I'm currently taking an intro course in nanotechnology and was introduced to Euler's formula while reading ($V + F - E = 2$). The formula was introduced in the context of bucky balls (carbon based structures with 20 hexagonal faces and 12 pentagonal faces), but another example was given to prove why the pentagonal faces were necessary. The example reads as follows:
"If one imagines starting to build [a closed shape using only hexagons] with the first three hexagons brought together along their edges, it is easy to see that each pair shares an edge ($E = F/2$) and that each triple shares two verticies ($V = 3F/2$) so that $V - E + F = 0$, proving that completing a closed shape without gaps is impossible unless one also uses tiles with a different shape." (Introduction to Nanoscience and Nanotechnology, Chris Binns, pg. $58$)
Now, I definitely have a lot to learn when it comes to geometry, but for whatever reason I've been having such a hard time visualizing what is meant here and understanding where the relations such as $E = F/2$ are coming from. Could someone possible explain it to me in a different way or provide a visual of some sort?