Triangles and Squares

There are two Archimedean tiling using triangles and squares.

Archi34 01

Both of them use twice as many triangles than squares. I find the first one is more interesting, maybe because it is chiral. There are still many other ways to tile the plane say periodically with just triangles and squares. There are three different ways to assemble two triangles and a square, and all of them give polyforms that can be used as a single subtile for the first Archimedean tiling:

Architile 01

Among these three polyforms I like the middle one best, maybe because it cannot be used to subtile the second Archimedean tiling. It is an amusing exercise to doodle around and find other tilings of the plane with this tile. Here, for instance, are two small turtles and a giant caterpillar, all part of a big creation.

TrisqueArt 01

I find it amusing how this simple polyform lends it self easily to organic shapes and abstract designs.

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There are (I think) 10 ways to combine two of them into a single polyform, not counting mirror images. At least two look like cats.

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Confusing as they look, almost all of them tile the plane. The two exceptions are shown below. It is not difficult to find an argument why these two do not tile.

Nottileduo 01

More interestingly the other eight tile, even though they look much more complicated. Typically one needs for each tile its mirror, suitably rotated. Here are two pretty examples. Homework is to find the others.

Duotileex 01

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