A188595 Decimal expansion of Brocard angle of side-golden right triangle.

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

Offset: 0

Author

Clark Kimberling, Apr 05 2011

Keywords

• nonn
• cons

Comments

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.)

Also <3_5> in Conway et al. (1999). - Eric W. Weisstein, Nov 06 2024

Examples

			Brocard angle: 0.420534335283965127888262515913215373 approx.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• John H. Conway, Charles Radin, and Lorenzo Sadun, On Angles Whose Squared Trigonometric Functions are Rational, arXiv:math-ph/9812019, 1998. See also Discr. Computat. Geom. (1999) Vol. 22, 321-332.
• Eric Weisstein's World of Mathematics, Dehn Invariant.
• Index entries for transcendental numbers.

Crossrefs

Cf. A188594, A152149, A188615, A228496.

Programs

Magma

[Arctan(Sqrt(1/5))]; // G. C. Greubel, Nov 21 2017

Mathematica

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 *)

PARI

atan(sqrt(1/5)) \\ G. C. Greubel, Nov 21 2017

Formula

Brocard angle: arccot((a^2+b^2+c^2)/(4*area(ABC))) = arccot(sqrt(5)).

Equals A228496/2. - Hugo Pfoertner, Nov 06 2024