Steve Trettel · with Nadir Hajouji

weierstrass hex · 2025

Hexagonal-lattice elliptic curve embeded in R4 by the Weierstrass functions.

The Weierstrass $\wp$ -function for the hexagonal lattice $\Lambda = \mathbb{Z} + \omega\mathbb{Z}$ (where $\omega = e^{2\pi i/3}$ ) uniformizes an elliptic curve $E = \mathbb{C}/\Lambda$ with extra $\mathbb{Z}/3$ symmetry. The map $\wp: E \to \mathbb{CP}^1 \cong S^2$ combined with the Hopf fibration $S^3 \to S^2$ lifts the branch locus and other features of $\wp$ to curves and surfaces in $S^3$ , which are then stereographically projected to $\mathbb{R}^3$ . The resulting geometry is rendered using three-gpu-pathtracer for physically-based global illumination, capturing the interplay of the hexagonal symmetry with the topology of the Hopf fibration. Controls: Click and drag to orbit, scroll to zoom.

Steve Trettel · with Nadir Hajouji

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