# Clifford_torus

Steve Trettel

Medium | Computer generated using a custom-built shader material and THREEjs. |
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The Art | A flat torus embedded in the 3-torus, viewed under stereographic projection from a point on the surface of the torus itself. The torus is parameterized by taking the preimage of a great circle on the 2-sphere under the Hopf map. |

The Math | Perhaps the simplest flat metric on the torus is constructed by rolling up a square to identify pairs of opposing sides: a single such roll (into a tube) can be accomplished in 3 dimensions, but to do both simultaneously requires four dimensions of ambient space. The resulting surface can be embedded on the surface of the 3-sphere, and divides this 3-manifold into two isometric solid tori. |

Categories | Differential Geometry, Computer Graphics, Math Illustration, Stereographic Projection |