A093603 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).

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

Offset: 0

Author

Lekraj Beedassy, May 14 2004

Keywords

• easy
• nonn
• cons

Comments

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).

Examples

			0.67338684354429918030954011877308216677216770182700......
		

References

• P. Halmos, Problems for Mathematicians Young and Old, Math. Assoc. of Amer. Washington DC 1991.
• C. W. Triggs, Mathematical Quickies, Dover NY 1985.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Scott Carr, Bisecting an arbitrary triangular cake (with a straight cut of shortest length)

Crossrefs

Cf. A093604.

Programs

Mathematica

RealDigits[Sqrt[Pi]/(2*3^(1/4)), 10, 50][[1]] (* G. C. Greubel, Jan 13 2017 *)

PARI

sqrt(Pi/sqrt(3))/2 \\ G. C. Greubel, Jan 13 2017

Formula

This is sqrt(Pi)/(2*3^(1/4)).