# Intrinsic_hex

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 as it would appear intrinsically, if light traveled to our eyes along geodesics of the 3-sphere metric. The torus is parameterized by drawing a closed curve on the 2-sphere, and then taking the preimage under the Hopf map. |

The Math | The torus admits a flat metric (one can be constructed abstractly by identifying opposing sides of a square sheet of paper), but such flat tori cannot be constructed in 3-dimensional space. However, a simple construction, due to Ulrich Pinkall, shows that all flat tori can be embedded in 4-dimensional space, on the surface of the 3-sphere. |

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