7-spheres-3d

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# 7-Sphere 3D

Python scripts to generate 3D models of some complex curves, as described in
_[A Construction for Computer Visualization of Certain Complex Curves](https://homes.luddy.indiana.edu/hansona/papers/CP2-94.pdf)_.

This repository contains scripts for generating models of the following curves:

1. The Fermat curve: $z_1^n + z_2^n = 1$
2. 2D cross-sections of the Brieskorn complex 4-fold:
   $z_1^{6n - 1} + z_2^3 + z_3^2 + z_4^2 + z_5^2$. These cross sections are
   given by taking $z_4 = z_5 = 0$ and fixing $z_3$ with $0 < |z_3| < 1$.

All such curves are embedded in ℝ⁴ and then projected into ℝ³ using different
functions (see `normalized_p` and `riemman_sphere_p`). A 3D model is then
generated by taking explicit triangulations of such surfaces.