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# Triangle Packing

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

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

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

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.

 max. area reference max. area reference 1 1 Stewart 1997 11 495 Stewart 1997 2 2 Stewart 1997 12 860 Stewart 1998 3 3 Stewart 1997 13 1559 Stewart 1998 4 7 Stewart 1997 14 2831 Stewart 1998 5 11 Stewart 1997 15 4782 Stewart 1999 6 20 Stewart 1997 16 8559 Stewart 1998 7 36 Stewart 1997 17 14279 Stewart 1998 8 71 Stewart 1997 9 146 Stewart 1997 10 260 Stewart 1997

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.

Triangle Packing

## Cite this as:

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