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Mathematics Illuminated

The Beauty of Symmetry

They say, beauty is in the eye of the beholder. What we consider to be beautiful in nature, art, or music often differs from culture to culture. But somehow, there seem to be constants — commonalities in how we as human beings "see" beauty. Where does that "sense" of beauty and order come from? And what does algebra or geometry have to do with it?

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Wallpaper Pattern

In mathematics, symmetry has more than just a visual or geometric quality. Mathematicians comprehend symmetries as motions—motions whose interactions and overall structure give rise to an important mathematical concept called a “group.” This unit explores Group Theory, the mathematical quantification of symmetry, which is key to understanding how to remove structure from (i.e., shuffle) a deck of cards or to fathom structure in a crystal.

Unit Goals

  • Symmetry, in a mathematical sense, is a transformation that leaves an object invariant.
  • Some symmetries are understood as geometric motions.
  • Some symmetries are understood as algebraic operations.
  • When symmetries are combined, another symmetry is the result.
  • Symmetries form what is known as a group, which allows mathematicians to perform a sort of “arithmetic” with things that are not numbers.
  • A “frieze group” is the set of symmetries of an infinite frieze (a frieze is a pattern that repeats along a line — i.e., in one dimension).
  • All frieze groups fall into one of seven general types.
  • A “wallpaper group” is the set of symmetries of an infinite wallpaper pattern — that is, a pattern that “repeats” in two dimensions.
  • Wallpaper groups fall into one of 17 general categories.
  • All groups of symmetries can be expressed as permutations, which also form groups.
  • The ways in which a polynomial’s roots behave under a certain collection of permutations determine whether or not the roots have a simple algebraic formula (such as the quadratic formula) in terms of the coefficients of the polynomial.
  • Every conserved physical quantity is based on a continuous symmetry.




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