Are there rep-tiles (shapes that can be dissected into smaller copies of the same shape) which contain sharp points on their boundary? By a sharp point, I mean a cusp on the boundary, a location where a point moving along the boundary must reverse direction. For example:
Looking online, I cannot find any examples of rep-tiles with this property. There are plenty of rep-tiles with fractal boundaries, but I haven't seen any containing cusps.
For a rep-tile to have a cusp on it's boundary, it cannot be a polygon and it cannot be a convex shape. It seems to me that if the boundary is non-convex, then the kind of self-similarity it inherits from the rep-tile will force it to be fractal, but I don't know how to prove this.
The image of each cusp in each sub-tile will be a cusp or it will lie on the boundary between sub-tiles. If there are rep-tiles with cusps, they are much less "well-behaved" than ones without (with the exception of tiles with fractal boundaries).
So, are there rep-tiles with cusps on their boundaries?
Edit: For $2$-d reptiles whose sub-tiles are not of identical size, there are no solutions.
If the rep-tile is not a polygon, but does not have a fractal boundary, then suppose there exists the point of maximum curvature $p$ of its boundary.
All images of $p$ will have curvature greater than $c$, so they cannot lie on the boundary of the rep-tile. Therefore, they must lie within the shape on the border between sub-tiles. They must meet with points of equal curvature, however different tiles have different scaling values, so they cannot meet with images of $p$. The largest image of $p$ must adjoin a point of equal curvature, but whose preimage has greater curvature than $p$, contradicting our assumption.
Can this be adapted for the case where all sub-tiles of the rep-tile are similar? Can this be adapted for dimensions higher than $2$?