## Mathematics Illuminated

# Geometries Beyond Euclid

## We live in a world — a reality — ruled by straight lines. Our streets, houses, cubicles — virtually all of our space is parceled into rectilinear grids. Mathematicians were also ruled by straight lines — some would say imprisoned by them — for two thousand years. But what is a straight line? And when is a straight line not "straight"?

Our first exposure to geometry is to that of Euclid; in which all triangles have 180 degrees. As it turns out, triangles can have more or less than 180 degrees. This unit explores these curved spaces that are at once otherworldly, yet firmly of this world—and present the key to understanding the human brain.

### Unit Goals

- Geometry is the mathematical study of space.
- Euclid’s postulates form the basis of the geometry we learn in high school.
- Euclid’s fifth postulate, also known as the parallel postulate, stood for over two thousand years before it was shown to be unnecessary in creating a self-consistent geometry.
- There are three broad categories of geometry: flat (zero curvature), spherical (positive curvature), and hyperbolic (negative curvature).
- The geometry of a space goes hand in hand with how one defines the shortest distance between two points in that space.
- Stereographic projection and other mappings allow us to visualize spaces that might be conceptually difficult.
- Einstein showed that curved geometry is a way to model gravitational attraction.
- The recently proven Geometrization Theorem states that if we live in a randomly selected universe with a uniform geometry, then it is probably a hyperbolic universe.

### Unit Glossary

### ADDITIONAL UNIT RESOURCES: BIBLIOGRAPHY

# Bibliography

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Aste, Tomaso. “The Shell Map: The Structure of Froths Through a Dynamic Map.” (arXiv:cond-mat/9803183v1), (1998). http://arxiv.org/ (accessed 2007.)

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.

Boyer, Carl B. (revised by Uta C. Merzbach). *A History of Mathematics*, 2^{nd} ed. New York: John Wiley and Sons, 1991.

Burton, David M. *History of Mathematics: An introduction*, 4^{th} ed. USA: WCB/ McGraw-Hill, 1999.

Cannon, James W., William J. Floyd, Richard Kenyon, and Walter R. Parry. “Hyperbolic Geometry,” in No. 31 of *Mathematical Sciences Research Institute Publications, Flavors of Geometry*, edited by Silvio Levy, 59-115. New York: Cambridge University Press, 1997.

Conway, J.H. and S. Torquato, “Packing, Tiling, and Covering with Tetrahedra” *Proceedings of the National Academy of Sciences*, USA, vol. 103, no. 28. (July 2006).

Coxeter, H.S.M. *Non-Euclidean Geometry*, 6^{th }ed. Washington, DC: Mathematical Association of America, 1998.

Delman, Charles and Gregory Galperin. “A Tale of Three Circles,” *Mathematics Magazine*, vol. 76, no.1 (February 2003).

Deza, Michel and Mikhail Shtogrin. “Uniform Partitions of 3-Space, Their Relatives and Embedding,” *European Journal of Combinatorics*, vol. 21, no. 6 (August 2000).

Euclid. *The Thirteen Books of Euclid’s Elements*, translated from the text of Heiberg, with introduction and commentary by Sir Thomas L. Heath, 2^{nd} ed. (unabridged). New York: Dover Publications, 1956.

Eves, Howard. *An Introduction to the History of Mathematics*, 5^{th} ed. (The Saunders Series) Philadelphia, PA: Saunders College Publishing, 1983.

Gauglhofer, Thomas and Hugo Parlier. “Minimal Length of Two Intersecting Simple Closed Geodesics,” *Manuscripta Mathematica*, vol. 122, no. 3 (2007).

Goe, George, B.L. van der Waerden, and Arthur I. Miller. “Comments on Miller’s “The Myth of Gauss’ Experiment on the Euclidean Nature of Physical Space,” *Isis*, vol. 65, no. 1 (March 1974).

Goodman-Strauss, Chaim. “Compass and Straightedge in the Poincaré Disk,” *American Mathematical Monthly*, vol. 108, no. 1 (January 2001).

Greenberg, Marvin Jay. *Euclidean and Non-Euclidean Geometries: Development and History*. 2^{nd} ed, New York: W.H. Freeman and Co., 1980.

Greene, Brian. *The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory*. New York: W.W. Norton and Co., 1999.

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

Luminet, Jean-Pierre and Boudewijn F. Roukema. “Topology of the Universe: Theory and Observations.” Cornell University Library. http://fr.arxiv.org/abs/astro-ph/9901364v3 (accessed 2007).

Miller, Arthur I. “The Myth of Gauss’ Experiment on the Euclidean Nature of Physical Space,” *Isis*, vol. 63, no. 3 (September 1972).

Monastyrsky, Michael. [Translated by James King and Victoria King. Edited by R.O. Wells Jr.] *Riemann, Topology and Physics*. Boston, MA: Birkhauser, 1979.

O’Shea, Donal. *The Poincaré Conjecture: In Search of the Shape of the Universe*. New York: Walker Publishing Company, 2007.

Paur, Kathy. “The Fenchel-Nielsen Coordinates of Teichmuller Space.” *MIT Undergraduate Journal of Mathematics*, vol. 1 (1999).

Peterson, Ivars. “Celestial Atomic Physics,” *Science News Online*. week of Sept 10, 2005; vol 168, no 11. http://www.sciencenews.org/view/generic/id/6555/description/Celestial_Atomic_Physics(accessed 2007).

ll, Lisa. *Warped Passages: Unraveling Mysteries of the Universe’s Hidden Dimensions*. New York: HarperPerennial. 2005.

Rockmore, Dan. *Stalking the Riemann Hypothesis The Quest To Find the Hidden Law of Prime Numbers*. New York: Vintage Books (division of Randomhouse), 2005.

Shackleton, Kenneth J. “Combinatorial Rigidity in Curve Complexes and Mapping Class Groups,” *Pacific Journal of Mathematics*, vol. 230, no. 1 (March 2007).

Thurston, William P. “The Geometry and Topology of Three-Manifolds.” Mathematical Sciences Research Institute. http://www.msri.org/publications/books/gt3m (accessed 2007).

Tobler, W.R. “Local Map Projections,” *The American Cartographer*, vol. 1, no. 1 (1974).

Weeks, Jeffrey. “The Poincaré Dodecahedral Space and the Mystery of the Missing Fluctuations,” *Notices of the AMS*, vol. 51, no. 6 (June/July 2004).