POLYHEDRA

  

A polyhedron is a solid that is constructed by polygons. There are many different types of polyhedra. The two most recognized groups are the platonic solids and the Archimedean solids.

Platonic Solids

 The five platonic solids consists of polyhedra that are constructed by congruent, regular polygons. Also, the figures formed at each vertex must be congruent, regular polygons. The five platonic solids are:

1. The tetrahedron has four faces, each of which is an equilateral triangle.

2. A cube, or hexahedron, consists of six square faces.

3. Eight faces of equilateral triangles construct the octahedron.

4. Twelve pentagons form the faces of the dodecahedron.

5. The icosahedron has twenty faces, each of which is an equilateral triangle.

 

Archimedean Solids

There is a group of polyhedra where all of the vertices of a particular polygon are identical and the faces consists of regular polygons, but the faces may be of different types of regular polygons. This group of polyhedra is called the Archimedean Solids. There are thirteen of this type of solid. All of the Archimedean polyhedra are alterations of a cube-octahedron pair and a dodecahedron-icosahedron pair, with the exception of the truncated tetrahedron.

(truncated cube, cuboctachedron, truncated octahedron, great rhombicuboctahedron, lesser rhombicuboctahedron)

(truncated dodecahedron, icosidodecahedron, truncated icosahedron, great rhombicosidodecahedron, lesser rhombicosidodecahedron)

(snub cube, snub dodecahedron, truncated tetrahedron)

Once a polyhedron is constructed, you can extend the edges of each polygon face until they meet. This process is called stellation. Surprisingly, there are no stellations of the cube or the tetrahedron to speak of. There does exist, however, one stellation of the octahedron (the stella octangula), three dodecahedron stellations (the small stellated, the great stellated, and the great dodecahedron), and 58 stellated icosahedron.

These stellated polyhedra give way to the Kepler-Poinsot Solids. The Kepler-Poinsot solids consist of four concave polyhedra with intersecting facial planes. Kepler discovered two of the four (the small stellated dodecahedron and the great stellated dodecahedron) around 1619, while Poinsot rediscovered those two and found two more (great dodecahedron and the great icosahedron) in 1809.

Kepler-Poinsot Solids

Great Dodecahedron

Great Icosahedron

 Great Stellated Dodecahedron

 Small Stellated Dodecahedron

Click HERE to see a table comparing the faces, edges, and vertices of the five platonic solids and the seventeen Archimedean solids.

Want to build your own polyhedra now?! Click HERE for templates and instructions on how to use paper to construct your own platonic and Archimedean solids, along with so many other types of polyhedra.

Links

The following links were extremely helpful in creating this webpage:

http://www.uwgb.edu/dutchs/symmetry/symmetry.htm

http://www.math.uga.edu/~clint/2003/5210/plato.htm

Return to ESCHER AND POLYHEDRA or to TESSELLATIONS AND ESCHER.