## 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?

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.

### Unit Glossary

### ADDITIONAL UNIT RESOURCES: BIBLIOGRAPHY

# Bibliography

Ash, Avner and Robert Gross. *Fearless Symmetry : Exposing the Hidden Patterns of Numbers*. Princeton, NJ: Princeton University Press, 2006.

Bashmakova, Isabella and Galina Smirnova. *The Beginnings and Evolution of Algebra*, trans. Abe Shenitzer. USA: Dolciani Mathematical Expositions, Number 23, Mathematics Association of America, 2000.

Bayer, Dave and Persi Diaconis. “Trailing the Dovetail Shuffle to its Lair,” *The Annals of Applied Probability*, vol. 2, no. 2 (May 1992).

Berlinghoff, William P. and Kerry E. Grant. *A Mathematics Sampler: Topics for the Liberal Arts*, 3^{rd} ed. New York: Ardsley House Publishers, Inc., 1992.

Byers, Nina. “E. Noether’s Discovery of the Deep Connection Between Symmetries and Conservation Laws.” Cornell University Library. http://arxiv.org/abs/physics/9807044 (accessed 2007).

Dolan, L. “The Beacon of Kac-Moody Symmetry for Physics,” *Notice of the AMS*, vol. 42, no. 12 (December 1995).

Fraleigh, John B. *A First Course in Abstract Algebra*, 6^{th} ed. Reading, MA: Addison- Wesley Publishing Company, 1997.

Gribben, John. *The Search for Superstrings, Symmetry, and the Theory of Everything.* Boston, MA: Little, Brown and Company, 1998.

Gross, David J. “The Role of Symmetry in Fundamental Physics,” *Proceedings of the National Academy of Sciences of the United States of America*, vol. 93, no. 25 (December 10, 1996).

Kostant, B. “The Graph of the Truncated Icosahedron and the Last Letter of Galois,” *Notice of the AMS,* vol. 42, no. 9 (September 1995).

Johnston, Bernard L. and Fred Richman. *Numbers and Symmetry: An Introduction to Algebra*. Boca Raton, FL: CRC Press, 1997.

Lederman, Leon M. and Christopher T. Hill. *Symmetry and the Beautiful Universe*. Amherst, NY: Prometheus Books, 2004.

Livio, Mark. *The Equation That Couldn’t Be Solved: How Mathematical Genius Discovered the Language of Symmetry*. New York: Simon & Schuster, 2005.

Miller, Gerald A. “Big Break for Charge Symmetry.” IOP Publishing. http://physicsweb.org/articles/world/16/6/3 (accessed 2007).

Morris, S. Brent. *Magic Tricks, Card Shuffling, and Dynamic Computer Memories.* Washington, DC: Mathematical Association of America, 1998.

Rockmore, Dan. *Stalking the Riemann Hypothesis.* New York: Vintage Books (division of Randomhouse), 2005.

Tannenbaum, Peter. *Excursions in Modern Mathematics*, 5^{th} ed. Upper Saddle River, NJ: Pearson Education, Inc.,2004.

Weyl, Hermann. *Symmetry (Princeton Science Library) *Princeton, NJ: Princeton University Press, 1989.