How would you define symmetry?
One thing mathematicians do that does not get much publicized is to find ways of defining concepts that have been around for ages but which have not been definitively specified. If you have ever taken a course in calculus then you have come across the delta-epsilon definition of continuity, which gives students all kinds of difficulties, but which is really a very compact way of defining the concept that was not discovered until the 19th century.
The defnition of symmetry is easier to understand, but still quite ingenious and I thought I would share it.
Consider the following:
The letter M
The letter N
The equation x^2 + xy + y^2
The description “A red lamp on a blue table next to a blue lamp on a red table”
Now see if you can come up with a definition of symmetry that covers all these cases. Kind of tough?
The definition of symmetry has 3 components: An object or set of objects, a property and an operation.
We say that a property of a set of objects is symmetic with respect to an operation if the property remains the same after the operation is performed.
For example, for the letter M, you could reflect the two different sides with respect to a verical mirror and the letter occupies the same space. So the letter M has reflective symmetry. For the letter N, you get similar results if you rotate it 180 degrees, so it has rotational symmetry.
The equation retains the same value if the symbols x and y are swapped and the description has the same meaning if the words red and blue are swapped.
Okay, I have gotten that out of my system. Any thoughts?(other than why in the world I would feel compelled to post this)
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