This blog post is (as will be several in the next weeks) inspired by the art of Matthew Shlian. Below you see ragged squares (or Aztec diamonds, if rotated by 45º), tiled by P-pentominos. On the right, we get by without using mirrored copies, and on the left we see all eight rotations and reflections of the P-pentomino.
The embossment on the left comes from Matt’s instagram pages where you can find a paper sculpture called Some Caterpillars Stay Caterpillars 42. He realizes this tiling using folded paper, and my different color are his different shades of grey, coming from natural light, and much more subtle in his sculpture than here.
The colors used in the embossment above add different esthetic and structural flavors. For instance, on the right the parallel slopes of the region are colored in different shades of the same color, emphasizing striped bands that completely eliminate the original tiling by P-pentominoes.
Matt’s original sculpture maintains a poetic balance between the mathematical rigor of the tiling and the more organic nature of a landscape of valleys and ridges that appear in his embossment.
Above is a tiling of the next larger Aztec diamond that can be tiled by P-pentominoes. It turns out that here you need the P-pentomino together with its mirror, but I don’t know a reason for this.