Great dodecahedron

Great dodecahedron
(3D model) (OFF file)
Rank	3
Type	Regular
Notation
Bowers style acronym	Gad
Coxeter diagram	o5/2o5x ()
Schläfli symbol	{5,5/2} {5,5∣3}
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
Skeleton	Icosahedral graph
Properties
Symmetry	H3, order 120
Flag orbits	1
Convex	No
Nature	Tame

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

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

Related polytopes

Alternative realizations

A fundamental domain of the great dodecahedron in {5,5}.

The great dodecahedron and the small stellated dodecahedron are conjugates. Thus they are both faithful symmetric realizations of the same abstract regular polytope, {5,5∣3}. This abstract polytope is a quotient of the order-5 pentagonal tiling which tessellates Bring's surface. There are in total 6 faithful symmetric realizations of the underlying abstract polytope. The great dodecahedron and the small stellated dodecahedron are the only pure faithfully symmetric realizations, the others are the results of blending those two along with {5,5∣3}/2.

Faithful symmetric realizations of {5,5∣3}

Dimension	Components	Name
3	Great dodecahedron	Great dodecahedron
3	Small stellated dodecahedron	Small stellated dodecahedron
6	Great dodecahedron Small stellated dodecahedron
8	Great dodecahedron {5,5∣3}/2
8	Small stellated dodecahedron {5,5∣3}/2
11	Great dodecahedron Small stellated dodecahedron {5,5∣3}/2

Compounds

Two uniform polyhedron compounds are composed of great dodecahedra:

• Pentagonal retrosnub pseudodisoctahedron (2)
• Pentagonal retrosnub pseudicosicosahedron (5)

In vertex figures

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}	$1$

External links

• Bowers, Jonathan. "Polyhedron Category 1: Regulars" (#6).

• Bowers, Jonathan. "Batch 2: Ike and Sissid Facetings" (#2 under ike).

• Klitzing, Richard. "Gad".
• Nan Ma. "Great dodecahedron {5, 5/2}".

• Wikipedia contributors. "Great dodecahedron".
• McCooey, David. "Great Dodecahedron"

• Hartley, Michael. "{5,5}*120".
• Wedd, N. S4:{5,5}

Bibliography

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