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A188543 Decimal expansion of the angle B in the doubly silver triangle ABC. 6


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

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

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

%N Decimal expansion of the angle B in the doubly silver triangle ABC.

%C There is a unique (shape of) triangle ABC that is both side-silver and angle-silver. Its angles are B, t*B and pi-B-t*B, where t is the silver ratio, 1+sqrt(2), at A014176. "Side-silver" and "angle-silver" refer to partitionings of ABC, each in a manner that matches the continued fraction [2,2,2,...] of t. For doubly golden and doubly e-ratio triangles, see A152149 and A188544. For the side partitioning and angle partitioning (i,e, constructions in which 2 triangles are removed at each stage, analogous to the removal of 1 square at each stage of the partitioning of the golden rectangle into squares) which match arbitrary continued fractions (of sidelength ratios and angle ratios), see the 2007 reference.

%H Clark Kimberling, <a href="http://www.heldermann.de/JGG/JGG11/JGG112/jgg11014.htm">Two kinds of golden triangles, generalized to match continued fractions</a>, Journal for Geometry and Graphics, 11 (2007) 165-171.

%F B is the number in [0,Pi] such that sin(B*t^2)=t*sin(B), where t=1+sqrt(2), the silver ratio.

%e B=0.4235466615478147887414222095779154 approximately.

%e B=24.2674 degrees approximately.

%t r = 1+2^(1/2); Clear[t]; RealDigits[FindRoot[Sin[r*t + t] == r*Sin[t], {t, 1}, WorkingPrecision -> 120][[1, 2]]][[1]]

%Y Cf. A152149, A014176, A188544.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Apr 03 2011

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