# Proving there can only be certain amounts of eq. triangles on sphere with same verte

1. Oct 14, 2012

### tainted

1. The problem statement, all variables and given/known data
Ok, I have this problem this week. (1)
$$Consider\ a\ tiling\ of\ the\ unit\ sphere\ in\ \mathbb{R}^{3}\ by\ N\ equilateral\\ triangles\ so\ that\ the\ triangles\ meet\ full\ edge\ to\ full\ edge\ (and\ vertex\ to\ vertex).\\ Show\ that\ the\ only\ possibilities\ for\ N\ are\ N = 4,\ N = 8,\ or\ N = 20.$$

Last week, we had the problem that follows (2)
$$Consider\ a\ tiling\ of\ the\ unit\ sphere\ in\ \mathbb{R}^{3}\ by\ equilateral\\ triangles\ so\ that\ the\ triangles\ meet\ full\ edge\ to\ full\ edge\ (and\ vertex\ to\\ vertex).\ Suppose\ n\ such\ triangles\ meet\ an\ one\ vertex.\ Show\ that\ the\ only\\ possibilities\ for\ n\ are\ n = 3,\ n = 4,\ n = 5.$$

Alright so my problem is that it seems to be nearly the exact same statement except (2) uses n while (1) uses N.

If that is true then wouldn't it be impossible to prove N can only be 4, 8, or 20 if I proved it was only 3, 4, or 5 last week.

2. Relevant equations
Area of each triangle = 3a - ∏

3. The attempt at a solution
Considering my question isn't for the solution rather than help understanding what it is saying/how this would be possible, I don't have any work yet.

If requested, I can show my answer to (2)

2. Oct 14, 2012

### Dick

Re: Proving there can only be certain amounts of eq. triangles on sphere with same ve

Think about using Euler's formula. http://en.wikipedia.org/wiki/Planar_graph#Euler.27s_formula n is the number of triangles meeting at a point, N is the total number of triangles on the whole sphere. They are two different things.

Last edited: Oct 14, 2012
3. Oct 15, 2012

### tainted

Re: Proving there can only be certain amounts of eq. triangles on sphere with same ve

Alright thanks man, that's all I needed, I got it!