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A polyiamond is a counterpart to a polyomino where the polygon used as the building block is an equilateral triangle rather than a square. The word polyiamond is a back-formation from diamond, motivated by the fact that this word is often used to describe the shape of a pair of equilateral triangles placed base to base.
Counting polyiamonds
The basic combinatorial question is how many different polyiamonds with a given number of triangles exist. If mirror images are considered identical, the number of possible n-iamonds for n = 1, 2, 3, … is:
- 1, 1, 1, 3, 4, 12, 24, 36, 66, 160, …
- (sequence A000577 (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000577) in OEIS)
| the moniamond: |
 |
| the diamond: |
 |
| the triamond: |
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the 3 tetriamonds: |
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the 4 pentiamonds: |
 |
Generalizations
Like polyominoes, but unlike polyhexes, polyiamonds have a three dimensional counterparts. These are formed by aggregating tetrahedra.
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