# Definition:Platonic Solid

## Contents

## Definition

A **platonic solid** is a convex polyhedron:

- $(1): \quad$ whose faces are congruent regular polygons
- $(2): \quad$ each of whose vertices is the common vertex of the same number of faces.

That is, a **platonic solid** is a convex regular polyhedron.

### Regular Tetrahedron

A **regular tetrahedron** is a tetrahedron whose $4$ faces are all congruent equilateral triangles.

### Cube

A **cube** is a hexahedron whose $6$ faces are all congruent squares.

### Regular Octahedron

A **regular octahedron** is an octahedron whose $8$ faces are all congruent equilateral triangles.

### Regular Dodecahedron

A **regular dodecahedron** is a dodecahedron whose $12$ faces are all congruent regular pentagons.

### Regular Icosahedron

A **regular icosahedron** is an icosahedron whose $20$ faces are all congruent equilateral triangles.

## Also known as

Some sources refer to a **Platonic solid** as a **regular polyhedron.**

However, the latter term technically also encompasses polyhedra that are not convex.

## Also see

## Source of Name

This entry was named for Plato.

## Historical Note

The platonic solids were all well-known to the ancient Greeks.

The compass and straightedge constructions of the Platonic solids forms the climax of Euclid's *The Elements*.

Some have suggested that to glorify the Platonic solids was its primary purpose.

Some sources suggest that the construction of the final two of these were the work of Theaetetus of Athens, while others suggest that Hippasus of Metapontum may have contributed.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.4$: Euclid (flourished ca. $300$ B.C.) - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $2$: The Logic of Shape: Euclid