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

  1. Oct 14, 2012 #1
    1. The problem statement, all variables and given/known data
    Ok, I have this problem this week. (1)
    [tex]
    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.[/tex]

    Last week, we had the problem that follows (2)
    [tex]
    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.[/tex]

    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. jcsd
  3. Oct 14, 2012 #2

    Dick

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    Science Advisor
    Homework Helper

    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
  4. Oct 15, 2012 #3
    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!
     
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