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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

 1/nsum_(j=1)^nsqrt(sum_(k=1)^d(x_(j,k)-y_k)^2)=a(E).
(1)

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

 m(E)=(a(E))/(diam(E)),
(2)

where

 diam(E)=max_(u,v in E)sqrt(sum_(k=1)^d(u_k-v_k)^2).
(3)

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

 1/2<=m(E)<1.
(4)

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

 m(I)=m(D)=1/2.
(5)

If C is a circle, then

 m(C)=2/pi=0.6366...
(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 T

 0.6675276<=m(T)<=0.6675284.
(7)

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

M(1)1/2
M(2)m(T)<=M(2)<=(2+sqrt(3))/(3sqrt(3))<0.7182336
M(d)d/(d+1)<=M(d)<=([Gamma(1/2d)]^22^(d-2)sqrt(2d))/(Gamma(d-1/2)sqrt((d+1)pi))<sqrt(d/(d+1))

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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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

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