Student Post: Variations in Size on General Petersen Graphs

Hello again!

One more blog post dealing with Petersen Graphs. Last week we saw several examples of embeddings of Petersen graphs. Since one of the main goals behind creating those realizations is to 3D print them. As such, it is important that the dimensions of our embeddings are reasonable. In the previous embeddings, the graphs were realized with edges of radius .02mm. The realizations from last time were created using code that hard coded most of the dimensions of the objects instead of using any of the inherent values of a Petersen graph. Since then, the code has been changed to be much more general (also allowing it to be more readable). I have also changed the code to include the vertices of the graphs as well to more effectively demonstrate the properties of a Petersen graph.

The following example is of GP(12, 4) where the edges have radius 2mm and the spherical vertices have radius 3mm. The radius of the approximated circle created by the outer ring (in blue) is 50mm. The height of the graph is also reasonable dimensioned in this embedding.

App Sized Double Thickness Centered Terraces GP 12 4

The code that produced this example is relatively robust in terms of what dimensions it can produce when realizing these embeddings of Petersen graphs. The smaller n/k is the more variability there is in how large the radius of the edges of the graph can be. I also took a suggestion to realize the graph with the base in the center, creating a beautiful embedding.

This example is of GP(40, 10) where the edges have radius 1mm and the base has radius 50mm. The height on this graph however is ~200mm.

Modified Steel Cable Structure GP 40 10

The biggest issue with this code is that it is not very robust in terms of the size of the radius of the edges. Last week we had an example of GP(60, 12) which was very tight and had a “steel cable” in it. However, this code was only able to produce an embedding with no intersections due to the tiny radius that was being used. This example however has edges that are just thick enough to print. Given the height of this graph, one might be interested in an embedding that doesn’t have such proportionally thin edges.

As stated last time, now that we have some relatively printable realizations of general Petersen graphs, I’m going to shift my focus to working with n-panelled and n-flat graphs for next time.

Thanks for reading!

 

-Ben Flint

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