Skew Line on a Torus

This animation was created using Maple software.

MAPLE logo

This animation shows how the “skew line” densely fills up the torus. First, we had MAPLE draw us a nice torus using a standard parametrization of form

with

Each curve on the torus is constructed by restricting the above parametrization to a line with a rational slope through the origin in the (t,s)-parameter space. For the n-th frame of animation we choose a line of form s = f(n)*t, where f(n) = ratio of two consecutive Fibonacci numbers. We chose these numbers because of the fact that Torus equation

where the golden ratio, Golden ration equation

, is an irrational number. The animation shows that as the slope of the line s = f(n)*t approaches an irrational number, the image of this line on the torus starts to densely fill it up. The differential topologists would say that the image of this line is an immersed submanifold of the torus.

References

On S. S. Chern, the differential geometer. . .
D. J. Albers and G. L. Alexanderson (eds.), Mathematical People: Profiles and Interviews (Boston, 1985), 33-40.

and . . .
< http://www-history.mcs.st-and.ac.uk/history/Biographies/Chern.html >.

For animations see . . .
John F. Putz, MAPLE ANIMATION, Chapman Hall CRC Press, 2003.

For those who have MATHEMATICA®, . . .
Alfred Gray, Modern Differential Geometry, 2nd ed., CRC Press, 1998, pp. 304-305.

Howard Eves, An Introduction to the History of Mathematics, 6th ed., Saunders College Publishing, p. 324.

Victor J. Katz, A History of Mathematics, 2nd ed., Addison Wesley Longman, 1998, pp. 768-771.

Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 1999, pp. 1816-1819.

Note: Sections of a torus that are taken as planes parallel to the axis (not skewed as in the case of our animation) are known as Spiric Lines of Perseus.