So: [not axioms just poorly worded thoughts with gaps]
If a point is construed as an axis of rotation;
if the point has a property that it is a center coordinate of 0,0;
if the point is non-empty; (pseudo)
If a point, when expanded, becomes a unit circle in its first step and stops;
if a unit circle is a circle with unit “radius” (it is – this one is true);
then that circle has a radius of 1.
If that circle’s radius is an axis of rotation;
a rotation of the radius is and also creates the unit circle;
If the circle is expanded beyond unit radius,
there is now a cone formed from point rotation to unit radius rotation to beyond unit radius rotation
But we can treat the cone as a circle
or we can also change have the point (or unit radius or beyond unit radius) circle ITSELF rotate around the axis of rotation, forming a sphere.
But regardless of expansions, in 2 dimensions, whether a cone formed from swinging a ball around a “point” along one plane or swinging the ball around a “point” along all possible planes in 3D given the constraint of being a fixed coordinate (moving itself but not moving from its spot on a grid)…
..it still forms a circle when viewed in 2D.
“A unit circle can be used to define right triangle relationships known as sine, cosine and tangent”
The unit circle can also be used in the complex plane.
So if we construe circles as constantly rotating over time, we have a basis for a physics, even if not entirely our physical universe.