Hamming, "n-Dimensional Space" (April 14, 1995)

Richard Hamming explains how geometric intuition about spheres and cubes breaks down in high-dimensional spaces.

Intro: When I became a professor, after 30 years of active research at Bell Telephone Laboratories, mainly in the math research department, I recalled that professors are supposed to think and digest past experiences. So I put my feet up on the desk and began to consider my past. In the early years I had been mainly in computing so naturally I was involved in many large projects that required computing. Thinking about how things worked out on several of the large engineering systems I was partially involved in, I began, now that I had some distance from them, to see that they had some common elements. Slowly I began to realize that the design problems all took place in a space of n-dimensions, where n is the number of independent parameters. Yes, we build three dimensional objects, but their design is in a high dimensional space, one dimension for each design parameter. The Art of Doing Science and Engineering: Learning to Learn" was the capstone course by Dr. Richard W. Hamming (1915-1998) for graduate students at the Naval Postgraduate School (NPS) in Monterey California. This course is intended to instill a "style of thinking" that will enhance one's ability to function as a problem solver of complex technical issues. With respect, students sometimes called the course "Hamming on Hamming" because he relates many research collaborations, discoveries, inventions and achievements of his own. This collection of stories and carefully distilled insights relates how those discoveries came about. Most importantly, these presentations provide objective analysis about the thought processes and reasoning that took place as Dr. Hamming, his associates and other major thinkers, in computer science and electronics, progressed through the grand challenges of science and engineering in the twentieth century.