(Now with a third dimension FOR FREE!!!)
This is a project to explore the fascinating mathematical concept of Circle Inversion which is a way to map every point inside a circle to every point outside the circle. Essentially it is the process of turning a circle inside out. What this means in practice is that a point on the outside edge of the circle will project onto the same point while a point on the center of the circle will project onto infinity. Every point in between will be mapped relative to infinity and the circle edge.
Shapes can also be mapped in the same way by mapping each point on the shape using the same method as the single point. When this is done with a square it gets distorted like so:
Interestingly though a circle is not distorted.
If you notice each point is the same color as the projected point making it easier to see how the shape is flipped upon projection. Becuase of the fact that the edge of the circle maps onto itself while the center maps onto infinity an interesting thing happens when the inner circle touches the center.
The projected circle has a radius of infinity causing it to look like a straight line.