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Polyiamond

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Polyiamonds

A polyiamond is a polyform consisting of a collection of equal-sized equilateral triangles arranged with coincident sides. Polyiamonds are sometimes simply known as iamonds.

The top row of hexiamonds in the above figure are known as the bar, crook, crown, sphinx, snake, and yacht. The bottom row of 6-polyiamonds are known as the chevron, signpost, lobster, hook, hexagon, and butterfly.

The number of two-sided (i.e., can be picked up and flipped, so mirror image pieces are considered identical) polyiamonds made up of n triangles are 1, 1, 1, 3, 4, 12, 24, 66, 160, 448, ... (OEIS A000577). The number of one-sided polyiamonds composed of n triangles are 1, 1, 1, 4, 6, 19, 43, 121, ... (OEIS A006534). One of the 160 9-polyiamonds has a hole (Gardner 1984, p. 174).

Polyiamond2-Trees

The polyiamonds illustrated above are 2-trees.

PolyiamondsWithHoles

The numbers of n-polyiamonds with holes for n=9, 10, 11, ... are 1, 4, 25, 108, 450, ... (OEIS A070764; Myers), the first few of which are illustrated above.

See also

Polyabolo, Polyhex, Polyiamond Tiling, Polyomino, Triangular Snake Graph

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References

Beeler, M. Item 112 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 48-50, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/polyominos.html#item112.Clarke, A. L. "Polyiamonds." http://www.recmath.com/PolyPages/PolyPages/Polyiamonds.htm.Gardner, M. "Mathematical Games: On Polyiamonds: Shapes That are Made Out of Equilateral Triangles." Sci. Amer. 211, Dec. 1964.Gardner, M. "Polyiamond." Ch. 18 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 173-182, 1984.Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems, and Packings, 2nd ed. Princeton, NJ: Princeton University Press, pp. 90-92, 1994.Keller, M. "Counting Polyforms." http://members.aol.com/wgreview/polyenum.html.Myers, J. "Polyomino Tiling." http://www.srcf.ucam.org/~jsm28/tiling/.O'Beirne, T. H. "Pentominoes and Hexiamonds." New Scientist 12, 379-380, 1961.Pegg, E. Jr. "Iamonds." http://www.mathpuzzle.com/iamond.htm.Reeve, J. E. and Tyrrell, J. A. "Maestro Puzzles." Math. Gaz. 45, 97-99, 1961.Sloane, N. J. A. Sequences A000577/M2374 and A006534/M3287 in "The On-Line Encyclopedia of Integer Sequences."Torbijn, I. P. J. "Polyiamonds." J. Recr. Math. 2, 216-227, 1969.Vichera, M. "Polyforms." http://www.vicher.cz/puzzle/polyforms.htm.von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 342-343, 1993.

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Polyiamond

Cite this as:

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

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