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A Disorienting Look at Euler's Theorem on the Axis of a Rotation

by Bob Palais (University of Utah) and Richard Palais (University of California Irvine) and Stephen Rodi (Austin Community College)

Award: Lester R. Ford

Year of Award: 2010

Publication Information: The American Mathematical Monthly, vol. 116, no. 10, December 2009, pp. 892-909.

Summary: Three hundred years ago Euler proved the theorem, "In whatever way a sphere be rotated around its own center, a diameter can always be chosen whose direction in the rotated configuration would coincide with the original configuration." This result is so famous and so important in both pure and applied mathematics that the general concept of rotation is often confused with rotation about an axis. As the authors point out, it is true, but not immediately obvious, that the composition of rotations about two different axes of the two dimensional sphere is again a rotation about a single axis. The authors lead us on a journey through the mathematics of spherical rotations during which (1) they present a new and constructive proof of Euler's theorem using only notions available from a muscular undergraduate first course in linear algebra, (2) they provide a translation (possibly the first) and explication of Euler's original proof, (3) they give another proof "using only the kind of classical spherical geometry arguments that Euler himself used, translated into a modern idiom," but which includes the orientation reversing case, (4) finally, they review some short modern (non-constructive) proofs using, respectively, linear algebra, topology, differential geometry and Lie theory.

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About the Authors: (From the Prizes and Awards booklet, MathFest 2010) 

Bob Palais received his B.A. from Harvard College in 1980 and his Ph.D. from the University of California Berkeley in 1986. He is currently a Research Professor at the University of Utah in Salt Lake City. He splits his time between the Math department (teaching and doing research on scientific computation and mathematical visualization) and the Pathology Department, where he works on DNA melting analysis and bioinformatics. His article -- Pi is Wrong! ( still generates the most spirited discussions. In his spare time, he enjoys mountaineering, often with other mathematicians. He and his father Richard recently co-authored Differential Equations, Mechanics, and Computation ( His mother Eleanor is a founding member of AWM, and had an exceptional streak with 100% of her BC calculus students receiving 5 on the AP exam for 9 years! His grandmother, Madeleine Galland, also taught mathematics, with Ronnie ('Christopher') Walken among her favorite students.

Richard S. Palais received his B.A. from Harvard College in 1952 and his Ph.D., also from Harvard, in 1956. After a long career in teaching and theoretical research at Brandeis, he partially retired in 1997 to work on developing a mathematical visualization program (see He moved to the University of California Irvine in 2004, where he teaches part time while continuing to program and write about novel algorithms for visualizing complex mathematics. He is proud of his many mathematical descendants (72 according to the Mathematical Genealogy Project) and also for entries (together with Luc Benard) in the 2006 and 2009 National Science Foundation/Science Magazine Scientific Visualization Challenge that both won First Prizes. The 2006 entry was the September 22, 2006 cover of Science.

 Stephen Rodi is a professor of mathematics at Austin Community College (Texas). He joined the infant college in 1976; it now boasts a booming 40,000 students. He has held various administrative posts, but returned exclusively to teaching in 1998. Rodi received a classical secondary education in New Orleans. He then spent seven years in Jesuit seminary training, including a philosophy curriculum taught in Latin. This explains his B.A in 1965 from Spring Hill College with a triple major in mathematics, philosophy, and Latin. He holds a Master‘s degree from Marquette University (1967) and a Ph.D. from The University of Texas at Austin (1974), both in mathematics.

Rodi has participated in many professional activities: president of the American Mathematical Association of Two-Year Colleges; chair of the Mathematical Association of America Committee on Two-Year Colleges; member of the Conference Board of the Mathematical Sciences (CBMS) and the Mathematical Sciences Education Board; chair of the advisory board for the calculus project at Harvard; co-editor in 2000 and 2005 of the CBMS statistical survey of undergraduate mathematics programs in the United States.

Subject classification(s): Algebra and Number Theory | Linear Algebra | Matrix Algebra | Geometry and Topology | Solid Geometry | Spheres
Publication Date: 
Friday, August 13, 2010