How Math Explains the World.pdf

If I denotes the identity of the group (the symmetry that leaves all the
vertices in their original positions), then we have the following relation-
ships between F and R.

F^2 I (recall that F^2 FF; two f lips of the triangle result in the origi-

nal position)

R^3 I (likewise for three successive 120-degree counterclockwise

rotations)

FR^2 RF

R^2 FFR

As shown in chapter 5, there are a total of six different symmetries of
the equilateral triangle, which can be produced by I, R, R^2 , F, RF, and FR.
Suppose that we used the above four rules to try to reduce lengthy words
using only the letters R and F to one of those six. Here’s an example.

RFR^2 FRF FR^2 R^2 F FR^2 (replaced first two and last two)

 F R^3 R F^2 R^2 (R^2 R^2 R^3 RR^4 )

 FIRIR^2 (R^3 F^2 I)

 FRR^2 (FI  F, R IR)

FR^3 FIF (Whew!)

It is easy to show that any “word” using just the letters R and F can be
reduced by using the three basic relationships to one of the six words cor-
responding to the symmetries of the equilateral triangle. Here’s the game
plan: let’s show that any string of three letters can be reduced to a string
of two or fewer letters. There are eight possibilities; I’ll just write out the
end result.

RRRI

RRFFR

RFRF

RFFR

FRR RF

FRFR^2

FFRR

FFFF

Since every string of three letters can be reduced to a string of two or
fewer letters, keep doing this until you get a word of two or fewer letters; it
must be one of the basic six that are the elements of the group. We say that
the group S 3 of the symmetries of the equilateral triangle is generated by
the two generators R and F subject to the four basic relationships.
There are many (not all) groups that are defined by a collection of gen-

128 How Math Explains the World