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A188595
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Decimal expansion of Brocard angle of side-golden right triangle.
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6
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4, 2, 0, 5, 3, 4, 3, 3, 5, 2, 8, 3, 9, 6, 5, 1, 2, 7, 8, 8, 8, 2, 6, 2, 5, 1, 5, 9, 1, 3, 2, 1, 5, 3, 7, 3, 3, 5, 1, 0, 3, 9, 3, 9, 2, 8, 1, 9, 9, 1, 9, 6, 0, 9, 8, 8, 9, 2, 6, 1, 4, 0, 2, 3, 4, 6, 0, 4, 4, 6, 5, 1, 7, 3, 8, 1, 6, 8, 6, 8, 0, 2, 5, 9, 2, 6, 7, 0, 0, 2, 4, 2, 5, 7, 9, 2, 5, 1, 6, 8, 9, 1, 4, 8, 9, 3, 4, 2, 6, 1, 8, 0, 1, 5, 2, 5, 8, 0, 2, 5, 2, 1, 1, 7, 7, 8, 2, 0, 6, 8
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OFFSET
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0,1
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COMMENTS
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The Brocard angle is invariant of the size of the side-golden right triangle ABC. The shape of ABC is given by sidelengths a,b,c, where a=r*b, and c=sqrt(a^2+b^2), where r=(golden ratio)=(1+sqrt(5))/2. This is the unique right triangle matching the continued fraction [1,1,1,...] of r; i.e, under the side-partitioning procedure described in the 2007 reference, there is exactly 1 removable subtriangle at each stage. (This is analogous to the removal of 1 square at each stage of the partitioning of the golden rectangle as a nest of squares.)
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LINKS
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FORMULA
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Brocard angle: arccot((a^2+b^2+c^2)/(4*area(ABC))) = arccot(sqrt(5)).
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EXAMPLE
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Brocard angle: 0.420534335283965127888262515913215373 approx.
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MATHEMATICA
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r=(1+5^(1/2))/2; b=1; a=r*b; c=(a^2+b^2)^(1/2); area=(1/4)((a+b+c)(b+c-a)(c+a-b)(a+b-c))^(1/2); brocard = ArcCot[(a^2+b^2+c^2)/(4 area)]; RealDigits[N[brocard, 130]][[1]]
RealDigits[ArcTan[Sqrt[1/5]], 10, 50][[1]] (* G. C. Greubel, Nov 21 2017 *)
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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