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Platonic and Archimedean Solids, and Euler's Formula f-e+v=2

There are five Platonic Solids and thirteen Archimedean Solids, which are convex polyhedra whose faces are all regular polygons.

Platonic Solids

A "Platonic Solid" is a Regular Polyhedron.  A regular polyhedron is convex, and its identical faces are all regular polygons.  There are only five such solids.  The tetrahedron is the smallest of them.  The others are the cube, the octahedron (with 8 equilateral triangle faces, made by gluing together the bases of two square pyramids with equal edge lengths), the icosahedron (with 20 equilateral triangular faces), and the dodecahedron (with 12 pentagonal faces).

From any regular polyhedron we can construct another one, called its dual, by joining the centers of its faces with line segments.  For example, if we join the centers of the faces of a cube, we get an octahedron sitting inside the cube, so the dual of a cube is an octahedron.  If we repeat the process for the octahedron, we obtain a cube sitting inside the octahedron, so the dual of an octahedron is a cube.  Similarly, the dodecahedron and the icosahedron are duals of each other, and the tetrahedron is its own dual.  A polyhedron and its dual have the same number of edges (12 for a cube and an octahedron, for example) but the numbers of vertices and faces are interchanged.

Name Figure Faces Vertices Edges
= Faces 
+ Vertices
 - 2 
Faces and
tetrahedron 4
4 6 tetrahedron
cube 6
8 12 octahedron
octahedron 8
6 12 cube
dodecahedron 12
20 30 icosahedron
icosahedron 20
12 30 dodecahedron

The relationship between Faces, Vertices, and Edges for any convex polyhedron is given by Euler's Formula:

v - e + f = 2

This is actually one of many formulas attributed to Leonhard Euler (pronouned "oiler"), a prolific Swiss mathematician who lived from 1707-1783.  It is believed that Euler discovered the formula in 1750.  It was first proven by Legendre in 1794.  Not quite a proof, but a way to see that it is true is by induction on faces.  It helps to imagine the polyhedron "flattened" by projecting it onto a plane, making a flat graph such that each region bounded by the edges corresponds to a face of the polyhedron, with the whole of the space outside the outer edge corresponding to one of the faces.  The "base case" is a solid with two faces glued back-to-back which is "flattened" to a single polygon.  Each of the two faces is bounded by the polygon, and shares its v vertices and e=v edges with the other face.  So v-e+f=2 in this case.  Now, for any convex polyhedron with more than two faces, you can reduce the number of faces by one by removing an edge, causing two faces to become one, and the formula still holds.  (This is essentially the second proof given in Seventeen Proofs of Euler's Formula.)

Archimedean Solids

There are 13 Archimedean Solids.  An Archimedean Solid is a convex polyhedron whose faces are regular polygons arranged the same way about each vertex.

Some are obtained by cutting off, or truncating, the corners of a regular polyhedron. Thus we obtain the truncated cube, the truncated tetrahedron, the truncated octahedron.  The cuboctahedron and the icosidodecahedron are obtained by taking the volume simultaneously enclosed by a regular polyhedron and its dual of the same radius.  (Performing the same process with a tetrahedron yields an octahedron).  Truncating a cuboctahedron and adjusting the resulting faces to make them squares gives the truncated cuboctahedron. The final model in this group is the rhombicuboctahedron, bounded by a cube, and octahedron, and rhombic dodecahedron.  The seven other Archimedean solids are the truncated dodecahedron, truncated icosahedron, cuboctahedron, rhombicosidodecahedron, truncated icosidodecahedron, snub cube, and snub dodecahedron.


Internet references

Mathworld -- Platonic Solids, Archimedean Solid

Mathematical Teaching Tools in the Department of Mathematics of The University of Arizona 

Seventeen Proofs of Euler's Formula 

Projective Geometry describes the meaning of "dual".

See Geometry from the Land of the Incas:  Platonic Solids for some animated rotating Platonic Solids and some nice MIDI music.

Related pages in this website

Go back to Geometry - Solids


Centroid describes what it is (the balancing point) and how to calculate it using integrals.

Solid of Rotation describes how to calculate it using integrals, and better yet, how to calculate it using Pappus' Centroid Theorem.  It also has a table of surface areas and volumes of a variety of simple solid shapes.



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