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A093603
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Bisecting a triangular cake using a curved cut of minimal length: decimal expansion of sqrt(Pi/sqrt(3))/2 = d/2, where d^2 = Pi/sqrt(3).
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2
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6, 7, 3, 3, 8, 6, 8, 4, 3, 5, 4, 4, 2, 9, 9, 1, 8, 0, 3, 0, 9, 5, 4, 0, 1, 1, 8, 7, 7, 3, 0, 8, 2, 1, 6, 6, 7, 7, 2, 1, 6, 7, 7, 0, 1, 8, 2, 7, 0, 0, 3, 9, 7, 3, 0, 9, 9, 8, 0, 1, 6, 6, 1, 3, 7, 3, 7, 9, 7, 9, 0, 1, 8, 2, 6, 2, 9, 5, 5, 0, 3, 2, 0, 0, 8, 2, 8, 3, 1, 5, 0, 3, 0, 7, 7, 5, 9, 6, 1, 5, 3, 8, 6, 4, 6
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OFFSET
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0,1
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COMMENTS
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A minimal dissection. The number d/2 = sqrt(Pi/sqrt(3))/2 = sqrt(Pi)/(2*3^(1/4)) gives the length of the shortest cut that bisects a unit-sided equilateral triangle. From A093602, it is plain that d^2 < 2, i.e., (d/2)^2 < 1/2 = square of the bisecting line segment parallel to the triangle's side. d/2 actually is the arc subtending the angle Pi/3 about the center of the circle with radius D/2, where D^2 = 3/d^2. Since Pi/3~1, d~D (see A093604).
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REFERENCES
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P. Halmos, Problems for Mathematicians Young and Old, Math. Assoc. of Amer. Washington DC 1991.
C. W. Triggs, Mathematical Quickies, Dover NY 1985.
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LINKS
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FORMULA
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This is sqrt(Pi)/(2*3^(1/4)).
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EXAMPLE
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0.67338684354429918030954011877308216677216770182700......
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MATHEMATICA
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RealDigits[Sqrt[Pi]/(2*3^(1/4)), 10, 50][[1]] (* G. C. Greubel, Jan 13 2017 *)
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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