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Figure 2.1. Frieze Patterns
26 2. GROUPS AND SYMMETRY
Example 2.23. Let †" R2 be an equilateral triangle with vertices A, B, C.
A
·
O
B C
A symmetry of is defined once we know where the vertices go, hence there are as many symme-
tries as permutations of the set {A, B, C}. Each symmetry can be described using permutation
notation and we obtain the six distinct symmetries
A B C A B C A B C
= ¹, = (A B C), = (A C B),
A B C B C A C A B
A B C A B C A B C
= (B C), = (A C), = (A B).
A C B C B A B A C
Therefore we have | Euc(2) | = 6. Notice that the identity and the two 3-cycles represent
rotations about O, while each of the three transpositions represents a reflection in lines through
O and a vertex.
Example 2.24. Let †" R2 be the square centred at the origin O and whose vertices are
at the points A(1, 1), B(-1, 1), C(-1, -1), D(1, -1).
B A
·
O
C D
Then a symmetry is defined by sending A to any one of the 4 vertices then choosing how to send
B to one of the 2 adjacent vertices. This gives a total of 4 × 2 = 8 such symmetries, therefore
| Euc(2) | = 8.
Again we can describe symmetries in terms of their effect on the vertices. Here are the eight
elements of Euc(2) described in permutation notation.
A B C D A B C D
= ¹, = (A B C D),
A B C D B C D A
A B C D A B C D
= (A C)(B D), = (A D C B),
C D A B D A B C
A B C D A B C D
= (B D), = (A D)(B C),
A D C B D C B A
A B C D A B C D
= (A C), = (A B)(C D).
C B A D B A D C
Each of the two 4-cycles represents a rotation through a quarter turn about O, while (A C)(B D)
represents a half turn. The transpositions (B D) and (A C) represent reflections in the diagonals
while (A D)(B C) and (A B)(C D) represent reflections in the lines joining opposite midpoints
of edges.
4. SYMMETRY GROUPS OF PLANE FIGURES 27
Example 2.25. Let R †" R2 be the rectangle centred at the origin O with vertices at A(2, 1),
B(-2, 1), C(-2, -1), D(2, -1).
B A
O·
C D
A symmetry can send A to any of the vertices, and then the long edge AB must go to the longer
of the adjacent edges. This gives a total of 4 such symmetries, thus | Euc(2)R| = 4.
Again we can describe symmetries in terms of their effect on the vertices. Here are the four
elements of Euc(2)R described using permutation notation.
A B C D A B C D
= ¹, = (A B)(C D),
A B C D B A D C
A B C D A B C D
= (A C)(B D), = (A D)(B C).
C D A B D C B A
(A C)(B D) represents a half turn about O while (A B)(C D) and (A D)(B C) represent
reflections in lines joining opposite midpoints of edges.
Example 2.26. Given a regular n-gon (i.e., a regular polygon with n sides all of the same
length and n vertices V1, V2, . . . , Vn), the symmetry group is a dihedral group of order 2n, with
elements
¹, ±, ±2, . . . , ±n-1, ², ±², ±2², . . . , ±n-1²,
where ±k is an anticlockwise rotation through 2Àk/n about the centre and ² is a reflection in
the line through V1 and the centre. In fact each of the elements ±2² is a reflection in a line
through the centre. Moreover we have
|±| = n, |²| = 2, ²±² = ±n-1 = ±-1.
In permutation notation this becomes the n-cycle
± = (V1 V2 · · · Vn),
but ² is more complicated to describe since it depends on whether n is even or odd.
For example, if n = 6 we have
± = (V1 V2 V3 V4 V5 V6), ² = (V2 V6)(V3 V5),
while if n = 7
± = (V1 V2 V3 V4 V5 V6 V7), ² = (V2 V7)(V3 V6)(V4 V5).
We have seen that when n = 3, Euc(2) is the permutation group of the vertices and so D6 is
essentially the same group as S6.
If we take the regular n-gon centred at the origin with the first vertex V1 at (1, 0), the
generators ± and ² can be represented as (A | 0) and (B | 0) using the matrices
cos 2À/n - sin 2À/n 1 0
A = , B =
sin 2À/n cos 2À/n 0 -1
In this case the symmetry group is the dihedral group of order 2n,
D2n = {¹, ±, ±2, . . . , ±n-1, ², ±², ±2², . . . , ±n-1²} O(2).
Notice that the subgroup of direct symmetries is
D+ = {¹, ±, ±2, . . . , ±n-1} SO(2).
2n
28 2. GROUPS AND SYMMETRY
More generally we have the following Theorem. A convex region of is a subset S †" R2 in
which for each pair of points x, y " S, the line segment joining them lies in S, i.e.,
{tx + (1 - t)y : 0 t 1} †" S.
Convex Non-convex
Theorem 2.27. If V1, . . . , Vn are the vertices in order of a polygon which bounds a convex
region P of R2 containing a point not on the boundary, then Euc(2)P can be identified with a
subgroup of the permutation group Perm{V1,...,Vn} of the vertices.
5. Similarity of isometries and subgroups of the Euclidean group
It is often the case that two subsets of the plane have the same symmetry subgroups. For
example, any two frieze patterns which only have translational symmetries are the same in this
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