Student Post: The Beginning of the Petersen Family of Graphs’ Journey into Space … 3-Space

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 Kbeing 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:

Petersen model perspective top

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. Petersen model persepectivetestpetersengif

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!codepetersen

K6

Now, even though Kdoesn’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 K,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, Kis a completely different beast in that regard. Here is a spin around the graph! As you’ll be able to see, Kin 3-D (or at least how I’ve represented it, there are probably cleaner ways) gets pretty intricate compared to the Petersen graph above!k6 gif

 

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

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