The semiregular, so-called Archimedean solids are the polyhedrons with less stringent demands on symmetry. Their vertices and edges are all equal, although the faces are made up of various regular polygons. This family also provides a finite list of thirteen solids and thirteen only.

These solids can be subdivided into smaller groups. The simple ones are produced by merely cutting across, i.e. truncating the corners of the five platonic solids. We thus have: the truncated tetrahedron (packable), the truncated hexahedron (packable), the truncated octahedron (packable), the truncated dodecahedron (non-packable) and the truncated icosahedron (non-packable).

The remaining eight archimedean solids are produced by successive truncation, firstly of the cuboctahedron (as truncation of the cube):

rhomb cuboctahedron

rhombitruncated cuboctahedron

snub hexahedron

and secondly, of the icosidodecahedron (as truncation of the icosahedron):

rhomb icosidodecahedron

rhombitruncated icosidodecahedron

snub dodecahedron.

In this way, all Archimedean solids are generated via the multiple truncation of platonic solids. All of these are contained in the Metaeder.

The last in each case of the cubic octahedron and icosidodecahedron series are not symmetric, but either dextrorotatory or laevorotatory: we here encounter for the first time the phenomena of chirality.