Volume 9, Issue 3

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Polyominoes and Related Families

Polyiamonds

Polyiamonds are figures built from congruent unit-side equilateral triangles in the same way that polyominoes are built from squares. Any triangle forming a polyiamond will be described by a pair , wherein codifies the complex coordinates of the leftmost vertex (anchor) of the triangle (we are assuming one of its sides rests horizontally) and t is its type taken from the set , corresponding to whether the apex is pointing up or down. The underlying grid on which to place the triangles is generated by all integral linear combinations of the numbers 1 and . Any triangle has three neighbors adjacent to it, namely, , , and . As with polyominoes, the canonical representation of a polyiamond moves the piece so that its leftmost vertex touches the origin.

To construct the corresponding equivalent versions of a polyiamond, only rotations of multiples of 60° are allowed (a rotation of 60° implies the type of a triangle is changed) so that point gets rotated to point . The reflection of a triangle is achieved simply by sending each of its vertices to . (In this representation a reflection gives the reverse negative of a rotation!) The type of a polyiamond is tested as follows.

Here are the polyiamond functions.

Here is an example of a polyiamond and its conversion to canonical form.

Here we obtain the polyiamonds equivalent to the previous one.

Finally, we generate all nonisomorphic polyiamonds. We make use of the fact that any triangle has neighbors , and , and that triangle has neighbors , and .

The number of different polyiamonds is given by the following table ([5, 6], seq. A000577).