A188616 Decimal expansion of angle B of unique side-golden and angle-silver triangle.

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, 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, 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

Offset: 0

Author

Clark Kimberling, Apr 05 2011

Keywords

• nonn
• cons

Comments

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.

Examples

			B=0.59106779970516487976323237419662 approximately
		

Links

• Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165-171.

Crossrefs

Cf. A152149, A188543.

Programs

Mathematica

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]]