A generalization of the
polyominoes using a collection of equal-sized equilateral triangles (instead
of squares) 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,
The number of two-sided (i.e., can be picked up and flipped, so
mirror image pieces are considered identical) polyiamonds made up of triangles are 1, 1, 1, 3, 4, 12, 24, 66, 160, 448, ... (OEIS
A000577). The number of one-sided polyiamonds
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).
The numbers of
with holes for ,
10, 11, ... are 1, 4, 25, 108, 450, ... (OEIS A070764;
Myers), the first few of which are illustrated above.
See also Polyabolo
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 Chicago, IL: University
of Chicago Press, pp. 173-182, 1984. The
Sixth Book of Mathematical Games from Scientific American. Golomb, S. W. Princeton, NJ: Princeton
University Press, pp. 90-92, 1994. Polyominoes:
Puzzles, Patterns, Problems, and Packings, 2nd ed. Keller, M. "Counting Polyforms."
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."
Seggern, D. Boca Raton, FL: CRC Press, pp. 342-343,
Standard Curves and Surfaces. Referenced on Wolfram|Alpha Polyiamond
Cite this as:
Weisstein, Eric W. "Polyiamond." From
--A Wolfram Web Resource. MathWorld https://mathworld.wolfram.com/Polyiamond.html