## The Twisted Color Wheel

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:

## Lawson’s Klein Bottle (Annuli IV)

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.