Can square circles exist in any meaningful sense?

Welcome to the debate arena! Today's topic: "Can square circles exist in any meaningful sense?". Your opponent's style: A person who thinks square circles can exist

Square circles cannot exist because it is a contradiction in terms. The definition of a circle is a plane curve everywhere equidistant from a given fixed point, the center. In other words, a circle is centered around a point, and every point on a circle is the same distance apart as any other point on that same circle. A square on the other hand is a plane figure with four equal sides. A shape with four equal sides cannot also have every point on it be the same distance away from it's central point. Take, for example a square at (0,0) with sides that are 2 units long each. The distance from point (-1,0) to the center (0,0) is 1. The distance from point (-1, 0.5) is about 1.12. This problem exists on every possible square.

You've trapped yourself in definitional rigidity. Yes, those Euclidean definitions contradict—but that's *precisely* why square circles can exist in non-Euclidean geometries where curvature is constant. On a sphere or hyperbolic plane, the definitions shift. What you've proven isn't that square circles are impossible, but that they're impossible *in flat space with your specific axioms*. That's not a logical barrier to their existence elsewhere. You've essentially argued "square circles can't exist because I'm using definitions that exclude them." That's circular reasoning, not refutation. Can you defend why Euclidean geometry is the only valid framework for this debate?

First off, pointing out contradictory definitions and showing how they are contradictory isn't circular reasoning. All I was doing was showing that the accepted definitions that mathematicians use contradict. You point out that I was specifically speaking on Euclidean terms and didn't mentioned sphere planes or anything like that. However, you haven't shown how square circles can exist on those planes. It's your burden to show that and you haven't done that.

You're right that I dodged the burden. But here's the trap: you defined "square circle" using *only* Euclidean properties, then demanded I prove it exists elsewhere using those same definitions. That's unfair. On a sphere, a "square" means four equal geodesic sides with equal angles, and a "circle" means equidistant points from a center along the surface. Both can coexist. You haven't actually defined what a square circle *means* in non-Euclidean space—you've just assumed your Euclidean definitions transfer universally. So the real question: are you defending square circles in Euclidean geometry specifically, or claiming they're impossible everywhere?