This week I’ll describe one strategy for representing 4-dimensional objects in 3-dimensions, using both modern and traditional media. While more than three spatial dimensions is a foreign concept to most people, Mathematicians have been studying theoretical objects in four dimensions and more for over a century. (Some contemporary Mathematicians even think about infinite-dimensional things, whatever that means!).
Each new dimension simply gives you more freedom to move. For example, if I draw a figure-8 curve on a piece of paper without lifting my pen, then it necessarily crosses itself. However, by introducing a break at the intersection, our brain tricks us into interpreting the 2-dimensional picture as a representation of a 3-dimensional loop with a twist.
In the same way one can imagine a four-dimensional object. Our familiar three dimensions become like the paper. By introducing breaks, we can interpret one part of the surface as passing right through another, without intersecting it. The surface is not actually broken: it has just been temporarily lifted into the fourth dimension and returned, like the breaks in the drawing above.
The Klein Bottle is a classic four-dimensional mathematical object. There are various models used to help us visualize it. The most common one looks a lot like a vase with a bent “nose” that seems to intersect itself. Here is an image of some beautiful glass Klein bottles that you can buy from Cliff Stoll at Acme Klein Bottle. I highly recommend checking out his website!
The deficit of these models is that they intersect themselves (as they must in three dimensions), while the actual four-dimensional object has no self-intersections. These models are analogous to drawing the twisted loop pictured above as a figure-8 curve, without lifting your pen.
To create a representation of the Klein Bottle with no self-intersections, I used a different model. Another way to think about the Klein Bottle is to sweep a figure-8 (with a break at the crossing to avoid the self-intersection) around a circle, while rotating 180 degrees. It’s hard to visualize, because this can only be done in four dimensions. If we try to construct it in three dimensions, it will intersect itself. However, by representing a skeleton of the model, and making the wires inter-weave, one can avoid self-intersections. Here’s a Rhino render of my design (modeled with Grasshopper):
Here’s the 3D printed version, available at Shapeways in small and large versions.
In the close-up on the right you can see how the individual wires that define the model are interwoven, so that there is no place in the model where it actually intersects itself.
This model got me thinking about making things with more traditional fabrication techniques. At the same time I had an undergraduate thesis student, Joi Chu-Ketterer, who wanted to explore knitting and crocheting mathematical surfaces. This kind of thing has been done a lot for hyperbolic surfaces. See, for example, the magnificent coral reef of the Institute for Figuring. I’d also seen many knitted tori and spheres. Cliff Stoll has some neat knitted Klein bottle hats on his website, but I had never seen anyone try to knit the figure-8 Klein bottle. The nice thing about yarn is that you can interweave the strands, to achieve the same kind of avoidance of self-intersections. It’s a painstaking process to thread the strands through each other while knitting them, but Joi did it!
