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.

Logspiral

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.

Archimedes

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.

Cycloid3b

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

Pentagram

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Avoiding Collisions (Helices I)

One of the simplest line configurations in space just utilizes the parallels to the coordinate axes that pass through the (red) points with integer coordinates.

Linepack

If we want to avoid the triple collisions at all these points, we can shift the lines one half unit each, like so:

Linepack2

This results in a dense packing of cylinders. Another possibility to avoid the collision is to let the lines spiral around the red points. I haven’t found a nice way to do this because the three helices would need to pass through the eight cubes surrounding a red point, meaning this is impossible in a symmetric way.

Linepack3

However, there is another line configuration where the lines pass through all the main diagonals. This is more complicated, because we have now four sets of parallel lines. Again we can shift the lines to avoid collisions.

Linepack6

Now, with four lines through each intersection, we can replace them by helices in a pretty symmetric fashion.

Spiralpack2