Polykites

Polykites are figures built from the quadrilaterals forming the following lattice. On top of this lattice, we have shown the arrangements corresponding to all the possible shapes for the 1-polykite, 2-polykites, and 3-polykites using different colors. Larger-order polykites offer a lovely resemblance of faces, birds, and animal shapes in a way similar to tangrams. Their different balance and aesthetic properties motivated the author to include their generation as a proper sibling of polyominoes; however, the other Penrose piece, the Dart, being nonconvex, resisted all attempts at generating polydarts.

Just as we have two types of triangles forming polyiamonds, we have six types of kites forming polykites and we will describe them in a similar fashion. The following comprises an adaptation of the functions we have previously designed. The type of a polykite is tested as follows.

Here are the polykite functions.

The polykite has neighbors depending on its anchor point and its type as shown in the following Switch instruction.

The number of different polykites is given by the following table ([5, 6], seq. A057786).

We invite the reader to generate all six polykites that, besides having beautiful shapes, present for the first time a hollowed piece. Here is the corresponding code.

In the next part of this work, we will tile rectangles with polyominoes and introduce the family of rep-tiles, which presents a fascinating way of tiling the whole plane with specific polyominoes and polyiamonds.