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 Check out our web site http://mathforum.org/dr.math/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.