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# Magic Geometric Constants

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)

(OEIS A060294). 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 -dimensional 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.

Magic Constant

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## References

Finch, S. R. "Rendezvous Constants." §8.21 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 537-542, 2003.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.Sloane, N. J. A. Sequence A060294 in "The On-Line Encyclopedia of Integer Sequences."Stadje, W. "A Property of Compact Connected Spaces." Arch. Math. (Basel) 36, 275-280, 1981.

## Referenced on Wolfram|Alpha

Magic Geometric Constants

## Cite this as:

Weisstein, Eric W. "Magic Geometric Constants." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MagicGeometricConstants.html