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

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, 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, 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

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

Eduard Baumann, Dissection of regular polygons in n equal area pieces with minimal cut length

Diophante, D447, Ce qui paraît évident n'est pas optimal, Jun. 2009 (in French).

Zhao Hui Du, Picture showing how to partition the square into 3 parts

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)

Index entries for transcendental numbers

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

Cf. A093603 (equilateral triangle in 2 pieces), A307235 (square into 4 pieces), A307237 (square into 5 pieces), A307238 (circle into 4 pieces).

Sequence in context: A055942 A127916 A165063 * A371875 A021620 A093408

Adjacent sequences: A307231 A307232 A307233 * A307235 A307236 A307237

Keyword

nonn,cons

Author

Zhao Hui Du, Mar 30 2019

Extensions

Edited by N. J. A. Sloane, Aug 16 2019

Status

approved