The Regular and Semi-Regular Polyhedra
Three of our projects here have to do with solid polyhedra. They are:
In light of this, we have a page (here) explaining what the regular and semi-regular polyhedra (also called the Platonic and Archimedean solids) are and what is so special about them.
The Regular Polyhedra (The Platonic Solids)
In the plane, a regular polygon is a polygon with equal sides and equal angles. There are an infinite number of such polygons, beginning with the triangle, square, regular pentagon, and so on. In three-dimensional space, however, there are only five polyhedra with the same regular polygon as all faces. There is a very neat proof of this fact. This was recognized by Plato, and so these solids are known as the Platonic Solids. They are:
- The Tetrahedron
- The Cube
- The Octahedron
- The Dodecahedron
- The Icosahedron
The Semi-Regular Polyhedra (The Archimedean Solids)
A Semi-Regular Polyhedron is one whose faces are not all the same regular polygon, but each vertex has the same number and type of polygons meeting. There are exactly 13 such solids, which were recognized by Archimedes, and so they are also known as the Archimedean Solids. They are:
- The Truncated Tetrahedron
- The Truncated Cube
- The Truncated Octahedron
- The Truncated Dodecahedron
- The Truncated Icosahedron
- The Cuboctahedron
- The Icosidodecahedron
- The Rhombicuboctahedron
- The Rhombicosidodecahedron
- The Rhombitruncated Cuboctahedron
- The Rhombitruncated Icosidodecahedron
- The Snub Cube
- The Snub Decahedron
Euler's Formula
Euler's formula is this simple equation:
F + V = E + 2
In this formula, F=faces of the polyhedron, V=vertices,
and E=edges. This formula holds for any polyhedron; try
it!
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