Is there a second, possibly non-real rectangle shape, whose shape remains the same upon removing a square?
The rectangle in the golden ratio retains its shape when a square is truncated from it. In fact, if we allow $n$ truncations, the base-$2$ Lyndon words of length $n$ classify and enumerate the rectangles which are periodic when cutting off rectangles.
With two exceptions: There is only one real rectangle of period one, whereas there are two length-one Lyndon words in base $2$. Hence OEIS A059966 differs from the number of Lyndon words by firstly not allowing for the one "empty string" and secondly only having a single fixed point whereas there are two Lyndon words which are constant when rotated.
Is there a pair of rectangle shapes which theoretically correspond with these two missing cases, i.e. the empty string and the other fixed point, e.g., such as the ratios $1:0$ and $0:0$?
Motivation
I have an interest in the truncation of binary strings, and in particular in the topology of the 2-adic numbers you get when you repeatedly write base-2 Lyndon words as binary numbers. I'd like to gain more insight around the differences between the topologies that underpin OEIS A05996 and OEIS A001037 – for example can one glue the ends of the logistic map together to arrive at a system with one fixed point rather than two?
If you remove an $x$ by $x$ square from a $1$ by $x$ rectangle, you're left with a $x$ by $1-x$ rectangle, so we're led to the equation $1/x=x/(1-x)$, which is $x^2+x-1=0$, which has solutions $(-1\pm\sqrt5)/2$. So there is a "non-real" rectangle shape (whatever that means), $1$ by $(-1-\sqrt5)/2$.