Tom Lowe
https://plus.google.com/+TomLowe/posts/b83BjJKLuZi?cfem=1
Self-Similarity and Fractals
May 29, 2:23 AM
Fractals are often described as self-similar shapes, in a strict sense this means they are built from smaller copies of the same shape. Is there such a thing as co-similar shapes? Where shapes in a set are each an arrangement of a smaller copy of that
set.

The answer is yes, and have been studied before under the topic of self-tiling tile sets:
https://en.wikipedia.org/wiki/Self-tiling_tile_set

Unfortunately their order 2 case only works if you allow reflections. If you don’t allow reflections then there are no known self-tiling tiles, excluding the trivial cases where two shapes in the set are identical (‘rep-tiles’).

But self-tiling tiles assume the shapes are dense. Without that assumption there is a wide and rich landscape of co-similar shapes, which I explore here:

https://tglad.blogspot.com.au/2018/04/co-similarity-in-fractals.html

After searching for one that is dense I’m pretty convinced there isn’t a dense co-similar pair of tiles. We can get arbitrarily close to dense just by approaching the rep-tiles but that is cheating. Aside from that, the densest co-similar shape of
order 2 is the posted pic. It shows the two shapes, this first is white, the second is shown in two positions as yellow or pink. Notice that yellow and white make a larger version of white, while pink and white make pink.

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