In

Euclidean geometry, a

**Platonic solid** is a

regular,

convex polyhedron. The

faces are

congruent,

regular polygons, with the same number of faces meeting at each

vertex. There are five Platonic solids; their names are derived from their numbers of faces.

The

aesthetic beauty and

symmetry of the Platonic solids have made them a favorite subject of

geometers for thousands of years. They are named for the

ancient Greek philosopher Plato, who theorized that the

classical elements were constructed from the regular solids.

The Platonic solids have been known since antiquity. Ornamented models of them can be found among the

carved stone balls created by the late

neolithic people of

Scotland at least 1000 years before Plato

^{}.

The

ancient Greeks studied the Platonic solids extensively. Some sources (such as

Proclus) credit

Pythagoras
with their discovery. Other evidence suggests he may have only been
familiar with the tetrahedron, cube, and dodecahedron, and that the
discovery of the octahedron and icosahedron belong to

Theaetetus,
a contemporary of Plato.

Euclid gave a complete mathematical description of the Platonic solids in the

*Elements*,
the last book (Book XIII) of which is devoted to their properties.
Propositions 13–17 in Book XIII describe the construction of the
tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that
order. For each solid Euclid finds the ratio of the diameter of the
circumscribed sphere to the edge length. In Proposition 18 he argues
that there are no further convex regular polyhedra.

^{} Much of the information in Book XIII is probably derived from the work of Theaetetus.

In the 16th century, the

German astronomer Johannes Kepler attempted to find a relation between the five extraterrestrial

planets known at that time and the five Platonic solids. In

*Mysterium Cosmographicum*, published in 1596, Kepler laid out a model of the

solar system
in which the five solids were set inside one another and separated by a
series of inscribed and circumscribed spheres. Kepler proposed that the
distance relationships between the six planets known at that time could
be understood in terms of the five Platonic solids, enclosed within a
sphere that represented the orbit of

Saturn. The six spheres each corresponded to one of the planets (

Mercury,

Venus,

Earth,

Mars,

Jupiter, and

Saturn).
The solids were ordered with the innermost being the octahedron,
followed by the icosahedron, dodecahedron, tetrahedron, and finally the
cube. In this way the structure of the solar system and the distance
relationships between the planets was dictated by the Platonic solids.
In the end, Kepler's original idea had to be abandoned, but out of his
research came his

three laws of orbital dynamics, the first of which was that

the orbits of planets are ellipses rather than circles, changing the course of physics and astronomy. He also discovered the

Kepler solids.