A rep-tile is a shape that can tile larger copies of the same shape.


**Question 1:** Are there any convex pentagons that are also rep-tiles?


**Remarks:** 15 convex pentagonal tiles of the plane are known and none of them *appears* to be a rep-tile. Assuming this observation is right, one can invoke a proof given in 2017 by Michel Rao - that these 15 are the *only* convex pentagonal tiles possible - to answer our question in the negative. However, I don't know if Rao's proof has been validated and if there is a simpler (elementary) proof that there are say no convex pentagonal rep-tiles. So, here one is actually asking if there is a simpler proof for a weaker claim.


**Definition:** Let us say a *multi-way rep-tile* is a polygon *P* with the property: if *P1* and *P2* are magnified copies of *P* and *P1* can be tiled with *m* units of *P* and *P2* can be tiled with *n* units of *P* with *n*> *m*, then, a layout of *n* units can form *P2* *without* *m* of the units in the layout together forming a *P1*. As shown on this page: <https://en.wikipedia.org/wiki/Rep-tile>, there are isosceles trapeziums with multi-way property (with *m*= 4 and *n* = 9). On the other hand, the square is obviously a rep-tile but *not* multi-way.


**Question 2:** Are there other convex polygons with this multi-way rep-tile property?



Are there any convex pentagonal rep-tiles?
Question 2 can be answered in the affirmative, at least: there are many triangles with the multi-way rep-tile property.


Every triangle has a simple tiling of itself with $k^2$ copies (just take the affine image of the standard equilateral tiling); these are called *quadratic tilings* by Michael Beeson in his 2010 [paper](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.180.4908) on tiling triangles by congruent triangular tiles. So for a triangle to have this property, it suffices to have *any* non-quadratic tiling of size $m$, since we can choose a larger quadratic tiling with some $n>m$ tiles: the quadratic tiling has no subtriangles that are not themselves tiled quadratically, as should be obvious from looking at which straight lines can be formed. (This condition is of course also necessary.)


So we can then consult the linked paper for a variety of triangles with non-quadratic tilings: this is true of the $(30^\circ,60^\circ,90^\circ)$ triangle and any right triangle whose legs are in a rational ratio, which I *think* is exhaustive though I haven't read the full paper in a while to confirm this (it focuses mostly on the realizable numbers of tiles, which is related but not quite identical to our question here).


If you want to revise your definition to the stricter condition that there are two self-tilings, neither of which contains *any* other nontrivial self-tiling, there are still triangular examples: consider the unique $3$-tiling of the $(30^\circ,60^\circ,90^\circ)$ triangle by itself and its $k=2$ quadratic tiling.




---


As an addendum, the question of convex *hexagonal* rep-tiles ought to be much easier to resolve, since there are only [three](https://en.wikipedia.org/wiki/Hexagonal_tiling#Monohedral_convex_hexagonal_tilings) classes of convex hexagon which can tile the plane, and I believe none of them can even tile a half-plane (which is a prerequisite to being a polygonal rep-tile, though the proof involves a somewhat messy compactness argument). You can at least use the fact that any rep-tile must have an angle of at most $90^\circ$ to narrow the scope of possible tiles a little. Combined with a negative answer to Question 1, this would effectively reduce the classification problem of Question 2 to the convex quadrilaterals, which seems like it might prove rather difficult - it might end up being easiest to just classify all convex quadrilateral rep-tiles and then work out which ones are multi-way.


