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

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

Author

Clark Kimberling, Apr 05 2011

Keywords

• nonn
• cons

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.

Examples

			B=0.285088730048610553714560913780216337024001 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. A188616, A152149, A188543.

Programs

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]]