## Quick Update on the 3D Printing Project

As the title suggests, this is just a quick update on the 3D printing project I’ve been doing this year.

First, after multiple failed attempts to print the Perko knots attached to pegs for the praxinoscope, I have finally created a model that passed all of Shapeways’ tests! The link to its page is here. Here are some changes from previous models:

• Constant diameter peg. The tapering of the edge where the knot connects seemed to 1) cause it to fail Shapeways’ test for object thickness, and 2) cause the knot to separate from the peg when exploring the Print-It-Anyways option.
• Knot loop embedded in the peg. Similar to the first point, having the knot just on the edge of the peg (which has the benefit of actually seeing the entirety of the knot) usually caused problems with printing. So this time, I put the strand completely inside the peg, both for stability, and I think it actually kind of looks better stylistically…you be the judge!
• Knot orientation/positioning. I aligned all of the knots to the same height, expanding the peg length when necessary, and made sure that the same strand was embedded in the peg at each stage of the isotopy. This will hopefully make the “animation” on the praxinoscope smoother.
• Instead of attaching physical numbers to the bottom of the pegs, I drilled the corresponding number of holes into the peg. Hopefully these will be visible upon printing!

Switching gears, let’s talk about the Shrikhande graph again! Recall that it is the only graph that shares the strongly-regular parameters with the square Rook’s graph $R_n$ (in the case where $n = 4$). Last post, I mentioned trying to recreate the Shrikhande graph as closely as possible to the Rook’s graph. My previous thought was this:

The left is the Rook’s graph for comparison, and the right is the Shrikhande graph. Note that the blue edges indicate upward (rising above the plane) loops and red edges indicate downwards (dipping below the plane) loops. For the Rook’s graph, this worked beautifully. But for the Shrikhande, as you can see, it became a bit of a mess.

Note that whenever two edges of the same color cross, we have to manipulate those loops so that they don’t actually cross in the 3D printed object. If two edges of different color cross, it’s no problem. So while the coloring of the Shrikhande graph on the right preserved the number of blue and red edges incident to each vertex, it also caused many troublesome crossing.

To alleviate this problem, I thought to introduce a third edge color, green, that indicated strands lying in the plane. Here is that graph:

Note that there are in fact NO edges of the same color crossing! I can’t help but get coerced by the mathematics lurking close by: What is the minimal number of colors you need to color the edges of a graph in a certain configuration so that no two crossing edges have the same color? What is the minimal number of colors you need to color the edges of a graph for any configuration? If two edges of the same color must cross, can you minimize the number of times it occurs? How?

But that is a topic I will have to wait to explore; for now, it’s time to design it and print it!