## Mathematics Illuminated

# Other Dimensions

## Is there such a thing as a higher dimension, a parallel universe where otherworldly things can happen? Over the years, artists, writers and filmmakers have tried to answer that question, creating some dazzling works of science fiction in the process. But are the higher dimensions we see in sci-fi really fiction?

The conventional notion of dimension consists of three degrees of freedom: length, width, and height, each of which is a quantity that can be measured independently of the others. Many mathematical objects, however, require more—potentially many more—than just three numbers to describe them. This unit explores different aspects of the concept of dimension, what it means to have higher dimensions, and how fractional or “fractal” dimensions may be better for measuring real-world objects such as ferns, mountains, and coastlines.

### Unit Goals

- Dimension is how mathematicians express the idea of degrees of freedom.
- Distance and angle are measurements that exist in many types of spaces.
- Lower-dimensional analogies extend qualitative understanding to spaces of four dimensions and higher.
- The techniques of projection and slicing help us to understand high-dimensional objects.
- High-dimensional space is one way to compare two people mathematically.
- Hausdorff dimension is a re-envisioning of our normal thinking of dimension due to behavior of objects under scaling.
- Fractal dimensions describe many real-world objects that exhibit statistical self-similarity.

### Additional Unit Resources: Bibliography

# Bibliography

## WEBSITES

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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.

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

Carter, Steve and Chadwick Snow. “Helping Singles Enter Better Marriages Using Predictive Models of Marital Success,” presented at the 16^{th} Annual Convention of the American Psychological Society, (May 2004).

Dantzig, Tobias. *Number: The Language of Science, The Masterpiece Science Edition.* New York: Pi Press, an imprint of Pearson Education, Inc., 2005.

Edgar, Gerald A. *Measure, Topology, and Fractal Geometry*. New York: Springer-Verlag, 1990.

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

Falk, Ruma and Arnold D. Well, “Many Faces of the Correlation Coefficient,” *Journal of Statistics Education*, vol. 5, no. 3 (1997).

Fiore, A.T. and Judith S. Donath. “Homophily in Online Dating: When Do You Like Someone Like Yourself?” Paper presented at the Conference on Human Factors in Computing Systems, Portland, Oregon, April 2-7, 2005

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

Kennedy, Randy. “Black, White and Read All Over Over.” NYTimes.com (December 15, 2006), http://www.nytimes.com/2006/12/15/arts/design/15serk.html?ex=1323838800&en=8388f00a8250bff2&ei=5088&partner=rssnyt&emc=rss (accessed 2007).

Mandelbrot, Benoit B. *The Fractal Geometry of Nature, Updated and Augmented.* New York: W.H. Freeman and Company, 1983.

Mandelbrot, Benoit B. “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension,” *Science*, 156 (1967).

McCallum, William G., Deborah Hughes-Hallett, Andrew M. Gleason, et al. *Multivariable Calculus.* New York: John Wiley and Sons, 1997.

Mureika, J.R. “Fractal Dimensions in Perceptual Color Space: A Comparison Study Using Jackson Pollock’s Art.” Cornell University Library. http://arxiv.org/abs/physics/0509152 (accessed 2007).

Mureika, J. R. et al. “Multifractal Structure in Nonrepresentational Art,” *Physical Review E*, vol. 72, issue 4 (2005).

Ouellette, Jennifer. “Pollock’s Fractals: That Isn’t Just a Lot of Splattered Paint on Those Canvases: It’s Good Mathematics” *Discover*, vol. 22, no. 11 (November 2001).

Randall, Lisa. *Warped Passages: Unraveling Mysteries of the Universe’s Hidden Dimensions.* New York: Harper Perennial, 2005.

Rucker, Rudy.* The Fourth Dimension: A Guided Tour of the Higher Universes.* Boston, MA: Houghton-Mifflin Co., 1984.

Salas, S.L. and Einar Hille. *Calculus: One and Several Variables,* 6^{th} ed. New York: John Wiley and Sons, 1990.

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

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Taylor, R.P. et al. “From Science to Art and Back Again!” *American Association for the Advancement of Science*. http://sciencecareers.sciencemag.org/career_magazine/previous_issues/articles/2001_04_27/noDOI.2477544795567000231 (accessed 2007).

Taylor, Richard P. “Order in Pollock’s Chaos,” *Scientific American*, 116 (December 2002).

Weeks, Jeffrey R.* The Shape of Space*, 2^{nd} ed. (Pure and Applied Mathematics). New York: Marcel Dekker Inc., 2002.

Weisstein, Eric W. “Hypercube.” Wolfram Research. http://mathworld.wolfram.com/Hypercube.html (accessed 2007).

Weisstein, Eric W. “Fractal.” Wolfram Research. http://mathworld.wolfram.com/Fractal.html (accessed 2007).