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A188617
Decimal expansion of angle B of unique side-silver and angle-golden triangle.
0
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, 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, 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
OFFSET
0,1
COMMENTS
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.
LINKS
Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165-171.
EXAMPLE
B=0.285088730048610553714560913780216337024001 approximately.
MATHEMATICA
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]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Apr 05 2011
STATUS
approved