Dodecahedral symmetry

Dodecahedral symmetry
Rank	3
Space	Spherical
Order	120
Info
Coxeter diagram
Elements
Axes	6 × (I2(10)×A1)/2, 10 × (G2×A1)/2, 15 × K3
Related polytopes
Omnitruncate	Great rhombicosidodecahedron

Dodecahedral symmetry, also known as icosahedral symmetry, doic symmetry, and notated as H₃, is a 3D spherical Coxeter group. It is the symmetry group of the dodecahedron and icosahedron.

Subgroups

• H₃+ (maximal)
• B₃/2 (maximal)
• A₃+
• (I₂(10)×A₁)/2 (maximal)
• (I₂(10)+×A₁)/2
• (G₂×A₁)/2 (maximal)
• (G₂+×A₁)/2
• (H₂×A₁)+
• H₂×I
• H₂+×I
• (A₂×A₁)+
• A₂×I
• A₂+×I
• K₃
• K₃+
• K₂×I
• K₂+×A₁
• K₂+×I
• ±(I×I×I)
• A₁×I×I
• I×I×I

Convex polytopes with H₃ symmetry

• Dodecahedron (regular)/Icosahedron (regular)
• Icosidodecahedron (isogonal)/Rhombic triacontahedron (isotopic)
• Truncated dodecahedron (isogonal)/Triakis icosahedron (isotopic)
• Truncated icosahedron (isogonal)/Pentakis dodecahedron (isotopic)
• Small rhombicosidodecahedron (isogonal)/Deltoidal hexecontahedron (isotopic)
• Great rhombicosidodecahedron (isogonal)/Disdyakis triacontahedron (isotopic)

Wythoffians with H₃ symmetry

o5o3o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Dodecahedron	doe	{5,3}	x5o3o
Truncated dodecahedron	tid	t{5,3}	x5x3o
Icosidodecahedron	id	r{5,3}	o5x3o
Truncated icosahedron	ti	t{3,5}	o5x3x
Icosahedron	ike	{3,5}	o5o3x
Small rhombicosidodecahedron	srid	rr{5,3}	x5o3x
Great rhombicosidodecahedron	grid	tr{5,3}	x5x3x
Snub dodecahedron	snid	sr{5,3}	s5s3s

o3o5/2o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Great icosahedron	gike	{3,5/2}	x3o5/2o ()
Truncated great icosahedron	tiggy	t{3,5/2}	x3x5/2o ()
Great icosidodecahedron	gid	r{3,5/2}	o3x5/2o ()
Truncated great stellated dodecahedron (degenerate, ike+2gad)		t{5/2,3}	o3x5/2x ()
Great stellated dodecahedron	gissid	{5/2,3}	o3o5/2x ()
Small complex rhombicosidodecahedron (degenerate, sidtid+rhom)	sicdatrid	rr{3,5/2}	x3o5/2x ()
Truncated great icosidodecahedron (degenerate, ri+12(10/2))		tr{3,5/2}	x3x5/2x ()
Great snub icosidodecahedron	gosid	sr{3,5/2}	s3s5/2s ()

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 ()

o5o5/2o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
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 ()

o5/3o5o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Small stellated dodecahedron	sissid	{5/3,5}	x5/3o5o ()
Quasitruncated small stellated dodecahedron	quit sissid	t{5/3,5}	x5/3x5o ()
Dodecadodecahedron	did	r{5,5/3}	o5/3x5o ()
Truncated great dodecahedron	tigid	t{5,5/3}	o5/3x5x ()
Great dodecahedron	gad	{5,5/3}	o5/3o5x ()
Complex ditrigonal rhombidodecadodecahedron (degenerate, ditdid+rhom)	cadditradid	rr{5,5/3}	x5/3o5x ()
Quasitruncated dodecadodecahedron	quitdid	tr{5,5/3}	x5/3x5x ()
Inverted snub dodecadodecahedron	isdid	sr{5,5/3}	s5/3s5s ()

o5/2o3o3*a truncations

Name	OBSA	CD diagram	Picture
Small ditrigonary icosidodecahedron	sidtid	x5/2o3o3*a ()
(degenerate, double cover of id)		x5/2x3o3*a ()
(degenerate, double cover of ike)		o5/2o3x3*a ()
Small icosicosidodecahedron	siid	x5/2o3x3*a ()
(degenerate, double cover of ti)		x5/2x3x3*a ()
Small snub icosicosidodecahedron	seside	s5/2s3s3*a ()

o3/2o3o5*a truncations

Name	OBSA	CD diagram	Picture
Great ditrigonary icosidodecahedron	gidtid	x3/2o3o5*a ()
(degenerate, 3ike+gad)		x3/2x3o5*a ()
(degenerate, double cover of gike)		o3/2x3o5*a ()
Great icosicosidodecahedron	giid	o3/2x3x5*a ()
Great ditrigonary icosidodecahedron	gidtid	o3/2o3x5*a ()
(degenerate, double cover of seihid)		x3/2o3x5*a ()
(degenerate, siddy+20(6/2))		x3/2x3x5*a ()
(degenerate, 5ike+gad)		s3/2s3s5*a

o5/3o5/2o3*a truncations

Name	OBSA	CD diagram	Picture
Great complex icosidodecahedron (degenerate, sissid+gike)	gacid	x5/3o5/2o3*a ()
Great dodecicosidodecahedron	gaddid	x5/3x5/2o3*a ()
(degenerate, double cover of gissid)		o5/3x5/2o3*a ()
(degenerate, ditdid+gidtid)		o5/3x5/2x3*a ()
Great complex icosidodecahedron (degenerate, sissid+gike)	gacid	o5/3o5/2x3*a ()
(degenerate, double cover of sidhei)		x5/3o5/2x3*a ()
(degenerate, giddy+12(10/2))		x5/3x5/2x3*a ()
Great snub dodecicosidodecahedron	gisdid	s5/3s5/2s2*a ()