# Networks on a Torus

1. Nov 4, 2004

### Zurtex

I'm asked to consider regular networks on a torus. I'm given that V - E + F = 0. I need to show it is impossible to have a regular network on a torus where the faces are pentagons; I don't understand that at all. Surely it is easy to have pentagons as faces… All you would need to is draw a pentagon on it, please tell me where I am not getting this.

2. Nov 4, 2004

### HallsofIvy

Staff Emeritus
I suspect that what they mean is a network that completely covers the torus: every point on the torus in on or inside some pentagon.

Suppose your network consisted of n pentagons. Then there are n faces. How many edges are there? (Each pentagon has 5 edges, but each edge is shared by two pentagons.) How many vertices are there? (Each pentagon has 5 vertices but each vertex is shared by 3 pentagons.)

Now plug those numbers into the Euler equation.

3. Nov 4, 2004

Thanks

4. Nov 7, 2004

### Zurtex

Erm writing this out, I'm confused again. How can all shapes be a pentagon in a regular network anyway? Pentagons don't tessellate.