Graphs can be a funny thing. When you say “graph”, some people will thing of a line or some other function in 2-D. Other people will think of marvelous planes and surfaces in 3-D. Fewer people will think of “standard” graph theory type graphs represented in 2-D. And even fewer people will think of those graph theory type graphs represented in 3-D. My goal is to change that.

Throughout the semester, I want to highlight the Y-Δ and Δ-Y transforms that allow you to go between graphs within the Petersen family of graphs. The names Y-Δ and Δ-Y are quite visual. In Y-Δ you are replacing 4 vertices that make a Y shape with 3 that make a triangle shape and in Δ-Y you are replacing 3 vertices that make a triangle shape with 4 that make a Y shape. Now, when the Petersen family of graphs are represented in 2-D, these Y-Δ and Δ-Y transforms are not quite apparent. I am hoping that representing these graphs in 3-D will make these transforms a bit clearer.

Since this is my first exposure to OpenSCAD (and really my first exposure to modeling something in 3-D all my own), I’m going to start off the semester by getting the Petersen family of graphs in 3-space and then go from there. This week we have the famous Petersen graph and K₆ being featured.

The Petersen Graph

Bringing the Petersen graph into 3 dimensions was quite exciting. The Petersen graph is classically portrayed as a big pentagon with a small star in the middle, like this:

As you might be able to guess, the Petersen graph is nonplanar. We can’t draw it on a plane without its edges overlapping. However, if we allow ourselves an extra dimension, we can represent the Petersen graph without any edges overlapping. 

Just like in 2-D, there are different ways to represent the Petersen graph in 3-D. Above is the one I chose to start my OpenSCAD intro with. I created it using an edge code my mentor, Dr. Taalman, sent me. Here’s a picture of my code!

K₆

Now, even though K₆ doesn’t look as complicated as the Petersen graph (it only has 6 vertices!), it is much more complicated to graph in 3-D while avoiding edges crossing over. The view from the top down doesn’t look like K₆ ,but when you move ever so slightly, you can see edges that weren’t visible before.

With the Petersen graph, I only had to adjust the height of a few points. However, K₆ is a completely different beast in that regard. Here is a spin around the graph! As you’ll be able to see, K₆ in 3-D (or at least how I’ve represented it, there are probably cleaner ways) gets pretty intricate compared to the Petersen graph above!

Hopefully next time I will have the other 5 graphs from the Petersen family to show you all, since I am much more familiar with OpenSCAD. I can tell that showing of the Y-Δ and Δ-Y transforms will be a challenge — but I’m excited!

Until next time,

Hannah Critchfield