The only real-world example of symmetries is when your mattress gets lumpy and you have a choice of ways to turn it. I appreciate where you're coming from and

The only real-world example of symmetries is when your mattress gets lumpy and you have a choice of ways to turn it. I appreciate where you're coming from and love the passion, but this couldn't be further from the truth. Symmetry is the basis of life, not to mention physics nor beauty! https://people.math.harvard.edu/~knill/teaching/mathe320_2017/blog17/Hermann_Weyl_Symmetry.pdf https://people.math.harvard.edu/~knill/teaching/mathe320_201... > Quantum mechanics represents the state of a physical system by a vector in a space of many, actually of infinitely many, dimensions. Two states that arise from each other, either by a virtual rotation of the system of electrons or by one of their permutations, are connected by a linear transformation associated with that rotation or that permutation. Hence *the profoundest and most systematic part of group theory*, the theory of representations of a group by linear transformations, comes into play here. I must refrain from giving you a more precise account of this difficult subject. But here symmetry once more has proved the clue to a field of great variety and importance. > From art, from biology, from crystallography and physics I finally turn to mathematics, which I must include all the more because the essential concepts, *especially that of a group*, were first developed from their applications in mathematics. - Hermann Weyl's Symmetry, p. 135