knot-complement

A notebook and experimental codebase for making mathematical spaces into places you can move through.

The guiding idea is not to paste mathematics onto a conventional game. The mathematical space itself is the world. The basic interaction can be as simple as walking, flying, or auto-walking and looking around.

Main directions

1. Walk inside knot complements and hyperbolic 3-manifolds

The central target is an actual inside view of a knot complement: start with the figure-eight knot complement, then other cusped hyperbolic 3-manifolds. The player should move through the quotient geometry rather than merely look at a knot embedded in ordinary Euclidean space.

Useful existing models include:

• SnapPy inside_view()
• Not Knot / Maniview
• Jeff Weeks's Curved Spaces
• Henry Segerman and collaborators' real-time ray-casting/ray-marching work

2. Walk through the eight Thurston geometries

Build a common movement/rendering experiment for

• Euclidean 3-space
• spherical 3-space
• hyperbolic 3-space
• (S^2 \times R)
• (H^2 \times R)
• (\widetilde{SL_2(R)})
• Nil
• Sol

The interesting part is perceptual: what does ordinary locomotion, distance, parallelism, horizon structure, and familiar scenery feel like when the geometry changes?

3. Fractals as traversable spaces

Do not stop at precomputed zoom videos. Treat a fractal as somewhere to move.

Starting references include cohomology fractals, Cannon--Thurston maps, classical Mandelbrot zooms, and the old Geometry Center visualization work.

4. Familiar landscapes inside unfamiliar geometry

Put recognizable things -- mountains, trees, paths, rooms, horizons -- into non-Euclidean geometry. The same scene should become a visual measuring instrument for the geometry.

An auto-walk/ride mode matters here: the user should be able to watch the geometry reveal itself without needing a complicated game mechanic.

5. Later: spaces built from mathematical operations

Possible later experiments include worlds organized around crossings, cancellations, transformations, operator-algebra-like composition, and knot-theoretic moves. These are intentionally less specified than the geometric fly-through work.

First experiment

Start with one small, inspectable target:

Walk inside the figure-eight knot complement (m004).

See FIRST_BUILD.md on the working branch for the concrete scope.

Notes

The reading/viewing list belongs in notes/references.md. It includes the papers, books, historical Geometry Center material, videos, and existing software that motivated the project.