Website owner: James Miller
The Dihedral Group of the Equilateral Triangle
Consider the equilateral triangle shown in Figure 1. There are six motions of this triangle which, when performed one after the other, form a group called the Dihedral Group of the Equilateral Triangle. They are:
I -- rotation of 0o (counterclockwise)
R1 – rotation of 120o
R2 – rotation of 240o
S1 – reflection about axis AD
S2 – reflection about axis BE
S3 – reflection about axis CF
The Dihedral Group of the Equilateral Triangle then is given by G = [ I, R1, R2, S1, S2, S3]. Multiplication in G consists of performing two of these motions in succession. Thus the product R1S2 corresponds to first performing operation R1, then operation S2.
The six motions I, R1, R2, S1, S2, S3 can be represented as permutations of the numbers 1, 2, and 3 where these numbers correspond to the three corners of the triangle as shown in Figure 2. Their permutation representation is
Symbol Permutation Operation
I (1) rotation of 0o (counterclockwise)
R1 (123) rotation of 120o
R2 (132) rotation of 240o
S1 (23) reflection about axis AD
S2 (13) reflection about axis BE
S3 (12) reflection about axis CF
Thus G can also be represented as the set
G = [ (1), (123), (132), (23), (13), (12)]
A multiplication table for G is shown below. Entries in the table contain the product XY where X corresponds to the row and Y corresponds to the column. Thus in the table R1S2 = S3.
|
I |
R1 |
R2 |
S1 |
S2 |
S3 |
I |
I |
R1 |
R2 |
S1 |
S2 |
S3 |
R1 |
R1 |
R2 |
I |
S2 |
S3 |
S1 |
R2 |
R2 |
I |
R1 |
S3 |
S1 |
S2 |
S1 |
S1 |
S3 |
S2 |
I |
R2 |
R1 |
S2 |
S2 |
S1 |
S3 |
R1 |
I |
R2 |
S3 |
S3 |
S2 |
S1 |
R2 |
R1 |
I |
The dihedral group of an equilateral triangle is the Symmetric group of order three, S3.
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