An Archimedean solid keeps one half of what makes a Platonic solid perfect and gives up the other half. Every face is still a regular polygon and every corner still looks exactly like every other, so the shape can be turned until any chosen corner takes the place of any other and nothing about it appears to change. What it gives up is the demand that all the faces be the same, and once two or three kinds of polygon are allowed the family opens up from five members to thirteen — plus the two endless families of prisms and antiprisms, which are usually counted separately because they are so plain.
Most of them are made by cutting a Platonic solid down. Slice the corners off a cube and the square faces become octagons while each removed corner leaves a triangle; cut deeper, until neighbouring cuts meet in the middle of every edge, and the cube and the octahedron both collapse into the same shape, the cuboctahedron. Other members come from pushing the faces of a solid outwards and filling the gaps, and the last two, the snub forms, come from giving those faces a twist before filling in — which is why they exist in left- and right-handed versions that cannot be turned into one another.
Archimedes described the whole set in a lost work that survives only through a summary by Pappus; the shapes had to be rediscovered piece by piece during the Renaissance, by Piero della Francesca, Luca Pacioli, Dürer and finally Kepler, who in 1619 published the complete list again and gave most of them the names still used today. One of them is now the most recognisable solid on earth: the truncated icosahedron, twelve pentagons among twenty hexagons, is the classic stitched football, the shape of the carbon-60 molecule named after Buckminster Fuller, and the pattern used for the mirror segments of large telescopes.