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: C_{n} cyclic group, D_{n} symmetry group of the regular ngone, S_{n} group of the n! permutations of a set of n elements (and it's subgroup A_{n} of the even permutations) ...
Groups of same order with the same structure (thus the same table) are isomorphic: D_{1} ≡ C_{2} ≡ S_{2}, D_{2} ≡ K, D_{3} ≡ S_{3} ...
Here are the tables of the simplest groups; the smallest non commutative group is D_{3} (the second group with six elements is C_{6}).
In these examples (geometric transformations of the plane) 1 denotes the identity, r a rotation, c the central symmetry and m a reflection.





A direct symmetry carries the model from one position to a different but indistinguishable position; it is a rotation, denoted r_{n} , about an axis (nfold axis) through an angle of 360°/n; after n rotations the model comes back to its initial position, we performed the identity r_{1} = 1.
A halfturn r_{2} is a reflection about an axis: r_{2}.r_{2} = 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 = 1. m.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 rotationreflection is a composition s_{2n} = r_{2n}.m = m.r_{2n}, the axis and the plane are orthogonal (notice that s_{2n}.s_{2n} = r_{n} ).
s_{2} = 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 subgroup of it. Italic characters are used for polyhedral groups.
We can introduce one or several indirect isometries (mostly plane reflections) in each of the six rotational groups C_{1} , C_{n} , D_{n} , 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.
C_{1} C_{s} C_{i}
C_{n} C_{nv} C_{nh} D_{n} S_{2n} D_{nv} D_{nh} T T_{d} T_{h} O O_{h} I I_{h} 
(prismatic types) (cubic types) (icosahedral types) 
Remark: D_{3} C_{3v} and C_{3h} have the same structure (they are not cyclic, and there are only two groups with six elements: C_{6} et D_{3 })
details about the groups of the regular polyhedra 
C_{6} 
C_{6v} 
C_{6h} 
D_{6} 
D_{3v} 
D_{6h} 
T 
T_{d} 
T_{h} 
O 
O_{h} 
The orbitstabiliser theorem: G=OrbxStab
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 D_{4} as stabiliser (eight symmetries), its eight equivalent vertices have D_{3} 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
D_{4h} x C_{2} = D_{4} x C_{2} x C_{2} = {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 289318) 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 to illustrate the 17 types of symmetry by David I. McCooey • 17 Types of Symmetry : web pages by Adrian Rossiter • the regular polyhedra of index two : web page by David A. Richter • les symétries en cristallographie (introduction to the cristallography)  www.kasuku.ch) in French • symmetry, a slide show (PPS, 4.3 Mb) by George Hart: groups, sculptures (M.C. Escher), 3D printing (spheres) 
March 2004 updated 13072008 