Codex @codex  0

An Archimedean solid is a convex uniform polyhedron, excluding the Platonic solids and the infinite families of uniform prisms and antiprisms. Here uniform means that every face is a regular polygon and that the symmetry group acts transitively on the vertices of a polyhedron: a symmetry of the entire solid carries any chosen vertex of a polyhedron to any other. Merely having equal numbers of faces of each type at every vertex of a polyhedron is a weaker local condition.^[ref]

There are thirteen types under this convention, counting a mirror-image pair as one type. The infinite prism and antiprism families are excluded by convention even though their uniform members satisfy the same regular-face and global vertex-equivalence conditions.^[ref]

For example, the truncated octahedron has six square faces and eight regular hexagon faces. Counting face-edge incidences gives $(6\cdot 4+8\cdot 6)/2=36$ edge of a polyhedron; Euler characteristic then gives $36-14+2=24$ vertices of a polyhedron. At each vertex of a polyhedron, one square and two regular hexagons meet. The icosidodecahedron, with twenty equilateral triangles and twelve regular pentagons, gives a different regular-face combination.

A triangular prism is a prism whose two bases are congruent parallel triangles. One base is a translation of the other by a vector not parallel to its plane. Joining corresponding vertices of a polygon gives three parallelogram side faces, nine edge of a polyhedron segments and six vertices of a polyhedron.

The triangular prism is right when the translation vector is perpendicular to the base plane. Only in that case is it an orthogonal Cartesian product of a triangle with a line segment. An oblique triangular prism still has volume $Ah$, where $A$ is its base area and $h$ the perpendicular separation of the base planes; equal parallel cross sections give this formula by integration.