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Decimal expansion of angle B of unique side-golden and angle-silver triangle.
2

%I #13 May 04 2016 12:11:10

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

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

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

%N Decimal expansion of angle B of unique side-golden and angle-silver triangle.

%C Let r=(golden ratio)=(1+sqrt(5))/2 and u=(silver ratio)=1+sqrt(2). A triangle ABC with sidelengths a,b,c is side-golden if a/b=r and angle-silver if C/B=u. There is a unique triangle that has both properties. The quickest way to understand the geometric reasons for the names is by analogy to the golden and silver rectangles. For the former, exactly 1 square is available at each stage of the partitioning of the rectangle into a nest of squares, and for the former, exactly 2 squares are available. Analogously, for ABC, exactly one 1 triangle of a certain kind is available at each stage of a side-partitioning procedure, and exactly 2 triangles of another kind are available for angle-partitioning. For details, 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.

%e B=0.59106779970516487976323237419662 approximately

%t Remove["Global`*"]; r=1+2^(1/2); u=(1+5^(1/2))/2; RealDigits[FindRoot[Sin[r*t+t]==u*Sin[t],{t,1}, WorkingPrecision->120][[1,2]]][[1]]

%Y Cf. A152149, A188543.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Apr 05 2011