Website owner: James Miller
The group of symmetries of the rectangle (the four group)
Consider the rectangle shown in Figure 1. There are four motions of the rectangle which, performed one after the other, carry it from its original position into itself. They are:
I – rotation of 0o
R – rotation of 180o
H – reflection about axis LN
V – reflection about axis KM
These four motions represent the group of symmetries of the rectangle and are commonly called the “four group”. If we re-label the vertices of the rectangle, replacing the designations A, B, C, D with 1, 2, 3, 4 as shown in Fig. 2, we can then represent the four motions as permutations of the numbers 1, 2, 3, and 4 as follows:
Symbol Permutation Operation
I (1) rotation of 0o
R (13)(24) rotation of 180o
H (14)(23) reflection about LN
V (12)(43) reflection about KM
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