Simplest Canonical Polyhedra of Each Symmetry Type
There are 17 possible types of symmetry that a polyhedron can have.
The following table shows the characteristics of these symmetry types.
Symmetry Types in 3 Dimensions
Schönflies
Orbifold
Symmetry Name
Rotation Axes
Highest Rotation Axis Order
Mirror Planes
Rotations
Improper Rotations
Invertible?
C1
1
No Symmetry
0
1
0
1
0
no
Ci=S2
x
Inversion
0
1
0
1
1
yes
Cs=C1h=C1v
*
Reflection
0
1
1
1
1
no
Cn
nn
Cyclic
1
n
0
n
0
no
Cnh (n odd)
n*
Cyclic + Orthogonal Reflection
1
n
1
n
n
no
Cnh (n even)
n*
Cyclic + Orthogonal Reflection
1
n
1
n
n
yes
Cnv
*nn
Pyramidal
1
n
n
n
n
no
S2n (n even)
nx
Rotoreflection
1
n
0
n
n
no
S2n (n odd)
nx
Rotoreflection
1
n
0
n
n
yes
Dn
22n
Dihedral
n+1
n
0
2*n
0
no
Dnh (n odd)
*22n
Prismatic
n+1
n
n+1
2*n
2*n
no
Dnh (n even)
*22n
Prismatic
n+1
n
n+1
2*n
2*n
yes
Dnv (n even)
2*n
Antiprismatic
n+1
n
n
2*n
2*n
no
Dnv (n odd)
2*n
Antiprismatic
n+1
n
n
2*n
2*n
yes
T
332
Chiral Tetrahedral
7
3
0
12
0
no
Td
*332
Full Tetrahedral
7
3
6
12
12
no
Th
3*2
Pyritohedral
7
3
3
12
12
yes
O
432
Chiral Octahedral
13
4
0
24
0
no
Oh
*432
Full Octahedral
13
4
9
24
24
yes
I
532
Chiral Icosahedral
31
5
0
60
0
no
Ih
*532
Full Icosahedral
31
5
15
60
60
yes
The simplest canonical polyhedra of each symmetry type can be accessed
using the links below. Simplicity is based on the number of edges in the
polyhedron.