Fibration Continued

In this lesson, we explore the concept of fibration and topology with a focus on the torus of revolution and families of circles on the torus. The torus is covered by four families of circles, each of which do not intersect each other. The Villarza theorem states that a plane that is tangent to the torus will cut it along two circles, and we can obtain four circles through each point on the torus. We also learn about hopped circles and the dupern cyclades, which are surfaces covered by four families of circles. Finally, we see how a simple rotation in the fourth dimension can switch the colors of the inner and outer faces of the torus.

More on fibration and topology. This lesson focuses on the torus of revolution and families of circles on the torus.

• What are fibrations?
• What do rotations of circles and lines in 4-space look like?
• What do families of circles on the torus look like?
• What does a section of a torus intersected by a plane look like?
• What families of circles are created when a torus of revolution is intersected by a plane?
• What are Hopf circles?
• What are Hopf fibrations?
• What are cycloids?
• What happens when a torus is rotated in the 4th dimension?