# Definition:Archimedean Polyhedron

## 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 isogonal
- $(3): \quad$ 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 isogonal it is usually excluded.

## Also known as

An **Archimedean polyhedron** is also known as a **semi-regular** (or **semiregular**) **polyhedron**, but $\mathsf{Pr} \infty \mathsf{fWiki}$ employs the definition for a wider category of polyhedra.

Some sources use the term **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):**Archimedean polyhedra** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Archimedean solid** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Archimedean solid** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Archimedean solid** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**semi-regular polyhedron**

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