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An Adventure in the Nth Dimension

On the mystery of a ball that fills a box, but vanishes in the vastness of higher dimensions

Who Done It?

A question I cannot answer with certainty is who first wrote down the n-ball formula. I have paddled up a long river of references, but I’m not sure I have reached the true source.

My journey began with the number 5.2569464. I entered the digits into the On-Line Encyclopedia of Integer Sequences, the vast compendium of number lore created by Neil J. A. Sloane. I found what I was looking for in sequence A074455. A reference there directed me to Sphere Packings, Lattices, and Groups, by John Horton Conway and Sloane. That book in turn cited An Introduction to the Geometry of N Dimensions, by Duncan Sommerville, published in 1929. The Sommerville book devotes a few pages to the n-ball formula and has a table of values for dimensions 1 through 7, but it says little about origins. However, further rooting in library catalogs revealed that Sommerville—a Scottish mathematician who emigrated to New Zealand in 1915—also published a bibliography of non-Euclidean and n-dimensional geometry.

The bibliography lists five works on “hypersphere volume and surface”; the earliest is a problem and solution published in 1866 by William Kingdon Clifford, a brilliant English geometer who died young. Clifford’s derivation of the formula is clearly original work, but it was not the first.

Elsewhere Sommerville mentions the Swiss mathematician Ludwig Schläfli as a pioneer of n-dimensional geometry. Schläfli’s treatise on the subject, written in the early 1850s, was not published in full until 1901, but an excerpt translated into English by Arthur Cayley appeared in 1858. The first paragraph of that excerpt gives the volume formula for an n-ball, commenting that it was determined “long ago.” An asterisk leads to a footnote citing papers published in 1839 and 1841 by the Belgian mathematician Eugène Catalan.

Looking up Catalan’s articles, I found that neither of them gives the correct formula in full, although they’re close. Catalan deserves partial credit.

Not one of these early works pauses to comment on the implications of the formula—the peak at n=5 or the trend toward zero volume in high dimensions. Of the works mentioned by Sommerville, the only one to make these connections is a thesis by Paul Renno Heyl, published by the University of Pennsylvania in 1897. This looked like a fairly obscure item, but with help from Harvard librarians, the volume was found on a basement shelf. I later discovered that the full text (but not the plates) is available on Google Books.

Heyl was a graduate student at the time of this work. He went on to a career with the National Bureau of Standards, and he was also a writer on science, philosophy and religion. (His best-known book was The Mystery of Evil.)

In the 1897 thesis Heyl derives formulas for both volume and surface area (which he calls “content” and “boundary”), and gives a lucid account of multidimensional geometry in general. He clearly appreciates the strangeness of the discovery that “… in a space of infinite dimension our locus can have no content at all.” I will allow Heyl to have the last word on the subject:

We might be pardoned for supposing that in a space of infinite dimension we should find the Absolute and Unconditioned if anywhere, but we have reached an opposite conclusion. This is the most curious thing I know of in the Wonderland of Higher Space.

Bibliography

• Ball, K. 1997. An elementary introduction to modern convex geometry. In Flavors of Geometry. Silvio Levy (ed.) Cambridge: Cambridge University Press.
• Bellman, R. E. 1961. Adaptive Control Processes: A Guided Tour. Princeton: Princeton University Press.
• Catalan, Eugène. 1839, 1841. Journal de Mathématiques Pures et Appliquées 4:323–344, 6:81–84.
• Cipra, B. 1991. Here’s looking at Euclid. In What’s Happening in the Mathematical Sciences, Vol. 1, p. 25. Providence: American Mathematical Society.
• Clifford, W. K. 1866. Question 1878. Mathematical Questions, with Their Solutions, from the “Educational Times” 6:83–87.
• Conway, J. H., and N. J. A. Sloane. 1999. Sphere Packings, Lattices, and Groups. 3rd edition. New York: Springer.
• Heyl, P. R. 1897. Properties of the locus r = constant in space of n dimensions. Philadelphia: Publications of the University of Pennsylvania, Mathematics, No. 1, 1897, pp. 33–39. Available online at http://books.google.com/books?id=j5pQAAAAYAAJ
• On-Line Encyclopedia of Integer Sequences, published electronically at http://oeis.org, 2010, Sequence A074455.
• Schläfli, L. 1858. On a multiple integral. The Quarterly Journal of Pure and Applied Mathematics 2:269–301.
• Sommerville, D. M. Y. 1911. Bibliography of Non-Euclidean Geometry, Including the Theory of Parallels, the Foundation of Geometry, and Space of N Dimensions. London: Harrison & Sons.
• Sommerville, D. M. Y. 1929. An Introduction to the Geometry of N Dimensions. New York: Dover Publications.
• Wikipedia. Deriving the volume of an n-ball. http://en.wikipedia.org/wiki/Deriving_the_volume_of_an_n-ball.