Mark B. Villarino wrote an elementary classification argument for Archimedean solids using Euler formula for a sphere. His paper proves the thirteen exceptional convex cases, treating prisms and antiprisms as separate infinite families.[ref]
On the Archimedean or Semiregular Polyhedra by Mark B. Villarino. Original byline, abstract and sections 1.2 and 5
The counting argument relates the faces meeting at a vertex to a global constraint on vertices, edges and faces. This makes a short topological identity constrain possible three-dimensional shapes. The classification must specify its convexity and vertex-uniformity assumptions to avoid including different notions of semiregularity.
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