OFFSET
0,1
COMMENTS
Known to be irrational. If m is a positive integer (1/Pi)*arccos(1/sqrt(m)) is rational iff m=1,2 or 4.
A constant giving the location (x = y) of the thermodynamic center (also called warmest point or "hot spot") of a unit isosceles right triangle. The thermodynamic center is the warmest point in a convex heat conductor with unitary initial temperature and with boundary grounded at zero temperature. The convex heat conductor here considered is the isosceles right triangle x>0, y>0, x+y <= 1. - Jean-François Alcover, Jul 26 2016
REFERENCES
M. Aigner and G. M. Ziegler, Proofs from The Book, Chap. 7, p. 40, Springer-Verlag, Berlin, 1999.
LINKS
Steven R. Finch, In limbo: Three triangle centers, arXiv:1406.0836 [math.HO] 2014 p. 11.
FORMULA
0 < x < 1/2 maximizing sin(Pi x)*sin(2Pi x), that is x = arctan(sqrt(2)) / Pi, solution to sin(Pi x) = 3 sin(3 Pi x). - Jean-François Alcover, Jul 26 2016
EXAMPLE
0.30408672...
MAPLE
arccos(1/sqrt(3))/Pi; evalf(%) ; # R. J. Mathar, Aug 02 2016
MATHEMATICA
Join[{0}, RealDigits[1/Pi ArcCos[1/Sqrt[3]], 10, 120][[1]]] (* Harvey P. Dale, Feb 22 2015 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Benoit Cloitre, Feb 09 2014
STATUS
approved