Great dodecicosidodecahedron

Great dodecicosidodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
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	${\frac {\sqrt {11-4{\sqrt {5}}}}{2}}\approx 0.71689$
Volume	$2{\frac {45-17{\sqrt {5}}}{3}}\approx 4.65790$
Dihedral angles	5/2–10/3: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
	3–10/3: $\arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }$
Central density	10
Number of external pieces	180
Level of complexity	13
Related polytopes
Army	Semi-uniform Ti, edge lengths ${\sqrt {5}}-2$ (pentagons), ${\frac {3-{\sqrt {5}}}{2}}$ (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

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.

Vertex coordinates

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

• $\left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),$

along with all even permutations of

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

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"