The Mathematics of Torus Triptych

A torus can be generated by rotating a circle around an axis in the same plane as the circle, but not intersecting it. We can produce parametric equations for such a torus of revolution as follows: if we consider a circle of radius b in the xz-plane, centered at the point (x, z) = (a, 0) on the x-axis, then the points on the circle are given by (x, z) = (a + b cos $\theta$ , b sin $\theta$ ). If we rotate this circle about the z-axis, then each point (x, z) on the original circle traces out a new circle in a plane parallel to the xy-plane; the radius of this new circle will be x (the distance of the original point from the z-axis), and the height of the plane containing the new circle will be z. This means the new circle can be parameterized by (x cos $\phi$ , x sin $\phi$ , z). As we let (x, y) vary over the entire original circle, we obtain a parameterization for the torus:

T( $\theta$ , $\phi$ ) = ((a + b cos $\theta$ ) cos $\phi$ , (a + b cos $\theta$ ) sin $\phi$ , b sin $\theta$ ).

In "Torus Triptych" we used a = $\sqrt{2}$ and b = 1. This basic torus was rotated to three different positions and then sliced by a horizontal plane at various heights to obtain the three sequences presented. The lower sequence (in blue) has a particularly interesting slice in the second image. Here, the horizontal plane intersects the torus in two overlapping circles of equal radius.