The Flat Torus in the 3Sphere
Projection of a torus in 4space 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 4space considered as the product of two planes, i.e. (cosu,sinu,cosv,sinv). This torus is a surface on the 3sphere of radius Ö2. We may project stereographically from (0,0,0,Ö2) to obtain a torus of revolution in 3space.Rotating the flat torus in the plane of the first and fourth coordinates produces a oneparameter family
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,Ö2) is on the rotated torus, the projected surface is a noncompact cyclide which separates all of 3space into two congruent parts. (cosa cosu + sina sinv, sinu, cosv, sina cosu + cosa sinv).This sequence was the first film we ever made, in 19689. Stills from the sequence have been recreated several times, in the paper with
David Laidlaw ,Fred Bisshopp , andHüseyin 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 3space that have the spherical twopiece property, i.e. any sphere separates them into at most two pieces. Their inverse stereographic projections are the only surfaces on the 3sphere 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 twentyfive years. A full account of the subject can be found in the volume edited by
ShiingShen Chern andThomas Cecil , including a posthumous article byNicolaas Kuiper and an extensive survey byWolfgang Kühnel and the author [12].Additional Reading:
SB3D: In and Outside the Torus The Flat Torus in the ThreeSphere

