by P. Adam Collier

A regular polyhedron is a three dimensional figure, comprised of a number of faces. For a polyhedron to be considered regular, each of it's faces must be a regular polygon.

Regular polygons are such that every interior angle is equal and every side is equal.

An important theorem concerning polygons is that concerning the measure of each interior angle.

The most important theorem concerning three dimensional solids is a theorm of Euclid (Proposition XI.21).

So we know that the sum of each angle touching at each vertex must be less then 360 degrees.

From this we can determine all possible polyhera.

** TRIANGLES **

First let us consider a polyhedron with regular triangles meeting at each vertex.

A regular triangle is such that all angles measure 60 degrees

The geometric figure with the fewest sides is the triangle - with three sides. You cannot construct a polygon with only two sides.

Additionally, at least three faces must meet at each vertex. You cannot construct a polyhedron so that only two faces meet at each vertex.

If three such triangles meet at each vertex then the figure is called a tetrahedron.

If four such triangles meet at each vertex then the figure is called an octahedron.

If five such triangles meet at each vertex then the figure is called a icosahedron.

Four triangles cannot be used to contruct a regular polyhedron.

** SQUARES **

Next let us consider a polyhedron with regular rectangles meeting at each vertex.

A regular square is such that all angles measure 90 degrees

If three such squares meet at each vertex then the figure is called a cube.

Four squares cannot be used to contruct a regular polyhedron.

** PENTAGONS **

Finally let us consider a polyhedron with regular pentagons meeting at each vertex.

A regular pentagon is such that all angles measure 108 degrees

If three such pentagonss meet at each vertex then the figure is called a dodecahedron.

Four pentagons cannot be used to contruct a regular polyhedron.

** HEXAGONS AND LARGER?
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** So polyhedra cannot be constructed from faces that are each a polygon with six or more sides, so our list of polyhedra is exhaustive.
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** James Joyce's translation of Euclid's Elements were used a source material for this project.
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