OFFSET
1,1
COMMENTS
This ratio 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.)
LINKS
Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165-171.
FORMULA
(circumradius)/(inradius) = abc(a+b+c)/(8*area^2), where area=area(ABC).
EXAMPLE
ratio=3.26197262739566856105805510300327465221450 approx.
MAPLE
a179260 := sqrt(2+sqrt(2)) ; a014176 := 1+sqrt(2) ; 1/(a014176/a179260-1) ; evalf(%) ; # R. J. Mathar, Apr 05 2011
MATHEMATICA
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); RealDigits[N[a*b*c*(a+b+c)/(8*area^2), 130]][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Apr 05 2011
STATUS
approved