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A188615
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Decimal expansion of Brocard angle of side-silver right triangle.
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9
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3, 3, 9, 8, 3, 6, 9, 0, 9, 4, 5, 4, 1, 2, 1, 9, 3, 7, 0, 9, 6, 3, 9, 2, 5, 1, 3, 3, 9, 1, 7, 6, 4, 0, 6, 6, 3, 8, 8, 2, 4, 4, 6, 9, 0, 3, 3, 2, 4, 5, 8, 0, 7, 1, 4, 3, 1, 9, 2, 3, 9, 6, 2, 4, 8, 9, 9, 1, 5, 8, 8, 8, 6, 6, 4, 8, 4, 8, 4, 1, 1, 4, 6, 0, 7, 6, 5, 7, 9, 2, 5, 0, 0, 1, 9, 7, 6, 1, 2, 8, 5, 2, 1, 2, 9, 7, 6, 3, 8, 0, 7, 4, 0, 2, 2, 9, 4, 4, 7, 4, 1, 5, 2, 3, 9, 3, 5, 7, 5, 6
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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-silver 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=(silver ratio)=(1+sqrt(2)). This is the unique right triangle matching the continued fraction [2,2,2,...] of r; i.e, under the side-partitioning procedure described in the 2007 reference, there are exactly 2 removable subtriangles at each stage. (This is analogous to the removal of 2 squares at each stage of the partitioning of the silver rectangle as a nest of squares.)
Archimedes's-like scheme: set p(0) = 1/(2*sqrt(2)), q(0) = 1/3; p(n+1) = 2*p(n)*q(n)/(p(n)+q(n)) (harmonic mean, i.e., 1/p(n+1) = (1/p(n) + 1/q(n))/2), q(n+1) = sqrt(p(n+1)*q(n)) (geometric mean, i.e., log(q(n+1)) = (log(p(n+1)) + log(q(n)))/2), for n >= 0. The error of p(n) and q(n) decreases by a factor of approximately 4 each iteration, i.e., approximately 2 bits are gained by each iteration. Set r(n) = (2*q(n) + p(n))/3, the error decreases by a factor of approximately 16 for each iteration, i.e., approximately 4 bits are gained by each iteration. For a similar scheme see also A244644. - A.H.M. Smeets, Jul 12 2018
This angle is also the half-angle at the summit of the Kelvin wake pattern traced by a boat. - Robert FERREOL, Sep 27 2019
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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(8)).
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EXAMPLE
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Brocard angle: 0.3398369094541219370963925133917640663882 approx.
Brocard angle: 19.471220634490691369245999 degrees, approx.
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MATHEMATICA
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r=1+2^(1/2);
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)/(4area)];
N[brocard, 130]
RealDigits[N[brocard, 130]][[1]]
N[180 brocard/Pi, 130] (* degrees *)
RealDigits[ArcCos[Sqrt[8/9]], 10, 50][[1]] (* G. C. Greubel, Nov 18 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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