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#symmetry

7 posts7 participants1 post today

In mathematics, we sometimes switch between two viewpoints without distinguishing them explicitly. Which causes confusion.
Think of symmetry transformations: The symmetry group of, say, a regular pentagon, consists of those geometrical transformations that leave it unchanged at the same position. So the pentagon does not change? And still there are different transformations?
The answer is that we view the pentagon in different ways when we speak about the symmetry and about the transformations, say with the light switched on and off. For the transformations, we think of the pentagon with markers written on its corners, say with the numbers 0, 1, 2, 3, and 4, so that we can speak of the rotation that maps the corner marked n to the place of corner (n + 2) mod 5. For the symmetry, we switch the light off and cannot see the markers: A symmetry transformation is then one in which we cannot feel any difference before and after the transformation.

The same idea is also in other contexts, for example with the bound variable in a definite integral: An expression like ∫ₐᵇ 𝑓(𝑥) 𝑑𝑥 is interpreted with “light off”, so that we do not see that x is the integration variable, but in our calculations, the “light“ is “on” and x is an ordinary variable that follows some special rules.