This article describes research results of Daniel Asimov, of NASA Ames Research Center in Mountain View, California. It gives an example of an open region in three-space continuously filled with hoops. It then explains the higher dimensional generalizations of continuous families of hoops. In order to set up the generalization to higher dimensions, this description of the circles of Villarceau uses the stereographic projection of the Hopf fibration of the three-sphere. In a previous article, I described this same configuration of hoops in a less technical manner.
Stereographic projection preserves circles and lines.
Thus projections of the Hopf circles are circles, except the circle throught the projection point, which projects to a line and a point at infinity. In order to get hoops rather than geometric circles of different radii, we apply a uniform shrinking to each individual torus. In order to preserve continuity, nearby tori from the Hopf fibration are shrunk by close shrinking constants. By choosing the shrinking constants wisely, after shrinking each torus, we get another series of nested tori filled now by hoops. The new nested tori all have one of the two generators equal to the unit circle in the xy-plane with the origin as center.