Colors are curious. Physics tells us that, like sound, they are just waves. But while we can hear sound waves of wave lengths between 17mm and 17m, the wave lengths of visible light range between 390 and 700 nm. That’s sad. Wouldn’t it be cool if we could see colors resonate by being able to see both red (700nm) and ultraviolet (350 nm). Of course matters are more complicated because we perceive colors differently, using three specific color sensors. Imagine being only able to hear three different sound frequencies.
A side effect of our biological limitation is that a color wheel makes sense to us, i.e. a continuous arrangement of the colors around a circle so that antipodal points represent complementary colors. This gave me the idea that one could color a Möbius strip continuously by hue so that points in the “front” and “back” are colored by complementary colors. Here is a 7-fold twisted rectangle as a ruled surface,
and here a minimal surface version based on a torus knot:
Finally, a Klein bottle, the immersion being obtained by rotating and revolving a figure 8 curve:
In mathematics, even the simplest things can have an astounding depth. Let’s for instance take the trefoil knot, the simplest knot there is:
One can replace the tube by a ribbon, like so:
This could be done with a simple ruled surface, but I like a challenge. To make this a minimal surface, one can use Björling’s formula. The game becomes tricky if one wants the surface to be of finite total curvature, but this can be done as well. Then it is not difficult to let the normal of the surface rotate once to get a knotted minimal Möbius strip.
Faster spinning normals create knotted helicoids.
Extending the surface beyond a small neighborhood of the trefoil knot makes things appear really complicated.
Of course the same can be done with more complicated knots.
I am sure we all have cut a Möbius strip in half, and been irritated by not getting two pieces but instead a doubly twisted strip.
The quickest way to parametrize a Möbius strip is as a ruled surface, letting line segments rotate by 180 degrees while moving them around the same circle where we usually cut. This raises the tantalizing question whether we could possible make a book whose pages are Möbius strips. For the moment I don’t know, I haven’t been able to find many explicit bendings of the ruled Möbius strip, except for its Doppelgänger:
This, however, is not a closed band but instead continues on periodically. There is a version of the Möbius strip as a minimal surface that also has a circle as a core curve.
As with ruled surfaces, you can twist these minimal surfaces more or less often around the circle. Here for instance is the triply twisted version.
All these minimal surfaces can be bent in their associate family. They stay minimal, and, surprisingly, also closed bands (after two turns), except for the case of a doubly twisted band. The conjugate of the triply twisted band looks like this.
Of course three is never enough, so here is the 40 fold twisted version. Amazingly enough, all these surfaces have explicit formulas.
This would be another possibility for a book project: One long twisted sheet of paper, bent into disk like pages…
This is what it might look like if you got stuck inside a highly reflective Klein Bottle. Wait – Klein Bottles don’t have an inside.
From the outside they can look like this:
Crawling through into the pipe at the bottom gets you from outside to inside. This is spooky, and responsible for this odd behavior is a Möbius strip. There are other versions of the Klein Bottle. Here is the figure eight version, obtained by rotating a figure 8 bout the vertical axis, and giving thereby the 8 a half twist.
Note that in both versions, the bottle cuts itself along a circular closed curve. That the two versions above are really quite different can be seen by looking at the bottles near these circles. In the first case, it looks like an annulus intersected with a cylinder, while in the second case we see two intersecting Möbius strips. The latter description helps to understand the geometry of the next version better.
As we know, a Möbius strip has just one boundary curve. The image above shows a Möbius strip where the boundary curve is a perfectly round circle. Taking a second copy of this Möbius strip and attaching it to the first along the boundary circles produces the stereographic projection of Lawson’s Klein Bottle, a minimal surface in the 3-dimensional sphere.
This is really complicated, so let’s look at the anatomy of this beast. The top translucent part, when turned around and after a paint job, reveals himself as a doubly twisted cylinder.
The other (bottom) part is still rather complicated. It consists of two pieces of the Klein Bottle that intersect along the orange circle.
One of them without its distracting sibling is once again a Möbius strip.
So Lawson’s Klein bottle is anatomically just a union of two intersecting Möbius strips and a doubly twisted cylinder.