History of the high-dimensional volume paradox

A related (and to me, when I first saw it, much more surprising) Fun Fact: Divide the n-dimensional cube in half in each of $n$ dimensions, to create $2^n$ smaller cubes of edge length 1/2. Inscribe a ball in each of these subcubes, and then construct the smallest ball tangent to each of those (and centered at the center of the original cube) like so:

      (source)

What happens to the diameter of the central ball as $n$ gets large?

This question received much attention at an algebraic K-theory conference in Boulder in the early 1980s, where each new arrival was presented with a multiple choice problem: Without stopping to compute, is the limit $-1$, $0$, $1/2$, $1$, $10$ or $\infty$? You were allowed to choose any three answers out of six, and place a bet on whether the right answer was among them. I can report that an overwhelming majority of algebraic K-theorists reason thusly: the answer can't be negative and can't be greater than 1 (the ball, after all, is obviously contained inside a box of side 1!); therefore it's safe to bet on the set $\lbrace 0,1/2,1 \rbrace $. Feel free to make money off this.

Brian Hayes wrote a column on the volume of the $n$-sphere for American Scientist a couple of years ago, available online here. It includes a bit of history, with bibliography, toward the end, which might be of help here.

Added 4/26/13: Here are a couple of pertinent passages from Brian's article:

"... 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."

and

"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."

Consideration of higher-dimensional spheres at least goes back to the 19th century.

In his paper: "Über verschiedene Theoreme aus der Theorie der Punktmengen in einem $n$-fach ausgedehnten stetigen Raume $G_n$. Zweite Mitteilung." Acta Mathematica 7 (1885) 105-124, Cantor uses "$n$-dimensionale Vollkugeln" ($n$-dimensional solid spheres) frequently. His calculation of the volume has first been mentioned in a letter to Felix Klein. See J. W. Dauben: "Georg Cantor His Mathematics and Philosophy of the Infinite", Princeton University Press (1990) p.326:

In a letter to Felix Klein of June 6, 1882, Cantor explained the details of his more accurate determination of the volume of the unit sphere of dimension $n$ in a space of dimension $n + 1$. It was true that the volume was always less than or equal to $2^n\pi$. But equality was true only for $n$ = 1, $n$ = 2.