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Calabi's Triangle

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CalabisTriangle

Calabi's triangle is the unique triangle other that the equilateral triangle for which the largest inscribed square can be inscribed in three different ways (Calabi 1997). Calabi's triangle is an isosceles triangle with base-to-side length ratio x=a/b=1.55138752454... (OEIS A046095), where

x=1/3+((-23+3isqrt(237))^(1/3))/(3·2^(2/3))+(11)/(3[2(-23+3isqrt(237))]^(1/3))

is the largest positive root of

2x^3-2x^2-3x+2=0.

It has continued fraction [1, 1, 1, 4, 2, 1, 2, 1, 5, 2, 1, 3, 1, 1, 390, ...] (OEIS A046096).

See also

Graham's Biggest Little Hexagon, Square Inscribing, Triangle

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References

Calabi, E. "Outline of Proof Regarding Squares Wedged in Triangle." Nov. 3, 1997. https://web.archive.org/web/20100830145434/http://algo.inria.fr:80/csolve/calabi.html.Conway, J. H. and Guy, R. K. "Calabi's Triangle." In The Book of Numbers. New York: Springer-Verlag, p. 206, 1996.Sloane, N. J. A. Sequences A046095 and A046096 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Calabi's Triangle

Cite this as:

Weisstein, Eric W. "Calabi's Triangle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CalabisTriangle.html

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