A reprint from American Scientist the magazine of Sigma Xi, The Scientific Research Society This reprint is provided for personal and noncommercial use. For any other use, please send a request Brian Hayes by electronic mail to [email protected]. C!"#$%&'( S)&*')* An Adventure in the Nth Dimension Brian Hayes +* ,-*, *').!/*0 by a circle is cal term for a solid spherical object. πr2. The volume inside a sphere “Sphere” itself is generally reserved 3 On the mystery T4 3 is ∕ πr . These are formulas I learned too for a hollow shell, like a soap bubble. early in life. Having committed them to More formally, a sphere is the locus memory as a schoolboy, I ceased to ask of a ball that fills of all points whose distance from the questions about their origin or mean- center is equal to the radius r. A ball ing. In particular, it never occurred to a box, but vanishes is the locus of points whose distance me to wonder how the two formulas from the center is less than or equal are related, or whether they could be in the vastness to r. And while I’m trudging through extended beyond the familiar world this mire of terminology, I should men- of two- and three-dimensional objects of higher dimensions tion that “n-ball” and “n-cube” refer to the geometry of higher-dimensional to an n-dimensional object inhabiting spaces. What’s the volume bounded n-dimensional space. This may seem by a four-dimensional sphere? Is there too obvious to bother stating, but some master formula that gives the mea- we played on a two-dimensional field. some branches of mathematics adopt sure of a round object in n dimensions? If we had lost our ball in a space of a different convention. In topology, a Some 50 years after my first expo- many dimensions, we might still be 2-sphere lives in 3-space.) sure to the formulas for area and vol- looking for it. ume, I have finally had occasion to look The mathematician Richard Bell- The Master Formula into these broader questions. Finding man labeled this effect “the curse of An n-ball of radius 1 (a “unit ball”) the master formula for n-dimensional dimensionality.” As the number of will just fit inside an n-cube with volumes was easy; a few minutes with spatial dimensions goes up, finding sides of length 2. The surface of the Google and Wikipedia was all it took. things or measuring their size and ball kisses the center of each face of But I’ve had many a brow-furrowing shape gets harder. This is a matter of the cube. In this configuration, what moment since then trying to make practical consequence, because many fraction of the cubic volume is filled sense of what the formula is telling me. computational tasks are carried out in by the ball? The relation between volume and di- a high-dimensional setting. Typically The question is answered easily in mension is not at all what I expected; each variable in a problem description the familiar low-dimensional spaces indeed, it’s one of the zaniest things I’ve is mapped to a separate dimension. we are all accustomed to living in. At ever come upon in mathematics. I’m A few months ago I was prepar- the bottom of the hierarchy is one- appalled to realize that I have passed ing an illustration of Bellman’s curse dimensional geometry, which is rather so much of my life in ignorance of this for an earlier Computing Science col- dull: Everything looks like a line seg- curious phenomenon. I write about it umn. My first thought was to show ment. A 1-ball with r = 1 and a 1-cube here in case anyone else also missed the ball-in-a-box phenomenon. Put an with s =2 are actually the same object— school on the day the class learned n- n-dimensional ball in an n-dimension- a line segment of length 2. Thus in one dimensional geometry. al cube just large enough to receive dimension the ball completely fills the it. As n increases, the fraction of the cube; the volume ratio is 1.0. Lost in Space cube’s volume occupied by the ball In two dimensions, a 2-ball inside a In those childhood years when I was falls dramatically. 2-cube is a disk inscribed in a square, memorizing volume formulas, I also In the end I chose a different and and so this problem can be solved with played a lot of ball games. Often the simpler scheme for the illustration. But one of my childhood formulas. With game was delayed when we lost the after the column appeared [“Quasi- r =1, the area πr2 is simply π, whereas ball in the weeds beyond right field. I random Ramblings,” July–August], I the area of the square, s2, is 4; the ratio didn't know it then, but we were lucky returned to the ball-in-a-box question of these quantities is about 0.79. out of curiosity. I had long thought that In three dimensions, the ball’s vol- 4 Brian Hayes is senior writer for American Scien- I understood it, but I realized that I had ume is ∕3π, whereas the cube has a vol- tist. Additional material related to the Comput- almost no quantitative data on the rela- ume of 8; this works out to a ratio of ing Science column appears at http://bit-player. tive size of the ball and the cube. approximately 0.52. org. Address: 11 Chandler St. #2, Somerville, MA (In this context “ball” is not just a On the basis of these three data points, 02144. E-mail: [email protected] plaything but also the mathemati- it appears that the ball fills a smaller and 442 American Scientist, Volume 99 © 2011 Brian Hayes. Reproduction with permission only. Contact [email protected]. smaller fraction of the cube as n increas- the gamma function, unlike the facto- But then I looked at the continuation es. There’s a simple, intuitive argument rial, is also defined for numbers oth- of the table: suggesting that the trend will continue: er than integers. For example, Γ(½) is – The regions of the cube that are left va- equal to √π. n V(n,1) cant by the ball are the corners. Each 1 2 time n increases by 1, the number of The Incredible Shrinking n-Ball 2 π ≈ 3.1416 corners doubles, so we can expect ever When I discovered the n-ball formula, 4 more volume to migrate into the nooks I did not pause to investigate its prov- 3 3 π ≈ 4.1888 and crannies near the cube’s vertices. enance or derivation. I was impatient 1 2 4 2 π ≈ 4.9348 To go beyond this appealing but non- to plug in some numbers and see what 5 8 π2 ≈ 5.2638 quantitative principle, I would have to would come out. So I wrote a hasty 15 1 3 ≈ calculate the volume of n-balls and n- one-line program in Mathematica and 6 6 π 5.1677 16 3 cubes for values of n greater than 3. began tabulating the volume of a unit 7 105 π ≈ 4.7248 The calculation is easy for the cube. An ball in various dimensions. I had defi- 8 1 π4 ≈ 4.0587 n-cube with sides of length s has vol- nite expectations about the outcome. 24 32 4 ≈ ume sn. The cube that encloses a unit I believed that the volume of the unit 9 945 π 3.2985 n 1 5 ball has s =2, so the volume is 2 . ball would increase steadily with n, 10 120 π ≈ 2.5502 But what about the n-ball? As I have though at a lower rate than the volume already noted, my early education of the enclosing s = 2 cube, thereby Beyond the fifth dimension, the vol- failed to equip me with the necessary confirming Bellman’s curse of dimen- ume of a unit n-ball decreases as n in- formula, and so I turned to the Web. sionality. Here are the first few results creases! I tried a few larger values of What a marvel it is! (And it gets better returned by the program: n, finding that V(20,1) is about 0.0258, all the time.) In two or three clicks I and V(100,1) is in the neighborhood of had before me a Wikipedia page titled n V(n,1) 10–40. Thus it looked very much like “Deriving the volume of an n-ball.” 1 2 the n-ball dwindles away to nothing as Near the top of that page was the for- 2 π ≈ 3.1416 n approaches infinity. mula I sought: 3 4 π ≈ 4.1888 n n 3 Doubly Cursed π 2 r V(n, r ) = . 4 1 π2 ≈ 4.9348 I had thought that I understood Bell- Γ( n + 1) 2 2 5 8 π2 ≈ 5.2638 man’s curse: Both the n-ball and the Later in this column I’ll say a few 15 n-cube grow along with n, but the cube words about where this formula came I noted immediately that the val- expands faster. In fact, the curse is far from, both mathematically and histori- ues for one, two and three dimensions more damning: At the same time the cally, but for now I merely note that the agreed with the results I already knew.