Well, the thing about a circle is that the very core of the definition has the same sort of "information" as whatever abstract "information" f'' = -f contains.
Yeah, a circle is basically a particular exchange between the velocity in the x and velocity in the y direction, offset by a phase equal to half of the period. With different kind of exchanges, you can get squircles and all types of other pseudo-circular shapes.
I guess this must be true, but from the sidelines it seems that “the set of points 1 metre from this particular point” and “the solution to this particular second-order differential equation” are quite different things. Any competent student of mathematics can connect them for sure, but it’s still quite remarkable they’re related isn’t it?