A152149 - OEIS (original) (raw)

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

COMMENTS

There is a unique (shape of) triangle ABC that is both side-golden and angle-golden. Its angles are B, C=t*B and A=pi-B-t*B, where t is the golden ratio. "Angle-golden" and "side-golden" refer to partitionings of ABC, each in a manner that matches the continued fraction [1,1,1,...] of t. (The partitionings are analogous to the partitioning of the golden rectangle into squares by the removal of exactly 1 square at each stage.)

For doubly silver and doubly e-ratio triangles, see A188543 and A188544.

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.

REFERENCES

Clark Kimberling, "A new kind of golden triangle," in Applications of Fibonacci Numbers, Proc. Fourth International Conference on Fibonacci Numbers and Their Applications, Kluwer, 1991.

FORMULA

B is the number in [0,Pi] such that sin(B*t^2)=t*sin(B), where t=(1+5^(1/2))/2, the golden ratio.

EXAMPLE

The number B begins with 0.65740548 (equivalent to 37.666559... degrees).

MATHEMATICA

r = (1 + 5^(1/2))/2; RealDigits[FindRoot[Sin[r*t + t] == r*Sin[t], {t, 1}, WorkingPrecision -> 120][[1, 2]]][[1]]

PROG

(PARI) t=(1+5^(1/2))/2; solve(b=.6, .7, sin(b*t^2)-t*sin(b)) \\ Iain Fox, Feb 11 2020

EXTENSIONS

Keyword:cons added and offset corrected by R. J. Mathar, Jun 18 2009