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%I #72 Jun 18 2023 15:43:31
%S 1,6,2,3,2,7,8,1,4,4,1,5,7,1,8,4,8,6,3,1,2,5,6,3,5,4,8,2,5,8,9,7,1,8,
%T 5,7,2,4,3,5,2,2,9,5,4,6,6,8,1,7,7,9,4,6,0,5,0,9,7,7,5,8,4,3,5,8,0,9,
%U 5,2,6,5,5,2,7,4,9,0,1,5,0,9,0,4,1,6,2,5,6,8,4,2,4,6,3,3,1,6,5,5,2,4,9,2,6,4,5,4,9,7,7,2
%N Decimal expansion of 2/3 + Pi/6 + sqrt(3)/4.
%C This is claimed to be the minimal cut length required to cut a unit square into 3 pieces of equal area.
%C 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.
%C From _Bernard Schott_, May 29 2019: (Start)
%C 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.
%C The web page of Eduard Baumann gives dissections of different regular polygons into equal area pieces with putatively minimal cut length.
%C Some calculations can be found in the Diophante link, see Problem D447. (End)
%H Jinyuan Wang, <a href="/A307234/b307234.txt">Table of n, a(n) for n = 1..10000</a>
%H Eduard Baumann, <a href="http://www.baumanneduard.ch/EqAreaOverview.htm">Dissection of regular polygons in n equal area pieces with minimal cut length</a>
%H Diophante, <a href="http://www.diophante.fr/problemes-par-themes/geometrie/d4-pavage-du-plan-et-de-l-espace-dissection/1215-d447-ce-qui-parait-evident-nest-pas-optimal">D447, Ce qui paraît évident n'est pas optimal</a>, Jun. 2009 (in French).
%H Zhao Hui Du, <a href="/A307234/a307234.png">Picture showing how to partition the square into 3 parts</a>
%H Frank Morgan,<a href="https://projecteuclid.org/euclid.pjm/1102621620">Soap bubbles in R^2 and in surfaces</a>, Pacific J. Math., Volume 165, Number 2 (1994), 347-361.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PlateausLaws.html">Plateau's laws</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Plateau%27s_laws">Plateau's laws</a>
%H Yi Yang, <a href="https://bbs.emath.ac.cn/thread-2745-2-1.html">A Chinese BBS</a> (in Chinese)
%H <a href="https://www.maths-forum.com/enigmes/tiers-carre-t121803.html">A French BBS</a> (in French)
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%e 1.623278...
%p evalf(2/3 + Pi/6 +sqrt(3)/4, 110); # _Bernard Schott_, May 29 2019
%t RealDigits[2/3+Pi/6+Sqrt[3]/4,10,120][[1]] (* _Harvey P. Dale_, Jun 18 2023 *)
%Y Cf. A093603 (equilateral triangle in 2 pieces), A307235 (square into 4 pieces), A307237 (square into 5 pieces), A307238 (circle into 4 pieces).
%K nonn,cons
%O 1,2
%A _Zhao Hui Du_, Mar 30 2019
%E Edited by _N. J. A. Sloane_, Aug 16 2019
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