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Calabi's Triangle

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CalabisTriangle

Calabi's triangle is the unique triangle up to similarity, other than an equilateral triangle, that admits three distinct placements of a largest square contained in it (Calabi 1997). Calabi's triangle is an isosceles triangle with base-to-side length ratio x=a/b=1.55138752454... (OEIS A046095). The exact value is

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

where the cube roots take their principal values. The ratio x, which satisfies sqrt(2)<x<2, 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. 18, 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."

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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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