Rank the Archimedean solids in order of prettiness. Of course this is subjective.
Is it quasi-regular? Edge-transitivity gives it a nice extra symmetry.
Is it chiral?
What is the discrepancy in area between its largest face and smallest face?
Consider its dual polyhedron. What is its most acute angle?
A ranking on the Archimedean solids induces a ranking on the Catalan solids.
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