Tetrahedral symmetry

Tetrahedral symmetry
Rank	3
Space	Spherical
Order	24
Info
Coxeter diagram
Elements
Axes	3 × (BC2×A1)/2, 4 × A2×I
Related polytopes
Omnitruncate	Great rhombitetratetrahedron

Tetrahedral symmetry, also known as tettic symmetry and notated A₃, is a 3D spherical Coxeter group. It is the symmetry group of the regular tetrahedron.

Subgroups

• Chiral tetrahedral symmetry (maximal)
• Triangular pyramidal symmetry (maximal)
• Chiral triangular pyramidal symmetry
• Digonal antiprismatic symmetry (maximal)
• Prodigonal antiprismatic symmetry
• Chiral digonal prismatic symmetry
• Rectangular pyramidal symmetry
• Chiral digonal pyramidal symmetry
• Reflection symmetry
• Identity symmetry

Convex polytopes with A₃ symmetry

• Tetrahedron (regular)
• Tetratetrahedron (isogonal)/Rhombic hexahedron (isotopic)
• Truncated tetrahedron (isogonal)/Triakis tetrahedron (isotopic)
• Rhombitetratetrahedron (isogonal)/Deltoidal dodecahedron (isotopic)
• Great rhombitetratetrahedron (isogonal)/Disdyakis hexahedron (isotopic)

Wythoffians with A₃ symmetry

o3o3o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Tetrahedron	tet	{3,3}	x3o3o ()
Truncated tetrahedron	tut	t{3,3}	x3x3o ()
Tetratetrahedron = Octahedron	oct	r{3,3}	o3x3o ()
Truncated tetrahedron	tut	t{3,3}	o3x3x ()
Tetrahedron	tet	{3,3}	o3o3x ()
Small rhombitetratetrahedron = Cuboctahedron	co	rr{3,3}	x3o3x ()
Great rhombitetratetrahedron = Truncated octahedron	toe	tr{3,3}	x3x3x ()
Snub tetrahedron = Icosahedron	ike	sr{3,3}	s3s3s ()