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On polyhedron, there’s Descartes Theorem besides Euler’s Formula.
Thinking of Polyhedron
Polyhedron is rounded by faces, whatever faces’ figure is. It means there’s a theorem. Now, let’s investigate regular polyhedrons.
About regular polyhedron, the summation of internal angle that gathers one vertex must be less than 360 degrees. If it equals to 360 degrees, vertex doesn’t exist.
Example
Calculate about regular tetrahedron.
A face of it is regular triangle and one of its angles is 60 degrees. At 1 vertex of tetrahedron, 3 faces gather, so summation of angle degrees gathering at one vertex is
\[ 60 times 3 = 180 . \]Difference from space, 360 degree, is
\[ 360 – 180 = 180 . \]The number of tetrahedron vertexes is 4, so sum of angle deficit for all faces are
\[ 180 times 4 = 720 . \]I calculated such angle deficit for all regular polyhedron. The result is below.
| Sum of internal angle on one vertex | Difference from 360 degrees (angle deficit) | Sum of angle deficit for all faces | |
|---|---|---|---|
| regular tetrahedron | 180 degrees | 180 degrees | 720 degrees |
| regular hexahedron | 270 degrees | 90 degrees | 720 degrees |
| regular octahedron | 240 degrees | 120 degrees | 720 degrees |
| regular dodecahedron | 324 degrees | 36 degrees | 720 degrees |
| regular icosahedron | 300 degrees | 60 degrees | 720 degrees |
See, the summation of angle deficit over the polyhedron is 720 degree.
Actually, it is true to every polyhedron. This theorem called Descartes theorem.
Now, let’s prove it.
Proof
Write vertex number, edge number and face number of a polyhedron as \( V \) , \( E \) , \( F \) . As I wrote on Think about Euler’s Formula, the following equation is true (Euler’s Formula).
\[ V – E + F = 2 \]Now, calculate summation of angle deficit. It is explained as ( 360 degree ) × ( vertex count ) – ( sum of internal angle for all faces ).
\[ (360 \textrm{degree} ) \times ( \textrm{vertex count} ) = 360 \times V \]is clear from definition of \( V \) .
Calculate summation of internal angle for each face.
The summation of internal angle for a face
Write edge number of a face as \( e \) . The summation of internal angle on this face is
\[ (e – 2) \times 180 . \]Calculate summation of the above value for all faces.
The summation of internal angle for all faces
With \( e \), it is written as the following.
\begin{eqnarray} & & \sum \left\{ (e – 2) times 180 \right\} & = & 180 ( \sum e – \sum 2 ) \end{eqnarray}\( \sum \) means summation.
\( \sum e \) is the summation of edge count for all faces. It means count twice for each edge, so \( \sum e = 2 E \) . Calculate summation for each face, then \( \sum 2 = 2 F \) .
Now, the value we want to know is
\begin{eqnarray} & & 180 ( sum e – sum 2 ) & = & 180 ( 2 E – 2 times F ) \\ & = & 360 ( E – F ) . \end{eqnarray}Then, the sum of angle deficit over polyhedron is
\begin{eqnarray} & & 360 V – \left\{ 360 ( E – F ) \right\} & = & 360 ( V – E + F) \\ & = & 360 \times 2 \\ & = & 720 . \end{eqnarray}
