Truncated great dodecahedron

Truncated great dodecahedron
	(3D model); (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Tigid
Coxeter diagram	o5/2x5x ()
Elements
Faces	12 pentagrams, 12 decagons
Edges	30+60
Vertices	60
Vertex figure	Isosceles triangle, edge lengths (√5–1)/2, √(5+√5)/2, √(5+√5)/2 ;
Measures (edge length 1)
Circumradius
Volume
Dihedral angles	10–5/2:
	10–10:
Central density	3
Number of pieces	72
Level of complexity	7
Related polytopes
Army	Semi-uniform Ti
Regiment	Tigid
Dual	Small stellapentakis dodecahedron
Conjugate	Quasitruncated small stellated dodecahedron
Convex core	Dodecahedron
Abstract properties
Euler characteristic	-6
Topological properties
Orientable	Yes
Genus	4
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The truncated great dodecahedron, or tigid, also called the great truncated dodecahedron, is a uniform polyhedron. It consists of 12 pentagrams and 12 decagons. Each vertex joins one pentagram and two decagons. As the name suggests, it can be obtained by the truncation of the great dodecahedron.

Vertex coordinates[edit | edit source]

A truncated great dodecahedron of edge length 1 has vertex coordinates given by all permutations of:

$\left(±\frac{\sqrt5-1}{4},\,±\frac{3+\sqrt5}{4},\,±\frac{3+\sqrt5}{4}\right),$

plus all even permutations of:

$\left(0,\,±\frac12,\,±\frac{5+\sqrt5}{4}\right),$
$\left(±\frac12,\,±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{2}\right).$

Related polyhedra[edit | edit source]

o5o5/2o truncations
Name	OBSA	Schläfli symbol	CD diagram
Great dodecahedron	gad	{5,5/2}	x5o5/2o ()
Truncated great dodecahedron	tigid	t{5,5/2}	x5x5/2o ()
Dodecadodecahedron	did	r{5,5/2}	o5x5/2o ()
Truncated small stellated dodecahedron (degenerate, triple cover of doe)		t{5/2,5}	o5x5/2x ()
Small stellated dodecahedron	sissid	{5/2,5}	o5o5/2x ()
Rhombidodecadodecahedron	raded	rr{5,5/2}	x5o5/2x ()
Truncated dodecadodecahedron (degenerate, sird+12(10/2))		tr{5,5/2}	x5x5/2x ()
Snub dodecadodecahedron	siddid	sr{5,5/2}	s5s5/2s ()