## Magic Geometric Constants

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

Let be a compact connected subset of -dimensional Euclidean Space. Gross (1964) and Stadje (1981) proved that there is a unique Real Number such that for all , , ..., , there exists with

 (1)

The magic constant of is defined by
 (2)

where
 (3)

These numbers are also called Dispersion Numbers and Rendezvous Values. For any , Gross (1964) and Stadje (1981) proved that
 (4)

If is a subinterval of the Line and is a circular Disk in the Plane, then
 (5)

If is a Circle, then
 (6)

An expression for the magic constant of an Ellipse in terms of its Semimajor and Semiminor Axes lengths is not known. Nikolas and Yost (1988) showed that for a Reuleaux Triangle
 (7)

Denote the Maximum value of in -D space by . Then

where is the Gamma Function (Nikolas and Yost 1988).

An unrelated quantity characteristic of a given Magic Square is also known as a Magic Constant.

References

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

Cleary, J.; Morris, S. A.; and Yost, D. Numerical Geometry--Numbers for Shapes.'' Amer. Math. Monthly 95, 260-275, 1986.

Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, 1994.

Gross, O. The Rendezvous Value of Metric Space. Princeton, NJ: Princeton University Press, pp. 49-53, 1964.

Nikolas, P. and Yost, D. The Average Distance Property for Subsets of Euclidean Space.'' Arch. Math. (Basel) 50, 380-384, 1988.

Stadje, W. A Property of Compact Connected Spaces.'' Arch. Math. (Basel) 36, 275-280, 1981.