2.3. THE DIHEDRAL GROUPS
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2.3. THE DIHEDRAL GROUPS
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2.3. The Dihedral Groups
In this section, we will work out the symmetry groups of regular and of the disk, which might be thought of as a “limit” of regular polygons as the number of sides increases. We regard these figures as thin plates, capable of rotations in three dimensions. Their symmetry groups are known collectively as the dihedral groups.
We have already found the symmetry group of the equilateral trian gle (regular 3-gon) in Exercise 1.3.1 and of the square (regular 4-gon) in Sections 1.2 and 1.3. For now, it will be convenient to work first with the disk
2x3
f y
4
5 W x2 C y2 1g;
0
whose symmetry group we denote by D.
Observe that the rotation rt through any angle t around the z–axis is a symmetry of the disk. Such rotations satisfy rt rs D rtCs and in particular rt r t D r0 D e, where e is the nonmotion. It follows that N D frt W t 2 Rg is a subgroup of D.
For any line passing through the origin in the .x; y/–plane , the flip over that line (i.e., the rotation by about that line) is a symmetry of the disk, interchanging the top and bottom faces of the disk. Denote by jt the flip over the line `t which passes through the origin and the point
2cos.t/3 4 sin.t / 5; and write j D j0 for the flip over the x–axis. Each jt generates
0
a subgroup of D of order 2. Symmetries of the disk are illustrated in Figure 2.3.1.
r
j
Figure 2.3.1. Symmetries of the disk.
3A regular polygon is one all of whose sides are congruent and all of whose internal angles are congruent.
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2. BASIC THEORY OF GROUPS Next, we observe that each jt can be expressed in terms of j and the rotation rt . To perform the flip jt about the line `t , we can rotate the disk until the line `t overlays the x–axis, then perform the flip j over the x– axis, and finally rotate the disk so that `t is returned to its original position.
Thus jt D rt jr t , or jt rt D rt j . Therefore, we need only work out how to compute products involving the flip j and the rotations rt .
2 cos.s/3 2 cos. s/3
Note that j applied to a point
4
sin.s/5 in the disk is 4 sin. s/5,
0
0
2 cos.s/3 2 cos.s C t/3 and rt applied to
4
sin.s/5 is 4 sin.s C t/5:
0
0
In the Exercises, you are asked to verify the following facts about the group D: 1. j rt D r t j , and jt D r2t j D jr 2t .
2. All products in D can be computed using these relations.
3. The symmetry group D of the disk consists of the rotations rt for t 2 R and the flips jt D r2t j . Writing N D frt W t 2 Rg, we have D D N [ Nj .
4. The subgroup N of D satisfies aN a 1 D N for all a 2 D.
Next, we turn to the symmetries of the regular polygons. Consider a
2cos.2k=n/3
regular n-gon with vertices at 4sin.2k=n/5 for k D 0; 1; : : : ;
0
n
1. Denote the symmetry group of the n-gon by Dn.
In the exercises, you are asked to verify the following facts about the symmetries of the n-gon:
1. The rotation r D r2=n through an angle of 2=n about the z– axis generates a cyclic subgroup of Dn of order n.
2. The “flips” jk=n D rk2=n j D rk j , for k 2 Z, are symme tries of the n-gon.
3. The distinct flip symmetries of the n-gon are r k j for k D 0; 1; : : : ; n 1.
4. If n is odd, then the axis of each of the flips passes through a vertex of the n-gon and the midpoint of the opposite edge. See Figure 2.3.2 on the next page for the case n D 5.
5. If n is even and k is even, then jk=n D rk j is a flip about an axis passing through a pair of opposite vertices of the n-gon.
6. If n is even and k is odd, then jk=n D rk j is a flip about an axis passing through the midpoints of a pair of opposite edges
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“bookmt” — 2006/8/8 — 12:58 — page 107 — #119
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2.3. THE DIHEDRAL GROUPS
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of the n-gon. See Figure 2.3.2 on the facing page for the case n D 6.
r
r
rj
j
j
rj
Figure 2.3.2. Symmetries of the pentagon and hexagon.
The symmetry group Dn consists of the 2n symmetries rk and rkj , for 0 k n 1. It follows from our computations for the symmetries of the disk that j r D r 1j , so jrk D r kj for all k. This relation allows the computation of all products in Dn.
The group Dn can appear as the symmetry group of a geometric figure, or of a physical object, in a slightly different form. Think, for example, of a five-petalled flower, or a star-fish, which look quite different from the top and from the bottom. Or think of a pentagonal card with its two faces of different colors. Such an object or figure does not admit rotational symmetries that exchange the top and bottom faces. However, a starfish or a flower does have reflection symmetries, as well as rotational symmetries that do not exchange top and bottom.
Figure 2.3.3.
Figure 2.3.4.
Object with D9 symmetry.
Object with Z5 symmetry.
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2. BASIC THEORY OF GROUPS
Consider a regular n-gon in the plane. The reflections in the lines passing through the centroid of the n-gon and a vertex, or through the centroid and the midpoint of an edge, are symmetries; there are n such reflection symmetries. We can show that the n rotational symmetries in the plane together with the n reflection symmetries form a group that is isomorphic to Dn. See Exercise 2.3.10.
Figure 2.3.3 (below) has D9 symmetry, while Figure 2.3.4 possesses Z5 symmetry, but not D5 symmetry. Both of these figures were generated by several million iterations of a discrete dynamical system exhibiting “chaotic” behavior; the figures are shaded according to the probability of the moving “particle” entering a region of the diagram — the darker regions are visited more frequently. A beautiful book by M. Field and M.
Golubitsky, Symmetry in Chaos (Oxford University Press, 1992), discusses symmetric figures arising from chaotic dynamical systems and displays many extraordinary figures produced by such systems.
Exercises 2.3 2.3.1. Show that the elements j and rt of the symmetry group D of the disk satisfy the relations j rt D r t j , and jt D r2t j D jr 2t .
2.3.2. The symmetry group D of the disk consists of the rotations rt for t 2 R and the “flips” jt D r2t j .
(a) Writing N D frt W t 2 Rg, show that D D N [ Nj .
(b) Show that all products in D can be computed using the relation jrt D r t j .
(c) Show that the subgroup N of D satisfies aN a 1 D N for all a 2 D.
2.3.3. The symmetries of the disk are implemented by linear transformations of R3. Write the matrices of the symmetries rt and j with respect to the standard basis of R3. Denote these matrices by Rt and J , respectively.
Confirm the relation JRt D R t J .
2.3.4. Consider the group Dn of symmetries of the n-gon.
(a) Show that the rotation r D r2=n through an angle of 2=n about the z–axis generates a cyclic subgroup of Dn of order n.
(b) Show that the “flips” jk=n D rk2=n j D rk j , for k 2 Z, are symmetries of the n-gon.
(c) Show that the distinct flip symmetries of the n-gon are r k j for k D 0; 1; : : : ; n 1.
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