Each of the Archimedean Solids can be constructed from operating on the Platonic Solids (or other Archimedean Solids).
For every polyhedron, there exists a dual polyhedron in which the faces and vertices occupy each other's places. The edges connecting the vertices of a solid become the edges connecting the faces of its dual.
The duals of the Platonic Solids are as follows:
The tetrahedron is self-dual
The cube and octahedron are a dual pair
The dodecahedron and icosahedron are a dual pair
Truncation is the operation of cutting the solid at each vertex to create new faces. In terms of Archimedean Solids, the vertices are cut uniformly to create regular polygonal faces.
The truncated cube is created by truncating the cube to make equilateral triangles, one at each vertex of the cube. Each new vertex is encompassed by one triangle and two regular octagons.
The truncated tetrahedron is created by truncating the tetrahedron to make equilateral triangles, one at each of the tetrahedron's vertices. Each new vertex is encompassed by one equilateral triangle and two regular hexagons.
The truncated octahedron is created by truncating the octahedron to make squares, one at each of the octahedron's vertices. Each new vertex is encompassed by a square and two regular hexagons.
The truncated dodecahedron is created by truncating the dodecahedron to make equilateral triangles, one at each of the dodecahedron's vertices. Each new vertex is encompassed by an equilateral triangle and two regular decagons.
The truncated icosahedron is created by truncating the icosahedron to create regular pentagons, one at each of the icosahedron's vertices. Each new vertex is encompassed by one regular pentagon and two regular hexagons.
Rectification is the operation of cutting the solid at the vertices of the midpoints of all of its edges.
The cuboctahedron is created by rectifying the cube or octahedron, making two equilateral triangles and two squares at each vertex.
The icosidodecahedron is created by rectifying the icosahedron or dodecahedron, making two equilateral triangles and two regular pentagons at each vertex.
The truncated cuboctahedron, with its slightly misleading name, can be created by rectifying the cuboctahedron, making a square, regular hexagon, and regular octagon at each vertex.
The truncated icosidodecahedron, with its slightly misleading name, can be created by rectifying the icosidodecahedron, making a square, regular hexagon, and regular decagon at each vertex.
Interestingly, the octahedron can be created by rectifying the tetahedron.
Expansion is the operation of displacing either the edges or faces of the solid and filling in the gaps with new faces.
The rhombicosidodecahedron is created by expanding the dodecahedron or icosahedron. Each new vertex is surrounded by an equilateral triangle, square, regular pentagon, and another square.
The rhombicuboctahedron is created by expanding the cube or octahedron. Each new vertex is surrounded by three squares and an equilateral triangle.
Snubification is the operation of expanding the faces of the solid, twisting them, and filling in the gaps with new faces.
The snub cube can be created from the snubification of the cube, making four equilateral triangles and one square at each vertex.
The snub dodecahedron can be created by the snubification of the dodecahedron, making four equilateral triangles and one regular pentagon at each vertex.