EXAMPLE 3 Symmetries of a Rectangle
What are the symmetries of a rectangle?
Consider the rectangle of Figure 19.15.
Figure 19.15: Figure 19.15 A rectangle, with reflection symmetries and ° rotation symmetry marked.
Its symmetries, the rigid motions that bring it back to coincide with itself (even as they interchange the labeled corners), are as follows:
- The identity symmetry , which leaves every point where it is
- A 180° (half-turn) rotation around its center
- A reflection in the vertical line through its center
- A reflection in the horizontal line through its center
Now, convince yourself that the symmetries fulfill the four properties above:
- Combining any pair by applying first one and then the other is equivalent to one of the others. It’s handy to have a notation for this combining; if we apply first and then , we will write the result as —in other words, we apply the sequence of actions from left to right. You can check that the result is the same as applying ; that is, . Check this by following where the corner goes under the symmetries. Practice combining symmetries by making yourself a “multiplication table” of them.
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The element is an identity element.
- Each element is its own inverse. For example, rotating by , then doing it again, gets the rectangle back to coincide with itself; in our notation, .
- Try some examples to verify that associativity holds. For instance, check that . In other words, applying then then , we get the same result if we combine the first two and then apply the third, or if we apply the first one and then apply the combination of the second two.