Polyhedron
A polyhedron is a three dimensional shape which is bounded by polygons.
Platonic Solids
A regular polyhedron is a polyhedron all of whose faces are identical regular polygons, and all of whose vertices have the same number of faces around each vertex. There are only five regular convex polyhedra—polyhedra in which the all of the faces are on the outside of the polyhedron.[1] They are:
- The tetrahedron ("tetra-" meaning "four")
- The cube or hexahedron ("hexa-" meaning "six")
- The octahedron ("octa-" meaning "eight")
- The dodecahedron ("dodeca-" meaning "twelve")
- The icosahedron ("icosa-" meaning "twenty")
Role-playing games often use dice in the shape of tetrahedra, octahedra, dodecahedra, and icosahedra, as well as traditional cubical dice and nonregular decahedra.
Because the dodecahedron has twelve sides, decorative calendars are sometimes made in the shape of a dodecahedron, with one month printed on each face.
Archimedean Solids
The first known record of the Archimedean Solids[2] is from Pappus of Alexandria[3] around 340 AD. In Book V of the Collection[4], he attributes Archimedes of Syracuse[5] (287-212 BC) as the discoverer of thirteenth semiregular polyhedra.
\begin{align*} \text{The }\color{Goldenrod}{Golden\,Ratio,\,Phi,}\text{ is defined as:}\quad\color{Goldenrod}{\varphi\,\equiv\, \frac{1+\sqrt{5}}{2}} \quad \end{align*}
| Archimedean Number |
Name | Faces | Edges | Vertices | |
|---|---|---|---|---|---|
| 1 | Truncated tetrahedron | 8 | 4 triangles 4 hexagons |
18 | 12 |
| 2 | Cuboctahedron (Rhombitetratetrahedron) |
14 | 8 triangles 6 squares |
24 | 12 |
| 3 | Truncated cube | 14 | 8 triangles 6 octagons |
36 | 24 |
| 4 | Truncated octahedron (Truncated tetratetrahedron) |
14 | 6 squares 8 hexagons |
36 | 24 |
| 5 | Rhombicuboctahedron (small Rhombicuboctahedron) |
26 | 8 triangles 18 squares |
48 | 24 |
| 6 | Truncated cuboctahedron (Great rhombicuboctahedron) |
26 | 12 squares 8 hexagons 6 octagons |
72 | 48 |
| 7 | Snub cube (Snub cuboctahedron) |
38 | 32 triangles 6 squares |
60 | 24 |
| 8 | Icosidodecahedron | 32 | 20 triangles 12 pentagons |
60 | 30 |
| 9 | Truncated dodecahedron | 32 | 20 triangles 12 decagons |
90 | 60 |
| 10 | Truncated icosahedron | 32 | 12 pentagons 20 hexagons |
90 | 60 |
| 11 | Rhombicosidodecahedron (Small rhombicosidodecahedron) |
62 | 20 triangles 30 squares 12 pentagons |
120 | 60 |
| 12 | Truncated icosidodecahedron (Great rhombicosidodecahedron) |
62 | 30 squares 20 hexagons 12 decagons |
180 | 120 |
| 13 | Snub dodecahedron (Snub icosidodecahedron) |
92 | 80 triangles 12 pentagons |
150 | 60 |
Notes
For a full description of polyhedra see: http://mathworld.wolfram.com/Polyhedron.html
- ↑ As opposed to four "star polyhedra" in which the "faces" of the polyhedron slice into the polyhedron
- ↑ Weisstein, Eric W. "Archimedean Solid." From MathWorld--A Wolfram Web Resource.
- ↑ O'Connor, J. J.; Robertson, E. F. "Pappus of Alexandria." MacTutor
- ↑ Pappus of Alexandria "first known mention of the thirteen Archimedean solids” Drexel University, 340 AD.
- ↑ O'Connor, J. J.; Robertson, E. F. "Archimedes." MacTutor