cp's OEIS Frontend

A307234 Decimal expansion of 2/3 + Pi/6 + sqrt(3)/4.

%I A307234 #73 Feb 16 2025 08:33:55
%S A307234 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 A307234 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 A307234 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 A307234 Decimal expansion of 2/3 + Pi/6 + sqrt(3)/4.
%C A307234 This is claimed to be the minimal cut length required to cut a unit square into 3 pieces of equal area.
%C A307234 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 A307234 From _Bernard Schott_, May 29 2019: (Start)
%C A307234 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 A307234 The web page of Eduard Baumann gives dissections of different regular polygons into equal area pieces with putatively minimal cut length.
%C A307234 Some calculations can be found in the Diophante link, see Problem D447. (End)
%H A307234 Jinyuan Wang, <a href="/A307234/b307234.txt">Table of n, a(n) for n = 1..10000</a>
%H A307234 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 A307234 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 A307234 Zhao Hui Du, <a href="/A307234/a307234.png">Picture showing how to partition the square into 3 parts</a>
%H A307234 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 A307234 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PlateausLaws.html">Plateau's laws</a>
%H A307234 Wikipedia, <a href="https://en.wikipedia.org/wiki/Plateau%27s_laws">Plateau's laws</a>
%H A307234 Yi Yang, <a href="https://bbs.emath.ac.cn/thread-2745-2-1.html">A Chinese BBS</a> (in Chinese)
%H A307234 <a href="https://www.maths-forum.com/enigmes/tiers-carre-t121803.html">A French BBS</a> (in French)
%H A307234 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%e A307234 1.623278...
%p A307234 evalf(2/3 + Pi/6 +sqrt(3)/4, 110); # _Bernard Schott_, May 29 2019
%t A307234 RealDigits[2/3+Pi/6+Sqrt[3]/4,10,120][[1]] (* _Harvey P. Dale_, Jun 18 2023 *)
%Y A307234 Cf. A093603 (equilateral triangle in 2 pieces), A307235 (square into 4 pieces), A307237 (square into 5 pieces), A307238 (circle into 4 pieces).
%K A307234 nonn,cons
%O A307234 1,2
%A A307234 _Zhao Hui Du_, Mar 30 2019
%E A307234 Edited by _N. J. A. Sloane_, Aug 16 2019