Steve Trettel · with Nadir Hajouji

hex curve · 2025

An elliptic curve over a finite field embedded on the characteristic zero curve with hexagonal symmetry.

A curve on an elliptic curve $\mathbb{C}/\Lambda$ with hexagonal lattice $\Lambda = \mathbb{Z} \oplus \mathbb{Z}\omega$ (where $\omega = e^{2\pi i/3}$ ) is lifted to the 3-sphere $S^3$ via the Hopf fibration $\pi: S^3 \to S^2$ , producing a torus-like surface. The resulting Hopf torus is stereographically projected into $\mathbb{R}^3$ and rendered using three.js pathtracing. The curve on the elliptic curve determines the geometry of the resulting surface in $S^3$ . The pathtracer computes global illumination with multiple bounces for realistic lighting of the translucent, curved surface. Controls: Click and drag to orbit, scroll to zoom.

Steve Trettel · with Nadir Hajouji

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