# Definition:Archimedean Polyhedron

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## Definition

An **Archimedean polyhedron** is a convex polyhedron with the following properties:

- $(1): \quad$ Each of its faces is a regular polygon
- $(2): \quad$ It is vertex-transitive
- $(3): \quad$ The faces are not all congruent.
- $(4): \quad$ It is not a regular prism or a regular antiprism.

## Also defined as

The pseudo-rhombicuboctahedron is also sometimes considered along with these, but as it is not vertex-transitive it is usually excluded.

## Also known as

An **Archimedean polyhedron** is also known as an **Archimedean solid**.

## Also see

- Results about
**Archimedean polyhedra**can be found here.

## Source of Name

This entry was named for Archimedes of Syracuse.

## Historical Note

The Archimedean polyhedra were originally classified by Archimedes of Syracuse in a work, now lost, that was discussed by Pappus of Alexandria.

The first of the modern mathematicians to describe them was Johannes Kepler in his $1619$ work *Harmonices Mundi*.

## Sources

- 1991: David Wells:
*Curious and Interesting Geometry*... (next): Entry:**Archimedean polyhedra** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Archimedean solid** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Archimedean solid** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Archimedean solid**

- Weisstein, Eric W. "Archimedean Solid." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/ArchimedeanSolid.html