A circular torus in three-space contains two "diagonal" families of Villarçeau circles. The circles in each family "foliate" the torus. The image shows the set of circles in one family.
Villarçeau circles may be viewed as great circle orbits in the three-sphere arising from multiplication of unit complex scalars acting on pairs of complex numbers. Write points of complex Cartesian two-space as \((z, w)\). A point lies in the unit three-sphere if \(|z|^{2} + |w|^{2} = 1\), and further lies on the "square Clifford torus" if \(|z|^{2} = |w|^{2} = \frac{1}{2}\). If \(\theta\) is real, then \(t = e^{i\theta}\) represents the general unit complex number. One family of Villarçeau circles arises from the Hopf action
\[
\chi_{t}^{+}(z, w) = (tz, tw).
\]
The other arises from the complex conjugate action
\[
\chi_{t}^{-}(z, w) = (tz, \overline{t}w).
\]
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