Cube Point Picking

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Let two points be picked randomly from a unit -D Hypercube. The expected distance between the points is then

The function satisfies
 (1)

(Anderssen et al. 1976).

Pick points , ..., randomly in a unit -cube. Let be the Convex Hull, so

 (2)

Let be the expected -D Volume (the Content) of , be the expected -D Surface Area of , and the expected number of Vertices on the Polygonal boundary of . Then

 (3)

 (4)

and

 (5)

(Rényi and Sulanke 1963, 1964). The average Distance between two points chosen at random inside a unit cube is
 (6)

(Robbins 1978, Le Lionnais 1983).

Pick points on a Cube, and space them as far apart as possible. The best value known for the minimum straight Line distance between any two points is given in the following table.

 5 1.1180339887498 6 1.0606601482100 7 1 8 1 9 0.86602540378463 10 0.74999998333331 11 0.70961617562351 12 0.70710678118660 13 0.70710678118660 14 0.70710678118660 15 0.625

See also Cube Triangle Picking, Discrepancy Theorem, Point Picking

References

Anderssen, R. S.; Brent, R. P.; Daley, D. J.; and Moran, A. P. Concerning and a Taylor Series Method.'' SIAM J. Appl. Math. 30, 22-30, 1976.

Bolis, T. S. Solution to Problem E2629. Average Distance Between Two Points in a Box.'' Amer. Math. Monthly 85, 277-278, 1978.

Finch, S. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/geom/geom.html

Ghosh, B. Random Distances within a Rectangle and Between Two Rectangles.'' Bull. Calcutta Math. Soc. 43, 17-24, 1951.

Holshouser, A. L.; King, L. R.; and Klein, B. G. Solution to Problem E3217, Minimum Average Distance Between Points in a Rectangle.'' Amer. Math. Monthly 96, 64-65, 1989.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 30, 1983.

Rényi, A. and Sulanke, R. Über die konvexe Hülle von zufällig gewählten Punkten, I.'' Z. Wahrscheinlichkeits 2, 75-84, 1963.

Rényi, A. and Sulanke, R. Über die konvexe Hülle von zufällig gewählten Punkten, II.'' Z. Wahrscheinlichkeits 3, 138-147, 1964.

Robbins, D. Average Distance Between Two Points in a Box.'' Amer. Math. Monthly 85, 278, 1978.

Santaló, L. A. Integral Geometry and Geometric Probability. Reading, MA: Addison-Wesley, 1976.

© 1996-9 Eric W. Weisstein
1999-05-25