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

  • #1
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.
 

Answers and Replies

  • #3
Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
4,175
333
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.
 

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

  • Last Post
Replies
3
Views
18K
Replies
15
Views
3K
  • Last Post
Replies
6
Views
2K
Replies
6
Views
708
Replies
16
Views
2K
Replies
3
Views
2K
Replies
7
Views
16K
Replies
8
Views
903
Replies
3
Views
2K
Top