I wonder which subsets of plane $X$ have a property that under any affine transformation $T$ there exists a similarity $S$ such that $S(T(X))=X$ so it essentially returns $X$ to its original form.
Couple observations:
1) If $X$ has this property then $X'$ also has it ($X'$ is a complement of $X$)
2) Any set that is a subset of some line has this property.
3) Parabola has it too
I have also some conjectures:
1) Every such set which is not a subset of any line has to be infinite.
2) Every such set which is not a subset of any line has to be unbounded.
My question is are these conjectures true? Is there any nice characterization of such sets? Or there are some "pathological" examples?
2026-03-26 04:23:33.1774499013