# Identities of Platonic Solids

From Portal

Platonic Solids Relationship Table^{[1]}

Polyhedron | Faces | Vertices | Edges | Formulas L=edge length a=surface area v=solid volume |
Radii cr=Circumradius mr=Midradius ir=Inradius |
Unit Vertices X, Y, Z Φ=1.61803… φ=0.61803… |
---|---|---|---|---|---|---|

Tetrahedron | 4 3 sides |
4 | 6 |
L=2√2 |
cr= |
+1, +1, +1 |

Hexahedron | 6 4 sides |
8 | 12 |
L=2 units |
cr= |
+1, +1, +1 -1, +1, +1 |

Octahedron | 8 3 sides |
6 | 12 |
L=2 units |
cr= |
+1, 0, 0 0, +1, 0 0, 0, +1 |

Dodecahedron | 12 5 sides |
20 | 30 |
L=2φ units |
cr=√3 |
0, +φ, +Φ +Φ, 0, +φ +φ, +Φ, 0 +1, +1, +1 -1, +1, +1 |

Icosahedron | 20 3 sides |
12 | 30 |
L=2 units |
cr= |
0, +φ, +1; +1, 0, +φ; +Φ, 0, +φ +φ, +Φ, 0 +1, +1, +1 -1, +1, +1 |

There are five Platonic Solids

- 1 The Tetrahedron
- Four faces of regular triangles

- 2 The Octahedron
- Eight faces of regular triangles

- 3 The Hexahedron (a cube)
- Six faces of regular squares

- 4 The Dodecahedron
- Twelve faces of regular pentagons

- 5 The Icosahedron
- Twenty faces of regular triangles

- Platonic Solids, Theaetetus's Theorem, Euler's Polyhedron Theorem
- “A platonic solid is a polyhedron all of whose faces are congruent regular polygons, and where the same number of faces meet at every vertex. A polyhedron is a solid figure bounded by plane polygons or faces.

- The Greek philosopher Plato described the solids in detail in his book "Timaeus" and assigned the items to the Platonic conception of the world, hence today they are well-known under the name "Platonic Solids."

- Euler's polyhedron theorem: F + V = E + 2, where F, V, E are the number of faces, vertices, and edges in the polyhedron.”

## See also

- Astro-Logix.com
- "Astro-logix is a simple yet ingenious construction toy, for building 3D frameworks, including DNA helixes, polyhedra, 3d stars, and spaceframes."

## References

- ↑ Pierce, Rod. "Platonic Solids" Math Is Fun. Ed. Rod Pierce. 20 Jun 2009. 7 May 2010 <http://www.mathsisfun.com/platonic_solids.html>