120
Related polytopes
Dual	Great ditrigonal dodecicosidodecahedron
Conjugate	Small ditrigonal dodecacronic hexecontahedron
Convex core	Triakis icosahedron
Abstract & topological properties
Flag count	480
Euler characteristic	–16
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The great ditrigonal dodecacronic hexecontahedron is a uniform dual polyhedron. It consists of 60 kites.

If its dual, the great ditrigonal dodecicosidodecahedron, has an edge length of 1, then the short edges of the kites will measure $3{\frac {\sqrt {3\left(145-62{\sqrt {5}}\right)}}{19}}\approx 0.68990$, and the long edges will be $3{\frac {\sqrt {6\left(85-31{\sqrt {5}}\right)}}{22}}\approx 1.32274$. ​The kite faces will have length $3{\frac {\sqrt {10\left(3517+585{\sqrt {5}}\right)}}{418}}\approx 1.57652$, and width $3{\frac {3-{\sqrt {5}}}{2}}\approx 1.14590$. The kites have two interior angles of $\arccos \left({\frac {5-3{\sqrt {5}}}{12}}\right)\approx 98.18387^{\circ }$, one of $\arccos \left({\frac {-25+{\sqrt {5}}}{60}}\right)\approx 112.29645^{\circ }$, and one of $\arccos \left({\frac {-5+19{\sqrt {5}}}{60}}\right)\approx 51.33580^{\circ }$.

Vertex coordinates

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

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

External links

• Wikipedia contributors. "Great ditrigonal dodecacronic hexecontahedron".
• McCooey, David. "Great Ditrigonal Dodecacronic Hexecontahedron"