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  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 

¨
¨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 ˜
Q
4 ˜
˜ ˜
˜ ˜
˜ ˜˜v
s

˜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?

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