This question is motivated by Escher's series of Metamorphosis woodcuts (see e.g. here), where one tesselating tile is gradually transformed into another. Basically, this is a precise way of asking the following: can we always "metamorphose" between any pair of tesselating shapes? (Now asked in modified form at MO.)
First, some definitions (everything is in $\mathbb{R}^2$ with the usual metric):
A shape is a compact connected set $X$ with $X=cl(int(X))$.
A tiling is a pair $(\mathscr{S},F)$ where $\mathscr{S}$ is a finite set of shapes and $F$ is a set of functions such that
each $f\in F$ is an isometric embedding of some $S\in\mathscr{S}$ into $\mathbb{R}^2$,
$\bigcup_{f\in F}ran(f)=\mathbb{R}^2$, and
if $f,g\in F$ are distinct with domains $S,T$ respectively and $x\in int(S)$, then $f(x)\not\in ran(g)$ ("shapes only meet at their boundaries").
A tile is a shape $X$ such that $(\{X\},F)$ is a tiling for some $F$.
$d_H$ is Hausdorff distance. (Note that we could replace $d_H$ with the modified version $d_{H*}(U,V)$ = the infimum over planar isometries $p$ of $d_H(p(U), V)$ without changing the question substantively; all that would change is that the number of tiles needed would shrink, but I'm not looking at that here.)
Now with apologies to Escher, given tiles $A,B$ and $\epsilon>0$ an $\epsilon$-metamorphism from $A$ to $B$ is a tiling $(\mathscr{S},F)$ such that
if $f\in F$ and $(x,y)\in ran(f)$ with $x<0$ then $dom(f)=A$,
there is some $N$ such that if $g\in F$ and $(x,y)\in ran(g)$ with $x>N$ then $dom(g)=B$ (call the least such $N$ the length of the $\epsilon$-metamorphism), and
if $f,g\in F$ and $ran(f)\cap ran(g)\not=0$ then $d_H(dom(f), dom(g))<\epsilon$.
Question: is there always an $\epsilon$-metamorphism between $A$ and $B$ for any tiles $A,B$ and any positive $\epsilon$?
I recall seeing a theorem that the answer is yes, and that moreover we can always find metamorphisms with length "close to" the naive guess $$\max\{diam(A),diam(B)\}\cdot d_H(A,B)\over \epsilon.$$ However, I haven't been able to track it down or prove it myself. I'm also interested in whether the situation changes as we appropriately tweak things to work in $\mathbb{R}^n$ for $n>2$.
