Quasitruncated great stellated dodecahedron

The quasitruncated great stellated dodecahedron, or quit gissid, also called the great stellated truncated dodecahedron, is a uniform polyhedron. It consists of 20 triangles and 12 decagrams. Each vertex joins one triangle and two decagrams. As the name suggests, it can be obtained by quasitruncation of the great stellated dodecahedron.

Quasitruncated great stellated dodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Quit gissid
Coxeter diagram	x5/3x3o ()
Elements
Faces	20 triangles, 12 decagrams
Edges	30+60
Vertices	60
Vertex figure	Isosceles triangle, edge lengths 1, √(5–√5)/2, √(5–√5)/2
Measures (edge length 1)
Circumradius	$\sqrt{\frac{37-15\sqrt5}{8}} ≈ 0.65755$
Volume	$5\frac{47\sqrt5-99}{12} ≈ 2.53966$
Dihedral angles	10/3–3: $\arccos\left(\sqrt{\frac{5-2\sqrt5}{15}}\right) ≈ 79.18768^\circ$
	10/3–10/3: $\arccos\left(\frac{\sqrt5}{5}\right) ≈ 63.43494^\circ$
Central density	13
Number of external pieces	120
Level of complexity	9
Related polytopes
Army	Semi-uniform Srid, edge lengths $\sqrt5-2$ (pentagons), $\frac{3-\sqrt5}{2}$ (triangles)
Regiment	Quit gissid
Dual	Great triakis icosahedron
Conjugate	Truncated dodecahedron
Convex core	Dodecahedron
Abstract & topological properties
Flag count	360
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

Vertex coordinates

A quasitruncated great stellated dodecahedron of edge length 1 has vertex coordinates given by all even permutations of:

• $\left(0,\,±\frac12,\,±\frac{3\sqrt5-5}{4}\right),$
• $\left(±\frac12,\,±\frac{3-\sqrt5}{4},\,±\frac{3-\sqrt5}{2}\right),$
• $\left(±\frac{3-\sqrt5}{4},\,±\frac{\sqrt5-1}{2},\,±\frac{\sqrt5-2}{2}\right).$

Related polyhedra

o5/3o3o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Great stellated dodecahedron	gissid	{5/3,3}	x5/3o3o ()
Quasitruncated great stellated dodecahedron	quit gissid	t{5/3,3}	x5/3x3o ()
Great icosidodecahedron	gid	r{3,5/3}	o5/3x3o ()
Truncated great icosahedron	tiggy	t{3,5/3}	o5/3x3x ()
Great icosahedron	gike	{3,5/3}	o5/3o3x ()
Quasirhombicosidodecahedron	qrid	rr{3,5/3}	x5/3o3x ()
Great quasitruncated icosidodecahedron	gaquatid	tr{3,5/3}	x5/3x3x ()
Great inverted snub icosidodecahedron	gisid	sr{3,5/3}	s5/3s3s ()

External links

• Bowers, Jonathan. "Polyhedron Category 2: Truncates" (#19).

• Klitzing, Richard. "Quit gissid".

• Wikipedia Contributors. "Great stellated truncated dodecahedron".
• McCooey, David. "Great Stellated Truncated Dodecahedron"