Great dodecicosidodecahedron

The great dodecicosidodecahedron, or gaddid, is a uniform polyhedron. It consists of 20 triangles, 12 pentagrams, and 12 decagrams. One triangle, one pentagram, and two decagrams join at each vertex.

Great dodecicosidodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Gaddid
Coxeter diagram	x5/3x5/2o3*a ()
Elements
Faces	20 triangles, 12 pentagrams, 12 decagrams
Edges	60+60
Vertices	60
Vertex figure	Isosceles trapezoid, edge lengths 1, √(5–√5)/2, (√5–1)/2, √(5–√5)/2
Measures (edge length 1)
Circumradius	$\frac{\sqrt{11-4\sqrt5}}{2} \approx 0.71689$
Volume	$2\frac{45-17\sqrt5}{3} \approx 4.65790$
Dihedral angles	5/2–10/3: $\arccos\left(-\frac{\sqrt5}{5}\right) \approx 116.56505^\circ$
	3–10/3: $\arccos\left(-\sqrt{\frac{5-2\sqrt5}{15}}\right) \approx 100.81232^\circ$
Central density	10
Number of external pieces	180
Level of complexity	13
Related polytopes
Army	Semi-uniform Ti, edge lengths $\sqrt5-2$ (pentagons), $\frac{3-\sqrt5}{2}$ (between ditrigons)
Regiment	Gaddid
Dual	Great dodecacronic hexecontahedron
Conjugate	Small dodecicosidodecahedron
Convex core	Truncated dodecahedron
Abstract & topological properties
Flag count	480
Euler characteristic	–16
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

Vertex coordinates

A great dodecicosidodecahedron of edge length 1 has vertex coordinates given by all permutations of

• $\left(±\frac{\sqrt5-2}{2},\,±\frac12,\,±\frac12\right),$

along with all even permutations of

• $\left(0,\,±\frac{3-\sqrt5}{4},\,±\frac{5-\sqrt5}{4}\right),$
• $\left(±\frac{\sqrt5-1}{4},\,±\frac{\sqrt5-1}{2},\,±\frac{3-\sqrt5}{4}\right).$

Related polyhedra

The great dodecicosidodecahedron is the colonel of a three-member regiment that also includes the quasirhombicosidodecahedron and the great rhombidodecahedron.

o5/3o5/2o3*a truncations

Name	OBSA	CD diagram	Picture
Great complex icosidodecahedron (degenerate, sissid+gike)	gacid	x5/3o5/2o3*a ()
Great dodecicosidodecahedron	gaddid	x5/3x5/2o3*a ()
(degenerate, double cover of gissid)		o5/3x5/2o3*a ()
(degenerate, ditdid+gidtid)		o5/3x5/2x3*a ()
Great complex icosidodecahedron (degenerate, sissid+gike)	gacid	o5/3o5/2x3*a ()
(degenerate, double cover of sidhei)		x5/3o5/2x3*a ()
(degenerate, giddy+12(10/2))		x5/3x5/2x3*a ()
Great snub dodecicosidodecahedron	gisdid	s5/3s5/2s2*a ()

External links

• Bowers, Jonathan. "Polyhedron Category 4: Trapeziverts" (#54).

• Klitzing, Richard. "gaddid".

• Wikipedia Contributors. "Great dodecicosidodecahedron".
• McCooey, David. "Great Dodecicosidodecahedron"