A307234 -id:A307234 - cp's OEIS frontend

A307238 This is claimed to be the minimal cut length required to cut a unit circle into 4 pieces of equal area after making certain assumptions about the cuts (compare A307234).

3, 9, 4, 5, 7, 0, 2, 9, 6, 7, 2, 6, 7, 1, 8, 5, 7, 1, 3, 8, 4, 2, 8, 9, 9, 5, 5, 2, 1, 1, 1, 7, 9, 9, 1, 8, 8, 8, 7, 4, 8, 3, 5, 4, 0, 1, 0, 7, 4, 7, 4, 1, 5, 2, 4, 2, 6, 8, 1, 6, 9, 6, 7, 1, 3, 1, 8, 7, 4, 3, 2, 9, 8, 3, 8, 1, 6, 2, 0, 0, 8, 4, 8, 7, 8, 5, 1, 4, 7, 7, 3, 8, 6, 0, 2, 1

Offset: 1

Zhao Hui Du, Mar 30 2019

It is assumed that:
all cut edges must be straight-line segments or circular arcs,
the angle between any two cut edges sharing the same point is 120 degrees,
the sum of the curvatures of three cut edges meeting at a point is 0, and
cut edges meeting the unit circle must be perpendicular to the circle.

			3.945702967267185713842899552111799188874835401074741524...

Zhao Hui Du, Picture showing the minimum cut length
J. Hu, A Chinese BBS discussing the problem

Cf. A307234, A207235, A307237.

p[x_]:=Sin[x]/(Sin[Pi/3]+Sin[Pi/3-x]); q[x_]:=Sin[Pi/3-x]/(Sin[Pi/3]+Sin[Pi/3-x]); R[x_]:=q[x]/Tan[x/2]; S[x_]:=(Pi/3 - x -p[x]*Sin[Pi/3 -x] + R[x]^2*(x-Sin[x]))/2; d := FindRoot[S[x] - Pi/8, {x, 0.1, 0.5}, WorkingPrecision -> 150]; RealDigits[2*(p[x] + 2*x*R[x])/.d, 10, 100][[1]] (* G. C. Greubel, Jul 02 2019 *)

default(realprecision, 100);
p(t)=sin(t)/(sin(Pi/3)+sin(Pi/3-t));
q(t)=sin(Pi/3-t)/(sin(Pi/3)+sin(Pi/3-t));
R(t)=q(t)/tan(t/2);
S(t)=( Pi/3 - t - p(t)*sin(Pi/3-t) + R(t)^2*(t-sin(t)) )/2;
d = solve(t=0.1,0.5, S(t)-Pi/8);
2*(p(d)+2*d*R(d))

Terms a(32) onward added by G. C. Greubel, Jul 02 2019

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

A307235 Decimal expansion of sqrt(2) + sqrt((3-3*sqrt(3)+Pi)/3).

Original entry on oeis.org

1, 9, 7, 5, 5, 9, 2, 8, 8, 4, 7, 8, 1, 5, 0, 0, 5, 1, 5, 9, 1, 6, 4, 6, 5, 2, 5, 8, 5, 1, 3, 5, 8, 9, 3, 4, 6, 5, 1, 6, 7, 4, 7, 9, 1, 6, 8, 4, 3, 2, 0, 8, 9, 8, 4, 5, 6, 0, 4, 2, 4, 3, 9, 1, 1, 7, 6, 6, 4, 7, 0, 9, 2, 8, 0, 5, 8, 4, 2, 8, 4, 7, 4, 2, 4, 6, 2, 5, 4, 2, 6, 4, 3, 1, 2, 1, 3

Offset: 1

Zhao Hui Du, Mar 30 2019

This is claimed to be the minimal cut length required to cut a unit square into 4 pieces of equal area after making certain assumptions about the cuts (compare A307234).

			1.975592884781500515916465258513589346516747916843208984560424391176647...

Eduard Baumann, Dissection of regular polygons in n equal area pieces with minimal cut length
Zhao Hui Du, Picture showing how to cut the square into 4 pieces
Paolo Licheri, f006 Tagliare una torta, (Cut a Cake, in Italian).
Yi Yang, A Chinese BBS
Index entries for transcendental numbers

Cf. A307234.

SetDefaultRealField(RealField(100)); R:= RealField(); Sqrt(2) + Sqrt((Pi(R)+3-3*Sqrt(3))/3); // G. C. Greubel, Jul 02 2019

RealDigits[Sqrt[2] + Sqrt[(Pi+3-3*Sqrt[3])/3], 10, 100][[1]] (* G. C. Greubel, Jul 02 2019 *)

default(realprecision, 100); sqrt(2) + sqrt((Pi+3-3*sqrt(3))/3) \\ G. C. Greubel, Jul 02 2019

numerical_approx(sqrt(2) + sqrt((pi+3-3*sqrt(3))/3), digits=100) # G. C. Greubel, Jul 02 2019

Terms a(32) onward added by G. C. Greubel, Jul 02 2019

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

A307237 Decimal expansion of 2 + (-6 + (1+sqrt(3))Pi)sqrt(2/(15(2Pi-3 +(Pi-3)*sqrt(3)))).

Original entry on oeis.org

2, 5, 0, 2, 1, 1, 2, 9, 3, 0, 4, 2, 7, 1, 8, 6, 2, 3, 2, 7, 0, 5, 5, 8, 5, 1, 9, 4, 0, 0, 8, 6, 9, 2, 2, 5, 1, 3, 9, 5, 8, 7, 5, 6, 2, 6, 2, 3, 0, 7, 7, 4, 5, 5, 3, 5, 3, 1, 9, 0, 1, 1, 9, 5, 5, 0, 2, 8, 0, 5, 9, 4, 0, 9, 4, 1, 9, 3, 1, 3, 5, 8, 0, 1, 1, 2, 0, 6, 5, 1, 5, 6, 8, 2, 8, 6

Offset: 1

Zhao Hui Du, Mar 30 2019

This is claimed to be the minimal cut length required to cut a unit square into 5 pieces of equal area after making certain assumptions about the cuts (compare A307234).

			2.5021129304271862327055851940086922513958756262307745535319011955...

Simon Cox, Minimal perimeter enclosing N cells of equal area with N no more than 42
Zhao Hui Du, Picture showing how square is cut into 5 pieces
M. Gardner, The Mathemagician and Pied Puzzler, Problem H14.

Cf. A307234, A307235, A307238.

SetDefaultRealField(RealField(100)); R:= RealField(); 2 +(-6 +(1+Sqrt(3))*Pi(R))*Sqrt(2/(15*(2*Pi(R)-3 +(Pi(R)-3)*Sqrt(3)))); // G. C. Greubel, Jul 02 2019

RealDigits[2 +(-6 +(1+Sqrt[3])*Pi)*Sqrt[2/(15*(2*Pi -3 +(Pi-3)*Sqrt[3]) )], 10, 100][[1]] (* G. C. Greubel, Jul 02 2019 *)

default(realprecision, 100); 2 +(-6 +(1+sqrt(3))*Pi)*sqrt(2/(15*(2*Pi-3 +(Pi-3)*sqrt(3)))) \\ G. C. Greubel, Jul 02 2019

numerical_approx(2 + (-6 + (1+sqrt(3))*pi)*sqrt(2/(15*(2*pi-3 +(pi-3)*sqrt(3)))), digits=100) # G. C. Greubel, Jul 02 2019

Terms a(32) onward added by G. C. Greubel, Jul 02 2019

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

cp's OEIS Frontend

A307238 This is claimed to be the minimal cut length required to cut a unit circle into 4 pieces of equal area after making certain assumptions about the cuts (compare A307234).

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A307235 Decimal expansion of sqrt(2) + sqrt((3-3*sqrt(3)+Pi)/3).

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A307237 Decimal expansion of 2 + (-6 + (1+sqrt(3))*Pi)*sqrt(2/(15*(2*Pi-3 +(Pi-3)*sqrt(3)))).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Extensions

A307237 Decimal expansion of 2 + (-6 + (1+sqrt(3))Pi)sqrt(2/(15(2Pi-3 +(Pi-3)*sqrt(3)))).