OFFSET
1,2
COMMENTS
This is claimed to be the minimal cut length required to cut a unit square into 3 pieces of equal area.
The minimal cut must satisfy the condition that all cuts are straight-line segments or circular arcs, the angle between any three cut edges sharing the same point is 120 degrees, and the sum of the curvatures of the three cut edges meeting at a point is 0. Also a cut edge meeting a side of the unit square must be perpendicular to the side.
From Bernard Schott, May 29 2019: (Start)
The comment that the angle between any three cut edges sharing the same point is 120 degrees follows from Plateau's laws for soap films.
The web page of Eduard Baumann gives dissections of different regular polygons into equal area pieces with putatively minimal cut length.
Some calculations can be found in the Diophante link, see Problem D447. (End)
LINKS
Jinyuan Wang, Table of n, a(n) for n = 1..10000
Diophante, D447, Ce qui paraît évident n'est pas optimal, Jun. 2009 (in French).
Frank Morgan,Soap bubbles in R^2 and in surfaces, Pacific J. Math., Volume 165, Number 2 (1994), 347-361.
Eric Weisstein's World of Mathematics, Plateau's laws
Wikipedia, Plateau's laws
Yi Yang, A Chinese BBS (in Chinese)
A French BBS (in French)
EXAMPLE
1.623278...
MAPLE
evalf(2/3 + Pi/6 +sqrt(3)/4, 110); # Bernard Schott, May 29 2019
MATHEMATICA
RealDigits[2/3+Pi/6+Sqrt[3]/4, 10, 120][[1]] (* Harvey P. Dale, Jun 18 2023 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Zhao Hui Du, Mar 30 2019
EXTENSIONS
Edited by N. J. A. Sloane, Aug 16 2019
STATUS
approved