## The Flat Torus in the 3-Sphere

*Projection of a torus in 4-space from a point on the surface, so that the projection extends to infinity, separating space into two congruent regions.*

*The flat torus is an embedding as a product of two circles in 4-space considered as the product of two planes, i.e. (cosu,sinu,cosv,sinv). This torus is a surface on the 3-sphere of radius Ã–2. We may project stereographically from (0,0,0,Ã–2) to obtain a torus of revolution in 3-space.*

*Rotating the flat torus in the plane of the first and fourth coordinates produces a one-parameter family*

(cosa cosu + sina sinv, sinu, cosv, -sina cosu + cosa sinv).

(cosa cosu + sina sinv, sinu, cosv, -sina cosu + cosa sinv).

*These rotated images of the flat torus project to a family of cyclides of Dupin, all conformally equivalent to the original torus. In particular when a = p/2 and the point (0,0,0,Ã–2) is on the rotated torus, the projected surface is a non-compact cyclide which separates all of 3-space into two congruent parts.*

This sequence was the first film we ever made, in 1968-9. Stills from the sequence have been recreated several times, in the paper withDavid Laidlaw, Fred Bisshopp, and Hüseyin Koçak on Hamiltonian Dynamical Systems, [18] in the related film by these authors andDavid Margolis, and on the cover of "Beyond the Third Dimension" [13]. The flat torus was the first example to be developed byDavide Cervone and the author at the Geometry Center of the University of Minnesota in 1994, and it is one of the featured items in both our Providence Art Club and Lisbon exhibits [16].

*The cyclides of Dupin and spheres are the only closed surfaces in 3-space that have the spherical two-piece property, i.e. any sphere separates them into at most two pieces. Their inverse stereographic projections are the only surfaces on the 3-sphere that are tight, i.e. every hyperplane separates them into at most two pieces [1].*

Tight and taut surfaces and their generalization have undergone extensive development over the past twenty-five years. A full account of the subject can be found in the volume edited by Shiing-Shen Chern and Thomas Cecil, including a posthumous article by Nicolaas Kuiper and an extensive survey by Wolfgang Kühnel and the author [12].

**In- and Outside the Torus**