Magic Geometric Constants

Let E be a compact connected subset of d-dimensional Euclidean space. Gross (1964) and Stadje (1981) proved that there is a unique real number a(E) such that for all x_1, x_2, ..., x_n in E, there exists y in E with


The magic constant m(E) of E is defined by



 diam(E)=max_(u,v in E)sqrt(sum_(k=1)^d(u_k-v_k)^2).

These numbers are also called dispersion numbers and rendezvous values. For any E, Gross (1964) and Stadje (1981) proved that


If I is a subinterval of the line and D is a circular disk in the plane, then


If C is a circle, then


(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 T


Denote the maximum value of m(E) in n-dimensional space by M(n). Then


where Gamma(z) is the gamma function (Nikolas and Yost 1988).

An unrelated quantity characteristic of a given magic square is also known as a magic constant.

See also

Magic Constant

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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.

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Magic Geometric Constants

Cite this as:

Weisstein, Eric W. "Magic Geometric Constants." From MathWorld--A Wolfram Web Resource.

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