Great quasitruncated icosidodecahedron

The great quasitruncated icosidodecahedron or gaquatid, also called the great truncated icosidodecahedron, is a uniform polyhedron. It consists of 12 decagrams, 20 hexagons, and 30 squares, with one of each type of face meeting per vertex. It can be obtained by quasicantitruncation of the great stellated dodecahedron or great icosahedron, or equivalently by quasitruncating the vertices of a great icosidodecahedron and then adjusting the edge lengths to be all equal.

Great quasitruncated icosidodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Gaquatid
Coxeter diagram	x5/3x3x ()
Elements
Faces	30 squares, 20 hexagons, 12 decagrams
Edges	60+60+60
Vertices	120
Vertex figure	Scalene triangle, edge lengths √2, √3, √(5–√5)/2
Measures (edge length 1)
Circumradius	$\frac{\sqrt{31-12\sqrt5}}{2} ≈ 1.02068$
Volume	$5(10\sqrt5-19) ≈ 16.80340$
Dihedral angles	10/3–4: $\arccos\left(-\sqrt{\frac{5-\sqrt5}{10}}\right) ≈ 121.71747^\circ$
	10/3–6: $\arccos\left(\sqrt{\frac{5-2\sqrt5}{15}}\right) ≈ 79.18768^\circ$
	6–4: $\arccos\left(\frac{\sqrt{15}-\sqrt3}{6}\right) ≈ 69.09484^\circ$
Central density	13
Number of external pieces	1140
Level of complexity	58
Related polytopes
Army	Semi-uniform Grid, edge lengths $\frac{3-\sqrt5}{2}$ (dipentagon-rectangle), $\sqrt5-2$ (dipentagon-ditrigon), $\frac{7-3\sqrt5}{2}$ (ditrigon-rectangle)
Regiment	Gaquatid
Dual	Great disdyakis triacontahedron
Conjugate	Great rhombicosidodecahedron
Convex core	Icosahedron
Abstract & topological properties
Flag count	720
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

It can be alternated into the great inverted snub icosidodecahedron after equalizing edge lengths.

Vertex coordinates

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

• $\left(±\frac12,\,±\frac12,\,±\frac{2\sqrt5-3}{2}\right),$

along with all even permutations of:

• $\left(±\frac12,\,±\frac{\sqrt5-2}{2},\,±\frac{4-\sqrt5}{2}\right),$
• $\left(±1,\,±\frac{3-\sqrt5}{4},\,±\frac{7-3\sqrt5}{4}\right),$
• $\left(±\frac{3-\sqrt5}{4},\,±3\frac{\sqrt5-1}{4},\,±\frac{3-\sqrt5}{2}\right),$
• $\left(±\frac{\sqrt5-1}{2},\,±\frac{3\sqrt5-5}{4},\,±\frac{5-\sqrt5}{4}\right).$

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 5: Omnitruncates" (#61).

• Klitzing, Richard. "gaquatid".

• Wikipedia Contributors. "Great truncated icosidodecahedron".
• McCooey, David. "Great Truncated Icosidodecahedron"