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Only Five Platonic Solids

Date: 03/05/98 at 10:56:55
From: Emre Demirel
Subject: Why only 5 Platonic Solids

I'm doing some research about platonic solids and Kepler's thoughts 
about them. I learned that there are only five platonic solids. I know 
that it has something to do with the interior angles and I did some 
searches on the Internet, but I could not find a specific solution to 
the question "Why only 5 platonic solids?" If there is a formula I 
would like to see that also. Thank you.

Emre Demirel, a Turkish student

Date: 03/05/98 at 13:20:09
From: Doctor Rob
Subject: Re: Why only 5 Platonic Solids

A platonic solid has for each face a regular polygon congruent to 
every other face, with each vertex touching the same number of faces.  
This means that we only have to look at one vertex to see what happens 
at every vertex. Each vertex must touch at least three faces.

Consider the number of edges of each face:

 1. 3 edges per face, each face is an equilateral triangle, with 
    interior angle 60 degrees. You can fit 3, 4, or 5 of these around 
    a vertex, but since 360/60 = 6, you cannot fit 6 or more.

    a. Each vertex touches 3 faces. This is a tetrahedron.
    b. Each vertex touches 4 faces. This is an octahedron.
    c. Each vertex touches 5 faces. This is an icosahedron.

 2. 4 edges per face, each face is a square, with interior angle 
    90 degrees. You can fit only 3 of these around a vertex, since 
    360/90 = 4.

    a. Each vertex toucnes 3 faces. This is a cube.

 3. 5 edges per face, each face is a regular pentagon, with interior 
    angle 108 degrees. Since 360/108 = 3.33..., you can only fit 3 
    of these around a vertex.

    a. Each vertex touches 3 faces. This is a dodecahedron.

 4. 6 or more edges per face, each face is a regular hexagon or more, 
    with interior angle 120 degrees or more.  Since 360/120 = 3, 
    we cannot fit three of these around a vertex. This means that 
    this case is impossible.

In all, there are just five cases possible.

-Doctor Rob,  The Math Forum
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Associated Topics:
High School Geometry
High School Polyhedra

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