Is there a convex polygon such that it cannot be tiled with some number of congruent connected pieces?

Question

So the title says it all.

I assume that polygons have straight line segments as their edges and that they have finite number of edges.

The number $n$ of pieces is, of course, $n>1$, to avoid triviality that every polygon tiles itself.

Answer 1

Any triangle can be divided into 4 congruent parts, so it's not a triangle.

I suggest that this quadrilateral cannot be divided into connected congruent parts. The sides are $\pi, e, K,$ and $\alpha$, with the last two being Khinchin and Feigenbaum. One diagonal has length $\sqrt{10}$.