I recently read Michèle Audin’s book Remembering Sofya Kovalevskaya and was surprised to learn about a mathematical object called the Kovalevskaya top (sometimes spelled Kovalevsky or Kowalevski). Why didn’t I know about a toy named after our mathematical foremother?! Well, it turns out it’s hardly a toy. Even though it’s called a Kovalevskaya top, you can’t exactly play with one on your table.

The top was the focus of the work that earned Kovalevskaya the Prix Bordin from the French Academy of Sciences in 1888. The prize went to work on a particular question. In this case, it asked one to “perfect in one important point the theory of the movement of a solid body round an immovable point.” The way an object spins depends not just on its size and mass but also the way its mass is distributed. Perhaps the flashiest demonstrations of this fact occur every four years when we watch Olympic figure skaters draw their arms and legs in to their bodies to speed up their rotations as they’re spinning on the ice.

Leonhard Euler and Joseph-Louis Lagrange had each worked out equations describing the motion of one special type of top a century prior to Kovalevskaya's work. Euler looked at tops where the fixed point is the center of gravity of the object, and Lagrange analyzed the case of a symmetric top in which the fixed point is where the object touches the ground, the case that's most like the toy we call a top. Kovalevskaya discovered another case in which the motion can be analyzed rather than being totally chaotic. In her case, two of the three moments of inertia are the same and the third is half their size. (The three moments of inertia relate to how weight is distributed relative to the three coordinate axes centered at one point in the object.)

There are several different ways to draw a top that has the properties Kovalevskaya required. I’ve used a picture from a 1997 video about the Kovalevskaya top by Peter Richter, Holger Dullin, and Andreas Wittek. (They also wrote a paper about the top (pdf). Both it and the video are aimed at readers/viewers with an extensive background in math.) Jump to 2:15 for a nice clip of the motion of the top.

There are a couple other pictures in Remembering Sofya Kovalevskaya, but no real-world versions of the Kovalevskaya top exist. It wouldn’t be so easy to make a toy you could play with at your desk, for example, because unlike with a regular top, the fixed point is not out on an extremity that could stay in one spot on the table.

I went to an amusement park a few days ago. I didn’t ride any of the rides this time, but I did enjoy watching some of them lurch around and marveling at the deep knowledge of physics the ride designers must have. Watching the Kovalevskaya Top video, I started wondering whether it would be possible to use the jerky, chaotic-looking movements of the Kovalevskaya Top as the basis for a ride. I’m officially throwing the idea out into the universe. I’ll keep my fingers crossed.

The Cantor Set
Fat Cantor Sets
The Topologist’s Sine Curve
Cantor's Leaky Tent
The Infinite Earring
The Line with Two Origins
The House with Two Rooms
The Fano Plane
The Torus
The Three-Torus
The Möbius Strip
The Long Line
Space-Filling Curves
The Wallis Sieve
Two Tori Glued along a Slit
The Empty Set
The Menger Sponge
The Connected Sum of Four Hopf Links
Borromean Rings
The Sierpinski Triangle
Lexicographic Ordering on the Unit Square
The SNCF Metric
The Mandelbrot Set
Fatou's Pancake
The Pseudosphere