Cylinders and Euler's Rule

Date: 09/06/2002 at 01:02:56
From: Joseph
Subject: Euler's Rule - how does it work for a cylinder?

How and why does Euler's rule work for cylinders?

Date: 09/06/2002 at 02:54:22
From: Doctor Mike
Subject: Re: Euler's Rule - how does it work for a cylinder?

Hi,

I suppose you mean the formula  V + F - E = 2.

A simple example is a cube, which has 8 vertex points, 6 faces, and 
12 edges, so 8 + 6 - 12 = 14 - 12 = 2.  The faces of a cube are flat, 
but this would also work if the faces or edges were somewhat curved, 
just so long as they don't intersect each other.  
  
The reason I mention this is that in the case of a cylinder, one of 
the faces will be the curved part of the cylinder.  Let's concentrate 
first on that curved face, because that is the "secret" about 
interpreting Euler's Law for a cylinder.  If you examine a 'tin can', 
common as a food container, you will see that there is a 'seam' from 
bottom to top.  That seam counts as an edge for the purposes of the 
formula.  Also, that edge has a vertex at each of its ends.

When we close up the cylinder 'tin can', the top is a circular face, 
with an edge around its circumference, and the same for the bottom 
face. 

Here is a list of all the faces, edges and vertices.
  Face 1 = the curved surface around the cylinder.
  Face 2 = the top, which is flat
  Face 3 = the bottom, which is also flat
  Edge 1 = the seam up the side of the curved face
  Edge 2 = the circle around the top face
  Edge 3 = the circle around the bottom face.
  Vertex 1 = the point at the top of the seam
  Vertex 2 = the point at the bottom of the seam
  
Note that the two vertices also serve double-duty as the points where 
the circular edges start and stop.
   
  So,  V + F - E  =  2 + 3 - 3  =  2 + 0  =  2 
  
This is not the only way to make a 'tin can'.  You could do it with 
two seams from bottom to top.  Think of it as making two half-cans, 
the front half and the back half, and then putting them together along 
the seams to enclose the cylindrical volume. How does this change 
things?  For one, there is an extra seam, which means an additional 
edge. But that's not all.  The extra seam has a vertex at each end, 
and that breaks the top and bottom circle edges into two parts each. 
So E = 6 instead of 3.

The curved face which previously went all around is now replaced by 
two faces, each of which goes half way around, so F = 4 instead of 3.  
There are two more vertices, namely, the points at the ends of the new 
seam edge, so V = 4 instead of 2.  So
  
    V + F - E  =  4 + 4 - 6  =  2      
  
The Law is verified under this configuration, too. I hope this helps 
you to understand how it works. 

- Doctor Mike, The Math Forum
  http://mathforum.org/dr.math/