Make Your Own Hyperbolic Tessellation
It's not too hard to make your own!
Summary:
- Select a (p,q) pair that satisfies (p-2)(q-2)>4. At the outset p=5 and q=4, which works.
- The radius of the circumcircle of a regular p-gon that tessellates with q copies at each vertex is automatically calculated and stored in the variable "radius".
- Create a point anywhere inside the Poincaré Disk. Rename it A (if needed).
- Create a "Hyperbolic Circle with a Given Radius" centered at A with radius "radius". Rename it d (if needed).
- Place a point anywhere on d. Rename it B (if needed).
- Use the Hyperbolic AngleWithGivenSize tool to rotate B around A by an angle "angle". Rename the new point C.
- Now rotate C around A by "angle" as well. Continue until there are p points on d. These are the vertices of a tessellating regular pentagon.
- Connect the vertices with Hyperbolic Segments. These are the edges of the hyperbolic tessellation.
- Reflect vertices through the hyperbolic segments (use
) to create new vertices. Connect them with Hyperbolic Segments. - Continue until you're satisfied (or crash your computer).