# The 5 Platonic Solids

On earlier pages such as when we were looking at graphs of platonic solids, we made note of the five platonic solids, the tetrahedron, icosahedron, dodecahedron, octahedron, and cube, as well as their properties:

Platonic Solid | Number of Vertices | Number of Faces | Number of Edges | $v + f – e$ | Dual | ||
---|---|---|---|---|---|---|---|

Tetrahedron | $4$ | $4$ | $6$ | $2$ | Tetrahedron | ||

Octahedron | $6$ | $8$ | $12$ | $2$ | Cube | ||

Icosahedron | $12$ | $20$ | $30$ | $2 �1481��1482� Dodecahedron �1483��1484��1485��1486�Cube �1487��1488� [[$ 8 �1489��1490� [[$ 6$ | $12$ | $2$ | Octahedron |

Dodecahedron | $20$ | $12$ | $30 �1507��1508� [[$ 2$ | Icosahedron |

We will now formally define what a platonic solid with the following two definitions:

Definition: A polygon is considered Regular if the inner angles of the polygon are all the same and the lengths of all sides of the polygon are the same. |

For example, a square is a regular polygon, however, a rhombus is not since the inner angles are not all the same, and similarly, a rectangle is not regular since the lengths of the sides are not the same.

Definition: A Platonic Solid is a solid in $\mathbb{R}^3$ constructed with only one type of regular polygon. |

We will now go on to prove that there are only 5 platonic solids.

Theorem 1: There exists only $5$ platonic solids. |

**Proof:**We will first note that we can only construct platonic solids using regular polygons. We will look at the first four regular polygons: the equilateral triangle, square, regular pentagon, and regular hexagon:

First let’s determine how a vertex of a platonic solid can be constructed. Clearly, we need at minimum of $3$ faces coming together to create a vertex, since only $2$ faces coming together will only be a bend. We will also note that the sum of the angles coming together to form a vertex must be less than $360$ degrees. If the sum of the angles coming together to form a vertex is greater than $360$ degrees, then a vertex cannot be constructed with them. Let’s look at the following vertex formation cases:

Type of Regular Polygon | Inner Angle of the Regular Polygon | Number of Faces Constructing Each Vertex | Sum of Angles | Sum of Angles Less Than 360 Degrees? (Y/N) | Constructable? (Y/N) | Visual Representation of Vertex |
---|---|---|---|---|---|---|

Equilateral Triangle | $60^{\circ}$ | $3$ | $180^{\circ}$ | Yes. $180^{\circ} | Yes | |

Equilateral Triangle | $60^{\circ}$ | $4$ | $240^{\circ}$ | Yes. $240^{\circ} | Yes | |

Equilateral Triangle | $60^{\circ}$ | $5$ | $300^{\circ}$ | Yes. $300^{\circ} | Yes | |

Equilateral Triangle | $60^{\circ}$ | $6$ | $360^{\circ}$ | No. $360^{\circ} = 360^{\circ}$ | No | – – – |

Square | $90^{\circ}$ | $3$ | $270^{\circ}$ | Yes. $270° | Yes | |

Square | $90^{\circ}$ | $4$ | $360^{\circ}$ | No. $360^{\circ} = 360^{\circ}$ | No | – – – |

Regular Pentagon | $108^{\circ}$ | $3$ | $324^{\circ}$ | Yes. $324^{\circ} | Yes | |

Regular Pentagon | $108^{\circ}$ | $3$ | $432^{\circ}$ | No. $432^{\circ} > 360^{\circ}$ | No | – – – |

Regular Octagon | $120^{\circ}$ | $3$ | $360^{\circ}$ | No. $360^{\circ} = 360^{\circ}$ | No | – – – |

- Note that no type of geometric vertex can be constructed with hexagons as specified in the table above, as $3$ hexagons would sum to $360$ degrees or greater which would result in the common honeycomb pattern (which is not a solid vertex) as all hexagons would have to lie on the same plane.

- In fact, there is no other way in which we can construct a vertex with regular polygons. Hence, there are only $5$ ways to construct a solid with a single type of regular polygon, each of which corresponds to one of the platonic solids. Respectively, the tetrahedron, octahedron, icosahedron, cube, and dodecahedron. $\blacksquare$