Great dodecahedron

Great dodecahedron
	(3D model); (OFF file)
Rank	3
Type	Regular
Notation
Bowers style acronym	Gad
Coxeter diagram
Schläfli symbol	;
Elements
Faces	12 pentagons
Edges	30
Vertices	12
Vertex figure	Pentagram, edge length (1+√5)/2 ;
Petrie polygons	10 skew hexagons
Holes	20 triangles
Measures (edge length 1)
Circumradius
Edge radius
Inradius
Volume
Dihedral angle
Central density	3
Number of external pieces	60
Level of complexity	3
Related polytopes
Army	Ike
Regiment	Ike
Dual	Small stellated dodecahedron
Petrie dual	Petrial great dodecahedron
φ 2	Icosahedron
Conjugate	Small stellated dodecahedron
Convex core	Dodecahedron
Abstract & topological properties
Flag count	120
Euler characteristic	–6
Schläfli type	{5,5}
Surface	Bring's surface
Orientable	Yes
Genus	4
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame
History
Discovered by	Johannes Kepler
First discovered	1613

The great dodecahedron, or gad, is one of the four Kepler-Poinsot solids. It has 12 pentagons as faces, joining 5 to a vertex in a pentagrammic fashion.

It is in the same regiment as the icosahedron, and comes from using the icosahedron's vertex figure pentagons as the faces.

It is the second stellation of the dodecahedron.

Great dodecahedra appear as cells in two star regular polychora, namely the great hecatonicosachoron and great grand hecatonicosachoron.

Vertex coordinates[edit | edit source]

Its vertices are the same as those of its regiment colonel, the icosahedron.

The great dodecahedron appears as a vertex figure of two Schläfli–Hess polychora.

Name	Picture	Schläfli symbol	Edge length
Grand stellated hecatonicosachoron		{5/2,5,5/2}	${\frac {{\sqrt {5}}-1}{2}}$
Faceted hexacosichoron		{3,5,5/2}	$1$

Abstractly the great dodecahedron is a quotient of the order-5 pentagonal tiling. Specifically it is $\{5,5\mid 3\}$ , a tessellation of Bring's surface. It is also abstractly equivalent to its conjugate, the small stellated dodecahedron.

Two uniform polyhedron compounds are composed of great dodecahedra:

↑ Earlier authors drew shapes resembling the great dodecahedron or similar shapes earlier, however Kepler was the first to recognize the great dodecahedron as regular, and explicitly describe it.

McMullen, Peter (1998). "The groups of regular star polytopes" (PDF). Canadian Journal of Mathematics. 50: 426–448. doi:10.4153/CJM-1998-023-7.

v t e truncations
Name	OBSA	Schläfli symbol	CD diagram	Image
Great dodecahedron	gad	${5,5/2}$
Truncated great dodecahedron	tigid	t ${5,5/2}$
Dodecadodecahedron	did	r ${5,5/2}$
Truncated small stellated dodecahedron = (degenerate, triple cover of doe)		t ${5/2,5}$
Small stellated dodecahedron	sissid	${5/2,5}$
Rhombidodecadodecahedron	raded	rr ${5,5/2}$
Truncated dodecadodecahedron = (degenerate, sird+12(10/2))		tr ${5,5/2}$
Snub dodecadodecahedron	siddid	sr ${5,5/2}$

Great dodecahedron
(3D model) (OFF file)
Rank	3
Type	Regular
Notation
Bowers style acronym	Gad
Coxeter diagram
Schläfli symbol	$\{5,5/2\}$ $\{5,5\mid 3\}$ ^[1]
Elements
Faces	12 pentagons
Edges	30
Vertices	12
Vertex figure	Pentagram, edge length (1+√5)/2
Petrie polygons	10 skew hexagons
Holes	20 triangles
Measures (edge length 1)
Circumradius	${\sqrt {\frac {5+{\sqrt {5}}}{8}}}\approx 0.95106$
Edge radius	${\frac {1+{\sqrt {5}}}{4}}\approx 0.80902$
Inradius	${\sqrt {\frac {5+{\sqrt {5}}}{40}}}\approx 0.42533$
Volume	${\frac {5+3{\sqrt {5}}}{4}}\approx 2.92705$
Dihedral angle	$\arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 63.43495^{\circ }$
Central density	3
Number of external pieces	60
Level of complexity	3
Related polytopes
Army	Ike
Regiment	Ike
Dual	Small stellated dodecahedron
Petrie dual	Petrial great dodecahedron
$φ 2$	Icosahedron
Conjugate	Small stellated dodecahedron
Convex core	Dodecahedron
Abstract & topological properties
Flag count	120
Euler characteristic	–6
Schläfli type	${5,5}$
Surface	Bring's surface
Orientable	Yes
Genus	4
Properties
Symmetry	H₃, order 120
Convex	No
Nature	Tame
History
Discovered by	Johannes Kepler^{[note 1]}
First discovered	1613