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 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 n-gone, S_n group of the n! permutations of a set of n elements (and it's sub-group A_n 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).
In these examples (geometric transformations of the plane) 1 denotes the identity, r a rotation, c the central symmetry and m a reflection.

C₁	1
1	1

C₂	1	c
1	1	c
c	c	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	c	m	m'
1	1	c	m	m'
c	c	1	m'	m
m	m	m'	1	c
m'	m'	m	c	1

Klein's group

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 r_n , 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 = 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 rotation-reflection 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 ).
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

cyclic symmetry: C_n contains n rotations, among which r_n and 1 ; there is only one n-fold axis.
C₁ is a special case: the only symmetry is the identity (the model is asymmetric).
remark: C_n is isomorphic to the group C_n of a rosette (where a rotation center takes the place of the axis) and to the quotient group Z/nZ = Z_n.
dihedral symmetry: D_n contains C_n and n half-turns. (Be careful! Don't mistake it with the symmetry group D_n of the regular n-gone.)
Besides the n-fold axis there are n reflection axis (2-fold axes) perpendicular to the n-fold axis, with angles of 360°/n.
When n is even the secondary axes separate into two sorts.
D₂ is a special case where the three reflection axes play the same role; they are two by two perpendicular (it's a Klein's group which may be considered as a spherical group).

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

tetrahedral symmetry: T (4x2+3+1=12 rotations), isomorphic to A₄.
The regular tetrahedron has four 3-fold axes (going through a vertex and the center of the opposite face) and three reflection axes two by two perpendicular (going through the midpoints of two opposite edges).
octahedral symmetry: O (3x3+4x2+6+1=24 rotations), isomorphic to S₄.
The regular octahedron has three 4-fold axes two by two perpendicular (going through two opposite vertices), four 3-fold axes (going through the centers of two opposite faces) and six reflection axes (going through the midpoints of two opposite edges).
icosahedral symmetry: I (6x4+10x2+15+1=60 rotations), isomorphic to A₅.
The regular icosahedron has six 5-fold axes (going through two opposite vertices), ten 3-fold axes (going through the centers of two opposite faces) and fifteen reflection axes (going through the midpoints of two opposite edges).

the other polyhedral groups

We can introduce one or several indirect isometries (mostly plane reflections) in each of the six rotational groups C₁ , 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_s contains only one plane reflection and the identity ("_s" comes from the German word "Spiegel", for mirror).
C_nv contains n plane reflections, each plane contains the n-fold principal axis.
It is the group of the regular n-pyramid (2n symmetries), isomorphic to D_n.
C_nh contains only one plane reflection, the plane is orthogonal to the n-fold principal axis (2n symmetries).
When n is even one of the rotation-reflections is the central inversion. C_2h is a Klein's group (see table above).
S_2n doesn't contain a plane reflection, but the n-fold principal axis is a 2n-fold rotation-reflection axis (2n symmetries).
When n is odd one of the rotation-reflections is the central inversion.
S₂ is a special case denoted C_i because, besides the identity, the only 2-fold rotation-reflection is the central inversion i.
C_i is the group of the parallelepiped (neither rectangular nor equifacial), isomorphic to Z₂.
D_nv contains n plane reflections, the planes contain the principal n-fold axis (which is also rotation-reflection axis) and are interleaved with the reflection axes. When n is odd the 2-fold rotation-reflection is the central inversion.
It is the group of the regular n-antiprism (4n symmetries), isomorphic to D_2n for n≠3.
D_nh contains (n+1) plane reflections (n planes contain the principal n-fold axis and one reflection axis, the last is orthogonal to them). When n is even the 2-fold rotation-reflection is the central inversion.
It is the group of the regular n-prism (4n symmetries),isomorphic to D_n x C₂ for n≠4.
T_h contains three reflections with respect to planes two by two orthogonal and the central inversion (24 symmetries); it is isomorphic to A₄ x C₂.
T_d contains six plane reflections, the planes contain one edge and are orthogonal to the opposite edge.
It is the group of the regular tetrahedron (24 symmetries), isomorphic to S₄.
O_h contains 3+6=9 plane reflections and the central inversion.
It is the group of the regular octahedron and of the cube (48 symmetries), isomorphic to S₄ x C₂.
I_h contains 5x3=15 plane reflections - five sets of three planes two by two orthogonal, each one contains two opposite edges - and the central inversion.
It is the group of the regular icosahedron and of the regular dodecahedron (120 symmetries), isomorphic to A₅ x C₂.

Thus there are 17 polyhedral symmetry types:

C₁ 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: D₃ C_3v and C_3h 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 D_nh D_nv T_d O_h and I_h

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

C_6v

C_6h

D₆

D_3v

D_6h

T_d

T_h

O_h

The orbit-stabiliser theorem: |G|=|Orb|x|Stab|
The number of symmetries is 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 stabilizer (eight symmetries), its eight equivalent vertices have D₃ as stabilizer (six symmetries) and its twelve equivalent edges have K as stabilizer; 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 the same color dark/light 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 center of the polyhedron, an isometry of the polyhedron maps one of these tetrahedra on an other because the center 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 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 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)

transitivity (second page)

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convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects

March 2004
updated 13-07-2008