Investigate: What are the ways that regularity of choice is expressed?
Relate the definition of geometry as "the regularity of choice" with Grothendieck's machinery, toposes, etc.
Collect and understand examples
- Fiber bundles, vector bundles, etc.
- Sheaves, schemes?
- The binomial theorem (x+h)^n can be interpreted as the context for a derivative of a polynomial power, distinguishing a volume and its boundary, where the second term is the derivative. How does this express the coordinate system? And how may taking the limit h->0 be considered as shifting to the asymmetric interpretation in terms of a simplex? And what does that say about the continuity and differentiability of nature, how it relates symmetry and asymmetry?
- Binomial theorem. Derivative. There is no volume, just the faces, as with the cross-polytopes. Also, the derivative has the boundary conditions as in homology.