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Kűrschák's Tile

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KurschaksTile

Kűrschák's tile is an attractive tiling of the square composed of two types of triangular tiles. It consists of 16 equilateral triangles and 32 15 degrees-15 degrees-150 degrees isosceles triangles arranged in the shape of a dodecagon.

KurschaksSquare

The composition of Kürschák's tile is motivated by drawing inward-pointing equilateral triangles on each side of a unit square and then connecting adjacent vertices to form a smaller square rotated 45 degrees with respect to the original square. Joining the midpoints of the square together with the intersections of the equilateral triangles then gives a dodecagon (Wells 1991) with circumradius

R=sin(pi/(12))=1/4(sqrt(6)-sqrt(2)).

See also

Dodecagon, Equilateral Triangle, Isosceles Triangle, Kűrschák's Theorem

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References

Alexanderson, G. L. and Seydel, K. "Kürschák's Tile." Math. Gaz. 62, 192-196, 1978.Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 30-32, 1985.Schoenberg, I. Mathematical Time Exposures. Washington, DC: Math. Assoc. Amer., p. 7, 1982.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, pp. 136-137, 1991.

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Kűrschák's Tile

Cite this as:

Weisstein, Eric W. "Kűrschák's Tile." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/KurschaksTile.html

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