Why can't a (5,5,4) solid exist?

[GeometryIsHARD]

So i was reading my geometry textbook and it said "and obviously there can be no solid where every vertex has a (5,5,4) arrangement"; unfortunately, this is not an obvious fact to me. Can somebody explain to me what makes this so obvious? This statement was the end of a proof by contradiction so i feel I should really understand why it's true.

 

[fresh_42]

I'm no geometer and not really firm in the definitions involved. But on the Wiki page there is an explanation that might help:
https://en.wikipedia.org/wiki/Verte...tex_configurations_of_regular_convex_polygons
(The or's are all either or's.)

 

[Office_Shredder]

This statement from the wikipedia article sums it up:

Specifically p.q.r implies that a p-gon is surrounded by alternating a q-gons and r-gons, so either p is even orq equals r.

If something is uniform 5.5.4 then focus on a pentagon. Each edge alternates being a pentagon or a square, but when you go all the way around the pentagon you will get two squares (or two pentagons) abutting each other.