The Sphere Game in n Dimensions

Document first posted 3/05/2002
Document last revised 5/15/2006
 
 

Al Lehnen	Gary E Wesenberg
Mathematics Department	Department of Biochemistry
Madison Area Technical College	University of Wisconsin
3550 Anderson Street	6606B Biochemistry
Madison, WI 53704	433 Babcock Drive
(608) 246-6567	Madison, WI 53706
alehnen@matcmadison.edu	(608) 263-5923
my.execpc.com/~aplehnen/al	gary@biochem.wisc.edu

Abstract:

This work reports a solution to the following problem: "To within a fixed tolerance, what is the most probable straight line distance between a fixed point and a second point picked at random from the surface of a sphere?". The surprising result in two and three dimensions are values near the diameter! In more than three dimensions, this most probable distance is less than the diameter and begins to approach the more intuitive "equatorial value",  the square root of two times the sphere's radius. The analysis presented solves the problem for a sphere in any dimension greater than one. In particular, as the dimension gets large the probability distribution approaches a normal distribution with a mean of the equatorial value and a standard deviation which goes to zero like the reciprocal of the square root of the dimension.
 
 

A Summary of the Results

Imagine fixing a point on the surface of an n-dimensional sphere (it is assumed that n > 1, the surface is a "hypersphere"

For even n,                            For odd n,

Both these results can be represented by the single formula: 

 

Figure 1	Figure 2
Figure 3	Figure 4
Figure 5	Figure 6

 
 

Details

.

However, theta is not a function of L since there are two possible angles for each non-zero L value less than the diameter. Imagine generating L values by letting theta range from to 0 . This is illustrated in Figure 7. The probability that a value
for the length would occur between L and L+dL would then be given by the following:

. Figure 7

This probability can be expressed in terms of L by using the chain rule.

.

This is entirely equivalent to using the fundamental transformation equation for a probability distribution [2, p. 137, 9 pp. 265-268, 14 p. 186]

For theta in the range from to 0 and from elementary trigonometry it follows that

.

Bertrand's Paradox

Figure 8

The probability that L would exceed S is given by .

with the domain. From these definitions it follows that

The element of volume in these coordinates requires calculating the following Jacobian.

Using an induction argument it is shown in Appendix A that

.

The total surface area is easily computed from the formula for the volume.

Orient the coordinate system so that the first "marked" point has coordinates x_n = -a, and x_j = 0 for j < n .

But this simplifies to the same result as on the circle! , with the difference that sinceis now

.

Thus, an explicit expression is obtained for the probability density function of L for any value of n.

.

In a definitive analysis published in 1954 R. D. Lord derived this same result [13]. In passing, we note that for a sphere in three dimensions the factor in brackets "magically" disappears and the probability distribution is linear. This has the "non-intuitive" result that in three dimensions, like in two, within a given tolerance the most probable value of L is close to the diameter. The results for a sphere in three dimensions seem were first brought to our attention  in a study by Christopher and Baldwin related to protein structures [8]. However, other researchers [4, 5, 6, 7] had noted them even earlier. The special case of three dimensions was also analyzed by us in detail in a previous publication [12].

From this recursion, one obtains the following:

For even n,                                                 For odd n, 

Calculating the first moment of the probability distribution of L gives the mean. This integral is a multiple of a Beta function which by a well known identity can be expressed as a ratio of Gamma functions.[1, p. 258, 15, pp. 253-255]

The results for n = 2 through n = 10 are summarized in Table 1.

Table 1

n	Vn	Sn	Mode of L	< L >
2
3
4
5
6
7
8
9
10

 

An Asymptotic Analysis of the L Distribution for Large n

The well known asymptotic formula for the gamma function is [1, p. 257, 15, p. 253, 17]

This equation provides a convenient starting point for a systematic analysis of the probability distribution of L for large n. For , and . For , A long, but straightforward calculation that uses the binomial series and the Maclaurin series for ln(1+ x) and exp(x) results in the following expansion.

.

From which it follows that

.

Since , an asymptotic formula for c(n) is obtained.

.

An asymptotic expansion for the mean follows from .

The variance of L is computed as  . This has the following asymptotic form.

The accuracy of these asymptotic expressions is demonstrated to ten decimal places in Table 2.

Table 2

n
10	1.3947761299	1.3947750312	0.0545995476	0.0546093750
15	1.4016613571	1.4016611894	0.0353454399	0.0353472222
20	1.4049475047	1.4049474597	0.0261225089	0.0261230469
25	1.4068707168	1.4068707004	0.0207147861	0.0207150000
50	1.4106109093	1.4106109086	0.0101768625	0.0101768750
75	1.4118268316	1.4118268315	0.0067449976	0.0067450000
100	1.4124291190	1.4124291190	0.0050439836	0.0050439844

As n increases the distribution of L narrows and becomes more symmetric with the mode converging to the median from above and the mean converging to the median from below. For large n the probability density function of L,

, rapidly approaches zero for L < a and L near 2a. Most of the probability is therefore concentrated near the mode of L, which for large n approaches the median value. To investigate the large n behavior of the probability density function in the neighborhood of the median, we define a new variable z as . The fundamental transformation equation for a probability distribution then implies the following result.

.

From the Maclaurin series of the logarithm and the asymptotic form of c(n), this results in the following expression.

.

So,  . Thus, for large n the probability distribution of L approaches a Normal Distribution with a mean of and a standard deviation . This result seems to have been first established for two points chosen at random within the hypersphere by Hammersley [10] and then presented in a generalized form by Lord which included two points chosen at random on the hypersphere[13].
 

Conclusion:

Intuitively the median value,, would seem to be the obvious answer to the question, "What is the most probable straight line distance between a fixed point and a second point picked at random from the surface of a sphere?". The rather humorous, if not surprising, conclusion of this analysis is that it is only in the limit of large dimensions that our intuition is realized!
Not only that, but for large dimensions,  is "almost the only" answer to "What is the straight line distance?".
 
 

Acknowledgements:

The reference to the work of R. D. Lord were kindly communicated to us by Mario N. Berberan-Santos of the

 
 

References:

1. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, 1964.