The design below was created using *inversive geometry*, combined with a twisting rotation.

Inversive geometry is a nice way to introduce the idea of switching back and forth between different geometries, which can lead to the study of this Poincaré disk. The ideas are accessible with using just geometry.

Inversive geometry requires that we have a circle of a given radius [latex]r[/latex]. Consider any point in the plane, at distance [latex]d[/latex], lying on a ray from the center of the circle. Then that point has an inverse point, at distance [latex]r^2/d[/latex] from the center of the circle, on the same ray. Taking the inverse of a point flips points from outside the circle to inside the circle, and vice versa. Imagine the plane being made out of flexible material, and inverting is like puncturing the plane at the center of the circle, and then turning the plane inside out, with the points on the edge of the circle remaining stationary.

You can play with this in **desmos** by clicking on the graph below. The orange disk is the unit circle. The red dot can be dragged. The green dashed line segment from the red dot towards the origin leads to the purple point, which is at a distance that is the reciprocal of the red dot's distance from origin.

What shape is traced out by the inverse point as we slide the red dot along the line? Is this always the case?

Now, consider infinite graph paper with integer coordinates. The lines consist of all vertical lines with integer [latex]x[/latex] values, and all horizontal lines with integer [latex]y[/latex] values. What do you get if you find the inverse of all those lines?

This mapping of the entire plane outside the disk into the unit disk (and vice versa) is a conformal mapping, which means it preserves angles. If we define distance between two points in the disk to be the same as the distance between the inverses of the points, then we are preserving distance. Angles are the same and distances are the same, so we could study the ideas in high school geometry, but staying entirely inside a disk. What project ideas does this give you?