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A188617 Decimal expansion of angle B of unique side-silver and angle-golden triangle.

%I A188617 #17 Oct 20 2024 00:27:40
%S A188617 2,8,5,0,8,8,7,3,0,0,4,8,6,1,0,5,5,3,7,1,4,5,6,0,9,1,3,7,8,0,2,1,6,3,
%T A188617 3,7,0,2,4,0,0,1,0,2,5,6,9,7,6,7,5,9,1,4,1,8,1,0,0,4,0,5,1,3,3,9,0,9,
%U A188617 0,3,9,6,5,6,7,1,4,0,1,1,5,5,4,0,7,0,3,8,4,5,0,1,3,8,3,1,0,8,0,1,6,1,4,0,7,1,6,0,9,8,8,9,3,6,8,9,1,7,6,9
%N A188617 Decimal expansion of angle B of unique side-silver and angle-golden triangle.
%C A188617 Let r=(silver ratio)=1+sqrt(2) and u=(golden ratio)=(1+sqrt(5))/2.  A triangle ABC with sidelengths a,b,c is side-silver if a/b=r and angle-golden 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 2 triangles of a certain kind are available at each stage of a side-partitioning procedure, and exactly 1 triangle of another kind are available for angle-partitioning.  For details, see the 2007 reference.
%H A188617 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 A188617 B=0.285088730048610553714560913780216337024001 approximately.
%t A188617 r=(1+5^(1/2))/2; u=1+2^(1/2); Clear[t]; RealDigits[FindRoot[Sin[r*t + t] == u*Sin[t], {t, 1}, WorkingPrecision->120][[1, 2]]][[1]]
%Y A188617 Cf. A188616, A152149, A188543.
%K A188617 nonn,cons
%O A188617 0,1
%A A188617 _Clark Kimberling_, Apr 05 2011