## From the 2-Sphere to the 3-Sphere

# Hopf Tori

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If $\gamma\colon S^1\to S^2$ is a simple closed curve on the sphere, its preimage under the hopf map $\pi\colon S^3\to S^2$ is topologically a torus, and naturally inherits a flat metric as a submanifold of the round metric on the 3-sphere.

It is a theorem of Ulrich Pinkall that all flat tori can be constructed in this way: he proved that if $L$ is the length of the curve on the 2-sphere and $A$ is the area it encloses (on the smaller side), then the resulting torus is isometric to the result of identifying opposing sides of a parallelogram with a vertex at the origin and sides $(2\pi,0)$ and $(A/2, L/2)$.

This program provides a simple graphing calculator to play around with this result, depicting the preimages of curves which oscillate sinusoidally about the equator of the 2-sphere with varying amplitudes.