A reptile is a creature, but a rep-tile is a shape that can tile a larger copy of the same shape. If you use that larger copy to tile still-larger copies, and so on, you get a tiling of the plane. It's often an aperiodic tiling, but not always: for instance, the square and the equilateral tiling are rep-tiles but generate periodic tilings when repped.

Another of the properties of the square and the equilateral triangle is that they are rep-tiles in many different way. Any square number of these tiles can be put together to make a larger copy. A rep-tile is said to be rep-k if it uses k tiles to tile its larger self; these shapes are rep-k for all square k. For tiles whose side lengths are all rational multiples of each other, that's the most versatile a rep-tile can be, because the side length of the larger copy is the square root of k. Let's say that a rep-tile is a pan-rep-tile if it has this same property, of being rep-k for all square k. Are there other pan-rep-tiles?

Over on the Wikipedia talk page for the rep-tile article, an anonymous (IP address) editor suggested this property as one that might actually be held by many rep-tiles, and gave some examples of tilings suggesting that the P pentomino and the sphinx might be examples of pan-rep-tiles. It turns out not to be particularly difficult to show that the P pentomino is, in fact, a pan-rep-tile: see the visual demonstration below.

Visual proof that the P pentomino is a pan-rep-tile

Is the same true for the sphinx?

ETA: Yoshio Okamoto informs me that his paper with Ryuhei Uehara and Takashi Horiyama at JCDCG^2 2015, "Ls in L and Sphinxes in Sphinx", proves that sphinxes are indeed pan-rep-tiles.

ETA 2: The results for the P-pentomino, sphinx, and several other rep-tiles are in Niţică, Viorel (2003), Rep-tiles revisited, MASS selecta, pp. 205–217, Amer. Math. Soc. Thanks to Gerhard Woeginger for the reference!



Could you spell out the induction step a bit more? For example, how do you get the rep-11^2 tiling?


To get rep-11^2, you start with rep-1 (i.e. a single P pentomino), which you can think of as being decomposed into a 1x1 square and a 2x2 square. You wrap the 1x1 square with the green central core of the odd L-shape to get an 11x11 square, and you wrap the 2x2 with two layers of even L-shapes to get 22x22.

The rep-7^2 already shown has the same type of construction — it's formed by wrapping the rep-3^2 with L-shapes, shown by the coloring — except that in that case the L-shapes are thinner than 10 units.