Hopf
Preimage of a loxodrome curve under the Hopf map
- Medium Print on metal
- Technique Mathematica
- Dimensions 12 × 18 in
- Year 2019
Exhibitions
The Mathematics
The Hopf map is a topologically nontrivial submersion from the three-sphere to the two-sphere, and the preimage of each point is a circle — any two of which are linked. This piece takes a loxodrome, the spiral crossing every meridian at a constant angle, and draws its preimage under the Hopf map: a family of linked circles sweeping out a nested, spiralling surface in the three-sphere, shown stereographically projected into R³.
Technique
I wrote a Mathematica program that takes coordinates on S² and parameterizes the Hopf fiber above them. Then, using the analog of a Frenet frame in S³, I parameterized a tube of constant width around each fiber, stereographically projected the tubes into R³, and rendered the result in Mathematica.