

A188544


Decimal expansion of the angle B in the doubly eratio triangle ABC.


2



3, 6, 8, 9, 3, 1, 2, 7, 4, 9, 4, 7, 8, 0, 5, 8, 4, 2, 6, 5, 1, 9, 1, 1, 2, 7, 2, 6, 8, 8, 6, 4, 0, 8, 5, 7, 1, 8, 6, 8, 3, 4, 4, 2, 8, 8, 3, 5, 2, 6, 1, 9, 0, 6, 5, 9, 8, 5, 4, 6, 2, 1, 2, 1, 1, 1, 1, 7, 6, 5, 9, 8, 7, 5, 6, 8, 4, 9, 0, 6, 0, 6, 7, 0, 1, 2, 1, 0, 6, 0, 4, 8, 5, 9, 8, 4, 2, 8, 4, 2, 8, 0, 9, 9, 1, 1, 8, 2, 1, 8, 8, 5, 9, 6, 9, 8, 4, 4, 2, 2, 9, 4, 8, 7, 3, 4, 6, 8
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OFFSET

0,1


COMMENTS

There is a unique (shape of) triangle ABC that is both sideeratio and angleeratio. Its angles are B, t*B and piBt*B, where t=e. "Sideeratio" and "angleeratio" refer to partitionings of ABC, each in a manner that matches the continued fraction [2,1,2,1,1,4,1,1,6,...] of t. For doubly golden and doubly silver triangles, see A152149 and A188543. For the side partitioning and angle partitioning (i,e, constructions) which match arbitrary continued fractions (of sidelength ratios and angle ratios), see the 2007 reference.


LINKS



FORMULA

B is the number in [0,Pi] such that sin(B*e^2)=e*sin(B).


EXAMPLE

B=0.36893127494780584265191127268864 approximately.
B=21.1382 degrees approximately.


MATHEMATICA

r = E; Clear[t]; RealDigits[FindRoot[Sin[r*t + t] == r*Sin[t], {t, 1}, WorkingPrecision > 120][[1, 2]]][[1]]


CROSSREFS



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AUTHOR



EXTENSIONS



STATUS

approved



