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Triangle Packing


TrianglesInTriangles

The best known packings of equilateral triangles into an equilateral triangle are illustrated above for the first few cases (Friedman).

TrianglesInCircles

The best known packings of equilateral triangles into a circle are illustrated above for the first few cases (Friedman).

TrianglesInSquares

The best known packings of equilateral triangles into a square are illustrated above for the first few cases (Friedman).

EquilateralPacking1-7
EquilateralPacking8-11
EquilateralPacking15
EquilateralPacking17

Stewart (1998, 1999) considered the problem of finding the largest convex area that can be nontrivially tiled with equilateral triangles whose sides are integers for a given number of triangles and which have no overall common divisor. There is no upper limit if an arbitrary number of triangles are used. The following table gives the best known packings for small numbers of triangles.

nmax. areareferencenmax. areareference
11Stewart 199711495Stewart 1997
22Stewart 199712860Stewart 1998
33Stewart 1997131559Stewart 1998
47Stewart 1997142831Stewart 1998
511Stewart 1997154782Stewart 1999
620Stewart 1997168559Stewart 1998
736Stewart 19971714279Stewart 1998
871Stewart 1997
9146Stewart 1997
10260Stewart 1997

See also

Circle Packing, Equilateral Triangle, Kenmotu Point, Packing, Square Packing

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References

Friedman, E. "Circles in Triangles." http://www.stetson.edu/~efriedma/cirintri/.Friedman, E. "Squares in Triangles." http://www.stetson.edu/~efriedma/squintri/.Friedman, E. "Triangles in Triangles." http://www.stetson.edu/~efriedma/triintri/.Graham, R. L. and Lubachevsky, B. D. "Dense Packings of Equal Disks in an Equilateral Triangle: From 22 to 34 and Beyond." Electronic J. Combinatorics 2, No. 1, F1, 1-39, 1995. http://www.combinatorics.org/Volume_2/Abstracts/v2i1f1.html.Stewart, I. "Squaring the Square." Sci. Amer. 277, 94-96, July 1997.Stewart, I. "Mathematical Recreations: Monks, Blobs and Common Knowledge. Feedback." Sci. Amer. 279, 97, Aug. 1998.Stewart, I. "Mathematical Recreations: The Synchronicity of Firefly Flashing. Feedback." Sci. Amer. 280, 106, Mar. 1999.

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Triangle Packing

Cite this as:

Weisstein, Eric W. "Triangle Packing." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TrianglePacking.html

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