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Hypercube Line Picking

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Hypercube line picking considers two points x and y chosen independently and uniformly from a unit n-dimensional hypercube, where n is a positive integer. Their expected distance Delta(n), i.e., the mean line segment length, is

Delta(n)=int_0^1...int_0^1_()_(2n)sqrt((x_1-y_1)^2+(x_2-y_2)^2+...+(x_n-y_n)^2)dx_1...dx_ndy_1...dy_n.
(1)

This multiple integral has been evaluated analytically only for small values of n. The case Delta(1) corresponds to line line picking between two random points in the interval [0,1].

HypercubeLinePlot

The first few values for Delta(n) are given in the following table.

nOEISDelta(n)
1--0.3333333333...
2A0915050.5214054331...
3A0730120.6617071822...
4A1039830.7776656535...
5A1039840.8785309152...
6A1039850.9689420830...
7A1039861.0515838734...
8A1039871.1281653402...

The function Delta(n) satisfies

1/3n^(1/2)<=Delta(n)<=(1/6n)^(1/2)sqrt(1/3[1+2(1-3/(5n))^(1/2)])
(2)

(Anderssen et al. 1976), plotted above together with the actual values.

HypercubeLinePickingIntegrands

M. Trott (pers. comm., Feb. 23, 2005) has devised an ingenious algorithm for reducing the 2n-dimensional integral to an integral over a 1-dimensional integrand I_n(x) such that

Delta(n)=int_0^inftyI_n(x)dx.
(3)

The first few values are

I_1=(2e^(-x^2))/(3sqrt(pi))
(4)
I_2=(e^(-x^2)erf(x))/(3x)+(4e^(-2x^2))/(15sqrt(pi))+(4e^(-x^2))/(15sqrt(pi))
(5)
I_3=-2/5e^(-x^2)sqrt(pi)erf^2(x)+(4e^(-2x^2)erf(x))/(5x)+(e^(-x^2)erf(x))/(5x)-(12e^(-3x^2))/(35sqrt(pi))+(68e^(-2x^2))/(105sqrt(pi))+(8e^(-x^2))/(105sqrt(pi))
(6)
I_4=-(2e^(-x^2)pierf^3(x))/(15x)-(136)/(105)e^(-2x^2)sqrt(pi)erf^2(x)-(32)/(105)e^(-x^2)sqrt(pi)erf^2(x)+(197e^(-3x^2)erf(x))/(210x)+(104e^(-2x^2)erf(x))/(105x)+(e^(-x^2)erf(x))/(14x)-(676e^(-4x^2))/(945sqrt(pi))+(16e^(-3x^2))/(35sqrt(pi))+(146e^(-2x^2))/(315sqrt(pi))+(16e^(-x^2))/(945sqrt(pi)).
(7)

In the limit as x->0, these have values for n=1, 2, ... given by 1/sqrt(pi) times 2/3, 6/5, 50/21, 38/9, 74/11, ... (OEIS A103990 and A103991).

This is equivalent to computing the box integral

Delta_n(s)=s/(Gamma(1-1/2s))int_0^infty(1-[d(u)]^n)/(u^(s+1))du,
(8)

for 0<Re(s)<2, where

d(u)=int_0^1int_0^1e^(-u^2(x-y)^2)dxdy
(9)
=(e^(-u^2)-1+sqrt(pi)uerf(u))/(u^2)
(10)

(Bailey et al. 2007).

These give closed-form results for n=1, 2, 3, 4, and 5:

Delta(1)=1/3
(11)
Delta(2)=1/(15)[sqrt(2)+2+5ln(1+sqrt(2))]
(12)
Delta(3)=1/(105)[4+17sqrt(2)-6sqrt(3)+21ln(1+sqrt(2))+42ln(2+sqrt(3))-7pi]
(13)
Delta(4)=(136)/(105)sqrt(2)tan^(-1)(1/2sqrt(2))-(34)/(105)pisqrt(2)+8/(105)sqrt(3)+(73)/(630)sqrt(2)+4/5Cl_2(alpha)-4/5Cl_2(alpha+1/2pi)+(197)/(420)ln3+1/(14)ln(1+sqrt(2))-4/5alphaln(1+sqrt(2))-1/5piln(1+sqrt(2))+(52)/(105)ln(2+sqrt(3))-(23)/(135)-(16)/(315)pi+(26)/(15)K
(14)
Delta(5)=(65)/(42)K-(380)/(6237)sqrt(5)+(568)/(3465)sqrt(3)-4/(189)pi-(449)/(3465)-(73)/(63)sqrt(2)tan^(-1)(1/4sqrt(2))-(184)/(189)ln2+(64)/(189)ln(sqrt(5)+1)+1/(54)ln(1+sqrt(2))+(40)/(63)ln(sqrt(2)+sqrt(6))-5/(28)piln(1+sqrt(2))+(52)/(63)piln2+(295)/(252)ln3+4/(315)pi^2+(3239)/(62370)sqrt(2)-8/(21)sqrt(3)cot^(-1)(sqrt(15))-(52)/(63)piln(sqrt(2)+sqrt(6))-5/7alphaln(1+sqrt(2))+5/7Cl_2(alpha)-5/7Cl_2(alpha+1/2pi)+(52)/(63)K_1,
(15)

where Cl_2(z) is Clausen's integral, K is Catalan's constant, and alpha=sin^(-1)((sqrt(2))/6-2/3). The n=4 case above was published for the first time in this work; the simplified form given above is due to Bailey et al. (2007). The expression for Delta(5) uses the auxiliary integral

K_1=pi[1/2ln(2+sqrt(3))-int_0^infty(e^(-2x^2)erf^3(x))/xdx]
(16)
=int_3^4(sec^(-1)x)/(sqrt((x-3)(x-1)))dx
(17)
=int_0^(Im[cos^(-1)2])sec^(-1)(2+coshtheta)dtheta
(18)

A closed form for K_1 in terms of Clausen functions was obtained by Bailey et al. (2010, eq. 61).

The value Delta(3) obtained for cube line picking is sometimes known as the Robbins constant.

See also

Cube Line Picking, Hypercube Point Picking, Mean Line Segment Length, Robbins Constant, Square Line Picking, Square Point Picking, Square Triangle Picking, Unit Square Integral

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References

Anderssen, R. S.; Brent, R. P.; Daley, D. J.; and Moran, A. P. "Concerning int_0^1...int_0^1sqrt(x_1^2+...+x_k^2)dx_1...dx_k and a Taylor Series Method." SIAM J. Appl. Math. 30, 22-30, 1976.Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, p. 272, 2007.Bailey, D. H.; Borwein, J. M.; and Crandall, R. E. "Box Integrals." J. Comput. Appl. Math. 206, 196-208, 2007.Bailey, D. H.; Borwein, J. M.; and Crandall, R. E. "Advances in the Theory of Box Integrals." Math. Comput. 79, 1839-1866, 2010.Finch, S. R. "Geometric Probability Constants." §8.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 479-484, 2003.Le Lionnais, F. Les nombres remarquables. Paris, France: Hermann, p. 30, 1983.Robbins, D. "Average Distance between Two Points in a Box." Amer. Math. Monthly 85, 278, 1978.Sloane, N. J. A. Sequences A073012, A091505, A103983, A103984, A103985, A103986, A103987, A103988, A103989, A103990, and A103991 in "The On-Line Encyclopedia of Integer Sequences."Trott, M. "The Area of a Random Triangle." Mathematica J. 7, 189-198, 1998.

Referenced on Wolfram|Alpha

Hypercube Line Picking

Cite this as:

Weisstein, Eric W. "Hypercube Line Picking." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HypercubeLinePicking.html

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