Bi-tilability

Some shapes can be tiled with two different sets of polyforms. This is called bi-tilability. True bi-tilability is very rare in the realm of polyforms but there are many cases where a set can tile a single piece from another set, sometimes in enlarged. Below are covered the following cases, true bi-tilability is in bold:

General considerations

All the cases that I treat here represent hexagons cut down into triangle (6 or 24 of them). When we tile a polyhex shape with polyiamonds, many of the edges of the latter have to coincide with the outer edge of the former. It is important therefore to know what maximal perimeter running on the hexagonal grid can be covered with polyiamonds. Let's call this number maximal hex congruency of the polyiamond set. Below is a visual application of this principle on heptiamonds and a side 1 hexagonal grid:

Some pieces can be put in other ways on the grid, but the maximal congruent perimeter is shown. The total for the whole set is 95. This means that no polyhex shape with a perimeter greater than 95 could be tiled with heptiamonds. In practice, this number is certainly never attained. It would be interesting to know what is the maximum perimeter a heptiamond bi-tilable polyhex can have.

The same calculations for the hexiamonds give 50 as the maximal hex congruent perimeter. This is shown below. In practice, the hexagon hexiamond would have to connect to the rest of the shape by an edge and the maximal perimeter drops do 48. The minimal perimeter for a dodecahex is 24, well below this limit. Dodecahexes with holes can have a perimeter of at least 36 which means that the theoretical maximum is too high.

Another way to know for sure that a particular polyhex isn't tilable with polyiamonds is to count its horns. I define a horn as a cell that is connected to the rest of the shape by a single edge. This greatly diminished the possible ways such a hexagonal cell can be tiled with polyiamonds. In the case of hexiamonds and heptiamonds (but also octiamonds), there is exactly one piece that can tile a horn. This means than only hornless and unicorn polyhexes can be tiled with hexiamonds or heptiamonds. If both sets are combined, bicorns can be tiled too.

Below is an illustration using the simplest tricorn whose horns are tiled with the hexagon hexiamond, the D heptiamond and the only possible octiamond.

Hex congruency and counting horns help to exclude polyhexes and thus save time on trying to tile them with polyiamonds, but both these criteria are negative and only tell us what is impossible, not what is possible.

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