Mathematical Experiment 3

Construct a torus with radius of the tube a and the distance from the center of the hole to the center of the tube c:

Construct one half of the torus by cutting it vertically.

Construct one half of the torus by cutting it horizontally.

Through every point on a torus, there are four circles passing though it! The two less obvious ones are called Villarceaux circles. Illustrate the Cillarceaux circles by dividing the torus into two halfs by the plane with equation bx = ax
Reference: Z.A. Melzak, Invitation to Geometry, pp. 63-68.

Rotate one of the Villarceaux circles [b cos(t), a + c sin(t), a cos(t)] about the z-axis by an angle of p to obtain the one half of a torus with the boundary consisting of two interlocking circles.

Construct the torus by pasting two patches as thus:

Construct a knot as thus:

Construct a twisted torus whose sections are formed by regular triangles: