It is well-known, that it's impossible to square a circle. There are good papers on the matter and good videos, like from Mathologer or Numberphile
So, if rules of the game are that we have a
- Ruler (NOT a measured one) for straight lines
- Compass for circles
it's proven to be an impossibility. I'm not a professional mathematician but I'm able to understand the material (I think) well.
My question is:
What is the "minimal" change in the rules of the game so that we can square a circle?
I put "minimal" into quotation marks because, while this is bugging me, I fail to come up with a proper definition of "minimal". I.e. I have no clear way of comparing one change of rules of the game with another to clearly answer which one does "fewer changes" to the rules of the game. But I still am curious about what could be possible answers to the question bearing in mind this "handwave-y" definition.
EDIT: (Bonus) if the construction process with the changed rules is the one we can reproduce in real life. For example, adding a new tool to the tool set and/or allowing other operations with tools.
As I noted in my comment, there are no really pretty ways to make squaring the circle possible. However, the following is as close as I imagine one can get to a loosening of rules to allow squaring the circle.
The new rule is to allow for infinite sums. For each $n\in\mathbb N$, we construct a segment of length $n^2$, and then one of length $\frac1{n^2}$ via this construction. Then, we line up all of these segments to construct a new segment of length $x$. Finally, we construct a segment with length $6x$, and then construct a segment of length $\sqrt[4]{6x}$ by repeating this construction. Finally, construct $\frac1{\sqrt[4]{6x}}$ by applying this construction.
Note that $\frac1{\sqrt[4]{6x}}=\frac1{\sqrt\pi}$, so a circle with radius $\frac1{\sqrt[4]{6x}}$ has area $1$, the same as that of the unit square.