Special reptiles - repeating shapes and fractals Let me first explain what a reptile is.

A reptile is a two-dimensional object, a shape, that can be dissected into smaller, equally sized copies of the same shape.

To illustrate this, see here a couple of reptiles:

A shape is called an $n$-reptile, or $n$-rep for short, if the shape can be dissected into $n$ smaller, equally sized copies. So above you see a $2$-rep (recognize the shape? Yup, it's A4 paper! Could be any of the A-series in fact) a $3$-rep (a Sierpinski triangle) and a random $4$-rep.
Now obviously, if a shape is an $n$-rep, it's also a $n^2$-rep; simply dissect it once in $n$ pieces, and dissect every piece you made again in $n$ pieces. One can repeat this pattern to see that in fact, if a shape is an $n$-rep, then it must be an $n^k$-rep for integer $k\geq 1$. These dissections in $n^k$ pieces aren't very interesting though, since they're all based on the base case with $n$ pieces. This got me thinking, and so here's my question:

Does there exist an $n$-reptile that is simultaneously an $m$-reptile with $n$ and $m$ coprime?


The Viper substitution tiling divides a particular triangle into 9 similar sub-triangles without reflection, in a non-trivial way.  The same triangle can also be divided into 4 similar sub-triangles without reflection, see the top half of the triangle in the rule image on that page.
