Taming the Snakes

Computer scientists, dog owners, parents, and most other generic humans are happy when their trained subjects behave as expected. Mathematicians are happy when things develop other then expected.

For instance, Rafael and I have built a machine that takes an explicit planar curve, lifts it to a space curve, and twirls an explicit minimal surface around it. The emphasis here is on explicit, because that allows to do all kinds of things to the minimal surface that would be hard to do otherwise.

So we started feeding curves to the machine that we hadn’t built it for.


As a first example, the logarithmic spiral is lifted to a space curve such that both ends of the spiral move up, and the speed with which the surface twists is much faster at the inner piece. We call this the cobra surface.

A few years after David, Mike, and I had shown that the genus one helicoid is embedded, I was contacted by a science freelance writer. She said that this helicoid with the handle had been pretty cool, whether we had maybe some new examples that looked very different and cool, too. We hadn’t. But here they come. The Archimedean spiral is next. Again, the surface spirals faster when the curve is more strongly curved.


If you liked the trefoil surfaces, you will like the next one, too: Here we start with a common cycloid, and the lifted curve becomes another trefoil knot.


Finally, the pentagram cycloid lifts to a knotted curve without cusps, and we can make another prettily knotted minimal surface.


What to Keep

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I have often been trying to capture personal time in this blog through old and new photographs. What you see above, is a recent (like 20 minutes ago) photo of an old toy of mine. I received this early edition of Spirograph when I was maybe 9 years old. You can now purchase a 50th anniversary edition (without the tasty pins …).

This is something that I (and my daughter) have used intermittently over all these years, and it has acquired a meaning for me way beyond its mere presence. Already back then I cherished it so much that I kept the products in the box. So this is, well, an ancient artifact:

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The lavishly illustrated instruction manual promised perfection that I never achieved. Too often one of the wheels started sliding instead of just rolling, or the pins didn’t quite hold. What counts, however, is the process. We are, truly, not interested in the ideal, the mathematical perfect curve, but in the process of getting there.

The curves that one can make with Spirograph are called Cycloids. You can get them abstractly by tracing a point on a wheel that is rolling along a curve. In its simplest form, you roll a circle along a line,

Cycloidline 01

and you learn that these curves can be found on the icy Saturn moon Europa, or as geodesics in the upper half plane when using the Riemannian metric 1/y ds (which is not quite the hyperbolic metric, of course). The ancient ones used them to model planetary orbits when popular belief pinned man into the center of the universe.

As my early Spirograph experiments show, the results make nice designs. Using contemporary software like Mathematica allows you to create these to perfection, you think? Unfortunately, plotting the true cycloids will result in images that are either inaccurate (not enough anchor points) or difficult to manipulate in Adobe Illustrator (too many anchor points). So, to make this:-:,

Cycloiddesign 01

I replaced the cycloidal arcs between intersections by cubic Bezier splines that have the same curvature as the cycloids at their end points. Again, this was just to find satisfaction in the process to approximate the ideal.