polyhedral symmetries  ( 1/2 - groups )

A model is symmetric if it looks the same from different viewpoints. The set of the symmetries of a polyhedron has a group structure for the composition (combining two symmetries means applying the first, then the second on the result, in a given order):
  •   each composition of two symmetries is a symmetry (the composition of a and b is denoted by  a.b, as a multiplication)
  •   the composition is associative (the order of the groupings doesn't matter): (a.b).c = a.(b.c)
       but not commutative (the order of the symmetries matter): in general a.b  is different from  b.a
  •   there is a symmetry which has no effect, the identity denoted by 1 (as for a multiplication): we have always  a.1 = 1.a = a
  •   for every symmetry s  there is a symmetry s' which neutralizes it: s.s' = s'.s = 1
A group is characterized by its table (analogue to a multiplication table) which summarized all the the compositions and which is a "Latin square" (each element appears exactly once in each line and in each column).
Classic group's notations are: Cn cyclic group, Dn symmetry group of the regular n-gone, Sn group of the n! permutations of a set of n elements (and it's sub-group An of the even permutations) ...
Groups of same order with the same structure (thus the same table) are isomorphic:  D1  C2  S2, D2  K, D3  S3 ...

Here are the tables of the simplest groups; the smallest non commutative group is D3 (the second group with six elements is C6).

C1 1
1 1

 
C2 1 r
1 1 r
r r 1
C3 1 r r2
1 1 r r2
r r r2 1
r2 r2 1 r
C4 1 r r2 r3
1 1 r r2 r3
r r r2 r3 1
r2 r2 r3 1 r
r3 r3 1 r r2
K 1 r m i
1 1 r m i
r r 1 i m
m m i 1 r
i i m r 1

Klein's group

C5 1 r r2 r3 r4
1 1 r r2 r3 r4
r r r2 r3 r4 1
r2 r2 r3 r4 1 r
r3 r3 r4 1 r r2
r4 r4 1 r r2 r3
D3 1 r r2 m m' m"
1 1 r r2 m m' m"
r r r2 1 m" m m'
r2 r2 1 r m' m" m
m m m' m" 1 r r2
m' m' m" m r2 1 r
m" m" m m' r r2 1
The complexity of these tables grows quickly with the number of elements (examples: tables of the tetrahedral groups).
We can also build direct products of groups  G x H = { gh | g∈G, h∈H } and quotient groups  G/H = { classes wrt a normal sub-group H }.

A direct symmetry carries the model from one position to a different but indistinguishable position; it is a rotation, denoted rn , about an axis (n-fold axis) through an angle of 360°/n; after n rotations the model comes back to its initial position, we performed the identity  r1 = 1.
A half-turn  r2  is a reflection about an axis: r2.r2 = 1.
The set of the rotations of a polyhedron is a group, and all the axes are concurrent.

The effects of indirect symmetries cannot be seen with a simple manipulation of the model but requires a mirror.
A plane reflection is denoted m, and we have  m.m = 1m.m'  is a rotation about the intersection of the two planes, through an angle double of the dihedral angle of the two planes.  m'.m  is the rotation in the opposition direction.
A rotation-reflection is a composition  s2n = r2n.m = m.r2n, the axis and the plane are orthogonal (notice that  s2n.s2n = rn ).
s2 = -1, denoted i, is the central inversion (reflection in a point), thus  i.i = 1.
The set of all symmetries is a group and the direct symmetries build a sub-group of it.  Italic characters are used for polyhedral groups.

polyhedral rotational groups

There are three other spherical groups (without preferred axis); they contain Klein's sub-groups.

the other polyhedral groups

We can introduce one or several indirect isometries (mostly plane reflections) in each of the six rotational groups C1 , Cn , Dn , T , O  and I  to obtain the others polyhedral groups. Models without mirror plane are chiral; they come in two forms (mirror images) called enantiomorphs.

Thus there are 17 polyhedral symmetry types:
C1   Cs   Ci
Cn   Cnv   Cnh   Dn   S2n   Dnv   Dnh
T   Td   Th   O   Oh
I   Ih

(prismatic types)
(cubic types)
(icosahedral types)
Remark: D3, C3v  and C3h  have the same structure (they are not cyclic, and there are only two groups with six elements: C6  and D3 )

details about the groups of the regular polyhedra

interesting results

Each finite space symmetry group is a sub-group of one of the five groups  Dnh   Dnv   Td   Oh   and  Ih
 
The best way to gain a good understanding of the different types of polyhedral symmetries is to use a set of models; decorating the lateral faces of an hexagonal prism (resp. the faces of a cube) with different patterns allows you to exhibit all the prismatic (resp. cubic) groups.


C6


C6v


C6h


D6


S6


D3v


D6h


T


Td


Th


O


Oh

The orbit-stabiliser theorem:   |G|=|Orb|x|Stab|
The number of symmetries is the the number of equivalent "objects" multiplied by the number of symmetries of each object.
example: the six equivalent faces of a cube have D4 as stabiliser (eight symmetries), its eight equivalent vertices have D3 as stabiliser (six symmetries) and its twelve equivalent edges have K as stabiliser; thus the cube has 6x8=8x6=12x4=48 symmetries.

The surface of a regular polyhedron can be covered with identical right angled triangles in number equal to the order of the corresponding polyhedral group (6x4=24 for the tetrahedron, 6x8=8x6=48 for the octahedron and the cube, and 6x20=10x12=120 for the icosahedron and the dodecahedron). For each couple of these triangles there is one symmetry in the group which transforms the first into the second (if the triangles are both dark/light coloured it is a rotations), and we get all the triangles by transforming any of them with all the symmetries of the group.
proof: A plane isometry is defined by a triangle and its image; likewise an isometry of the space is defined by a tetrahedron and its image.
If we cut the polyhedron into tetrahedra with bases the triangles and apex the centre of the polyhedron, an isometry of the polyhedron maps one of these tetrahedra on an other because the centre is invariant, a vertex is transformed into a vertex, a center of a face into a center of a face and a midpoint of an edge into a midpoint of an edge. Thus we have a bijection between the isometries of the polyhedron and the triangles.


Assemblages of polyhedra, like kaleidocycles, have symmetry groups which are often direct products of groups: the group of the regular kaleidocycle of order 8 is
            D4h x C2  = D4 x C2 x C2 = {1,r,r²,r³,m,mr,mr²,mr³} x {1,μ} x {1,ω} 
where r is a 90░ rotation around the axis δ of the kaleidocycle (shown in grey), m a reflection with respect to a plane going through δ and two opposite edges, μ the reflection with respect to a plane orthogonal to δ, and ω the 180░ rotation of the ring around itself (it "turns upside down" each tetrahedron but preserves the ring as a whole); thus there are 8x2x2=32 isometries.

references:   Polyhedra (pages 289-318) by Peter R. Cromwell (Cambridge University Press, 1997). A decision tree.
Point groups in three dimensions on www.answers.com
Point Groups and Space Groups in Geometric Algebra  by David Hestenes.
simplest examples of canonical polyhedra with one of the 17 types of symmetry,
17 Types of Symmetry : web pages by Adrian Rossiter
the regular polyhedra of index two : web page by David A. Richter
les symÚtries en cristallographie (introduction Ó la cristallographie - www.kasuku.ch) in French
symmetry, a slide show (PPS, 4.3 Mb) by George Hart: groups, sculptures (M.C. Escher), 3D printing (spheres)
transitivity (second page)

  summary   March 2004
updated 13-07-2008