Some interesting shapes are mentioned in Derrick Niederman’s Number Freak (2010). Using identical matchsticks, what’s the smallest fully connected shape you can make in which two matches meet at every vertex? That is, what is the smallest 2-regular matchstick graph?
It’s an equilateral triangle:
Now, what is the smallest fully connected shape you can make in which three matches meet at every vertex? That is, what is the smallest 3-regular matchstick graph? It uses twelve identical matches and looks like this:
And here is the smallest known 4-regular matchstick graph, discovered by the German mathematician Heiko Harborth and using 104 identical matches:
But Niederman says that “it’s impossible to create any arrangement in which five or more matchsticks meet at every vertex” (entry for “104”, pg. 230 of the 2012 paperback).