Torus Triptych
Title:Torus Triptych Artist:Tom Banchoff and Davide Cervone Medium:Computer Graphic (Ilfochrome print) Size:40"x12", 48"x10", and 40"x12"
This sequence of images appeared originally in the book Beyond the Third Dimension, part of the Scientific American Library. You can purchase it on line, or read a short review of it.
This sequence of images appeared originally in the book Beyond the Third Dimension, part of the Scientific American Library. You can purchase it on line, or read a short review of it.
The Artist's Comments on Torus Triptych
The "Torus Triptych" is an example of commercial art, being new illustrations from the second edition of the author's book Beyond the Third Dimension. The various images in each of the three parts indicate the "waterlevel curves" as a torus is gradually submerged into a liquid, giving quite different collections of curves depending on the way the torus is positioned with respect to a horizontal plane.
The "Torus Triptych" is an example of commercial art, being new illustrations from the second edition of the author's book Beyond the Third Dimension. The various images in each of the three parts indicate the "waterlevel curves" as a torus is gradually submerged into a liquid, giving quite different collections of curves depending on the way the torus is positioned with respect to a horizontal plane.
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 xzplane, centered at the point (x,z) = (a,0) on the xaxis, then the points on the circle are given by (x,z) = (a + bcos q, bsin q). If we rotate this circle about the zaxis, then each point (x,z) on the original circle traces out a new circle in a plane parallel to the xyplane; the radius of this new circle will be x (the distance of the original point from the zaxis), and the height of the plane containing the new circle will be z. This means the new circle can be parameterized by(xcos f, xsin f, z). As we let (x,y) vary over the entire original circle, we obtain a parameterization for the torus:
T(q,f) = ((a + bcos q) cos f, (a + bcos q) sin f,bsin q).
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. This sequence of slices was discussed recently in The College Math Journal.
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 xzplane, centered at the point (x,z) = (a,0) on the xaxis, then the points on the circle are given by (x,z) = (a + bcos q, bsin q). If we rotate this circle about the zaxis, then each point (x,z) on the original circle traces out a new circle in a plane parallel to the xyplane; the radius of this new circle will be x (the distance of the original point from the zaxis), and the height of the plane containing the new circle will be z. This means the new circle can be parameterized by(xcos f, xsin f, z). As we let (x,y) vary over the entire original circle, we obtain a parameterization for the torus:
T(q,f) = ((a + bcos q) cos f, (a + bcos q) sin f,bsin q).
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. This sequence of slices was discussed recently in The College Math Journal.
This image shows a torus that is being sliced by a plane perpendicular to its axis of rotational symmetry, much as a bagle would be sliced. You can think of it as a rising water level enveloping the torus. The slice starts as a single circle, and then splits into two circles which eventually recombine at the top of the torus.

Here the torus is being sliced as through it were being dunked into a cup of coffee. Here the slice curve starts as a circle which becomes "squeezed" at the center, and for a moment is a figure8. After this, it becomes two circles, which eventually recombine as a figure8 and then finishes up as a circle.

In this movie, the torus is twisted as a very special angle. Here, the initial circle folds over so that it touches itself at a point. At the same time, its middle squeezes together to touch itself at a second point. Both these touchings occur at the same instant, and at that moment, the slice is formed by two perfect circles. If you have the ability to step through the movie, you might want to do so to see this slice more clearly.

Interactive Versions of In and Outside the Torus