Semiregular polyhedra
uniform polyhedra, Archimedean solids
A uniform polyhedron is a polyhedron all faces of which are regular polygons, while any vertex is related to all the other vertices by symmetry operations. Thus, the convex uniform polyhedra consist of the five Platonic solids along with those given in the Table, where $ V $ is the number of vertices, $ E $ the number of edges, $ F $ the number of faces, $ F _ {k} $ the number of $ n _ {k} $ gonal faces, $ s $ the number of faces meeting at each vertex, namely $ s _ {1} $ $ n _ {1} $ gonal faces, $ s _ {2} $ $ n _ {2} $ gonal faces, etc.
Figure: s084300a
Figure: s084300b
Figure: s084300c
Figure: s084300d
Figure: s084300e
Figure: s084300f
Figure: s084300g
Figure: s084300h
Figure: s084300i
Figure: s084300j
Figure: s084300k
Figure: s084300l
Figure: s084300m
Figure: s084300n
Figure: s084300o
Figure: s084300p
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The first 13 polyhedra in the Table are attributed to Archimedes (Fig. a, Fig. b, Fig. c, 5–14).
In addition to the 13 solids of Archimedes and the convex prisms (Fig.15) and antiprisms (Fig.16), there are nonconvex prisms and antiprisms whose bases are "star polygons" , and 53 other nonconvex uniform polyhedra.
References
[1]  P.S. Alexandroff [P.S. Aleksandrov] (ed.) et al. (ed.) , Enzyklopaedie der Elementarmathematik , 4. Geometrie , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian) 
[2]  L.A. Lyusternik, "Convex figures and polyhedra" , Moscow (1956) (In Russian) 
[3]  M. Brückner, "Vielecke und Vielflache. Theorie und Geschichte" , Teubner (1900) 
[4]  M. Wenninger, "Polyhedron models" , Cambridge Univ. Press (1971) 
Comments
The individual names of the "Archimedean polyhedra" were coined by J. Kepler [a6], pp. 10–11, who referred to "fourteen Archimedean solids" (Archimedeis quattuordecim). The extra one (Fig. d) is the pseudorhombicubocahedron, which is not strictly "uniform" because, although all its polyhedral angles are congruent, they are of two different types from the standpoint of symmetry.
References
[a1]  H.S.M. Coxeter, "Regular and semiregular polytopes I" Math. Z. , 46 (1940) pp. 380–407 
[a2]  H.S.M. Coxeter, "Regular and semiregular polytopes II" Math. Z. , 188 (1985) pp. 559–591 
[a3]  H.S.M. Coxeter, "Regular and semiregular polytopes III" Math. Z. , 200 (1988) pp. 3–45 
[a4]  S.A. Robertson, "Polytopes and symmetry" , Cambridge Univ. Press (1984) 
[a5]  L. Fejes Toth, "Lagerungen in der Ebene, auf der Kugel und im Raum" , Springer (1972) 
[a6]  J. Kepler, "Strena: The sixcornered snowflake" , Oxford Univ. Press (1966) 
[a7]  H. Senechal (ed.) G. Fleck (ed.) , Shaping space , Birkhäuser (1988) 
[a8]  W.W.R. Ball, H.S.M. Coxeter, "Mathematical recreations and essays" , Dover, reprint (1987) pp. Chapt. 5 
Semiregular polyhedra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semiregular_polyhedra&oldid=48666