mirror in another mirror is a

mirror again. Notice that if

∆ is compact then all mirrors

intersect in a common point.

Figure 2.2: A closed system of mirrors.

has seen a mirror3 . We are interested in the study of ¬nite closed systems of

mirrors and other, closely related objects”root systems and ¬nite groups

generated by re¬‚ections.

Systems of re¬‚ections. If Σ is a system of mirrors, the set of all re-

¬‚ections in mirrors of Σ will be refered to as a closed system of re¬‚ections.

In view of Lemma 2.1.3, a set S of re¬‚ections forms a closed system of

re¬‚ections if and only if st ∈ S for all s, t ∈ S. Here st is the standard, in

group theory, abbreviation for conjugation: st = t’1 st. Recall that con-

jugation by any element t is an automorphism of any group containing t:

(xy)t = xt y t .

Lemma 2.2.1 A ¬nite closed system of re¬‚ections generates a ¬nite group

of isometries.

Proof. This result is a partial case of the following elementary group

theoretic property.

Let W be a group generated by a ¬nite set S of involutions such

that st ∈ S for all s, t ∈ S. Then W is ¬nite.

Indeed, since s ∈ S are involutions, s’1 = s. Let w ∈ W and ¬nd the

shortest expression w = s1 · · · sk of w as a product of elements from S. If

3

We cannot resist temptation and recall an old puzzle: why is it that the mirror

changes left and right but does not change up and down?

36

t t t

d

d

d

d

d

t t t

d

t t t

d d d

d d d

t

t t t

t

d

t t t d

d d

d d

t t t d

t t t

d d d

d d d

t t t

d

d

d

t t t

t

t t t

t t

t t

˜

A2 BC 2

¨

rr

t t

t

t

¨¨ t r t

t ¨ r

¨t t rr t

¨

¨ r

r ¨

tr

t

¨t

t t

¨

r t

t t ¨

r

t rr ¨ t

t¨

¨

¨rr

t

t

t

¨ t r t

t

¨ r

¨t t rr t

¨

¨ r

t

t

t

t

t t

t

˜ ˜ ˜

A1 + A1 G2

˜

A1 + A1

Figure 2.3: Examples of in¬nite closed mirror systems in AR2 with their tradi-

˜

tional notation: tesselations of the plane by congruent equilateral triangles (A2 ),

˜ ˜

isosceles right triangles (BC 2 ), rectangles (A1 + A1 ), triangles with the angles

˜ ˜

π/2, π/3, π/6 (G2 ), in¬nite half stripes (A1 + A1 ).

A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 37

the word s1 · · · sk contains two occurrences of the same involution s ∈ S

then

w= s1 · · · si ssi+1 · · · sj ssj+1 · · · sk

s1 · · · si (si+1 · · · sj )s sj+1 · · · sk

=

s1 · · · si ss · · · ss sj+1 · · · sk

= i+1 j

= s1 · · · si si+1 · · · sj sj+1 · · · sk ,

where all sl = ss belong to S and the resulting expression is shorter then

l

the original. Therefore all shortest expressions of elements from W in

terms of generators s ∈ S contain no repetition of symbols. Therefore the

length of any such expression is at most |S|, and, counting the numbers

of expressions of length 0, 1, . . . , |S|, we ¬nd that their total number is at

most

1 + |S| + |S|2 + · · · + |S||S| .

Hence W is ¬nite.

Finite re¬‚ection groups. A group-theoretic interpretation of closed

systems of mirrors comes in the form of a ¬nite re¬‚ection group, i.e. a

¬nite group W of isometries of an a¬ne Euclidean space A generated by

re¬‚ections.

Let s be a re¬‚ection in W and sW = { wsw’1 | w ∈ W } its conjugacy

class. Form the set of mirrors Σ = { Ht | t ∈ sW }. Then it follows from

Lemma 2.1.3 that Σ is a mirror system: if Hr , Ht ∈ Σ then the re¬‚ection

of Hr in Ht is the mirror Hrt . Thus sW is a closed system of re¬‚ections.

The same observation is valid for any normal set S of re¬‚ections in W ,

i.e. a set S such that sw ∈ S for all s ∈ S and w ∈ W . We shall show

later that if the re¬‚ection group W arises from a closed system of mirrors

Σ then every re¬‚ection in W is actually the re¬‚ection in one of the mirrors

in Σ.

Since W is ¬nite, all its orbits are ¬nite and W ¬xes a point by virtue

of Theorem 1.4.1. We can take this ¬xed point for the origin of an or-

thonormal coordinate system and, in view of Theorem 1.4.2, treat W as a

group of linear orthogonal transformations.

If W is the group generated by the re¬‚ections in the ¬nite closed system

of mirrors Σ then the ¬xed points of W are ¬xed by every re¬‚ection in a

mirror from Σ hence belong to each mirror in Σ. Thus we proved

Theorem 2.2.2 (1) A ¬nite re¬‚ection group in ARn has a ¬xed point.

(2) All the mirrors in a ¬nite closed system of mirrors in ARn have a point

in common.

38

Since we are interested in ¬nite closed system of mirros and ¬nite groups

generated by re¬‚ections, this result allows us to assume without loss of

generality that all mirrors pass through the origin of Rn . So we can forget

about the a¬ne space ARn and work entirely in the Euclidean vector space

V = Rn .

Exercises

Systems of mirrors.

2.2.1 Prove that if ∆ is a subset in ARn then the system Σ of its mirrors of

symmetry is closed.

Hint: If M and N are two mirrors in Σ with the re¬‚ections s and t, then, in

view of Lemma 2.1.3, the mirror image of M in N is the mirror of the re¬‚ection

st . If s and t map ∆ onto ∆ then so does st .

4˜

4 ˜

Q

4 ˜

˜ ˜

˜ ˜

˜ ˜˜v

s

k˜

˜f

˜

Figure 2.4: Billiard, for Exercise 2.2.2.

2.2.2 Two balls, white and black, are placed on a billiard table (Figure 2.4).

The white ball must bounce o¬ two cushions of the table and then strike the

black one. Find its trajectory.

2.2.3 Prove that a ray of light re¬‚ecting from two mirrors forming a corner will

eventually get out of the corner (Figure 2.5). If the angle formed by the mirrors

is ±, what is the maximal possible number of times the ray would bounce o¬

the sides of the corner?