Semiregular polyhedron
A semiregular polyhedron is a polyhedron with regular faces and a symmetry group which is transitive on its vertices. Or at least, that is what follows from Gossett's 1900 definition of the more general semiregular polytope.[1][2] These polyhedra include:
- The thirteen Archimedean solids.
- An infinite series of convex prisms.
- An infinite series of convex antiprisms (their semiregular nature was frist observed by Kepler).
These semiregular solids can be fully specified by a vertex configuration, a listing of the faces by number of sides in order as they occur around a vertex. For example 3.5.3.5, represents the icosidodecahedron which alternates two triangles and two pentagons around each vertex. 3.3.3.5 in contrast is a pentagonal antiprism. These polyhedra are sometimes described as vertex-uniform.
Since Gossett, other authors have used the term semiregular in different ways. Elte provided a definition which Coxeter found too artificial. Coxeter himself dubbed Gossett's figures uniform, with only a quite restricted subset classified as semiregular. [3]
Yet others have taken the opposite path, categorising more polyhedra as semiregular. These include:
- Three sets of star polyhedra which meet Gossett's definition, analogous to the three convex sets listed above.
- The duals of the above semiregular solids, arguing that since the dual polyhedra share the same symmetries as the originals, they too should be regarded as semiregular. These duals include the Catalan solids, the convex dipyramids and antidipyramids or trapezohedra, and their nonconvex analogues.
A further source of confusion lies in the way that the Archimedean solids are defined, again with different interpretations appearing.
Gossett's definition of semiregular includes figures of higher symmetry, the regular and quasiregular polyhedra. Some later authors prefer to say that these are not semiregular, because they are more regular than that - the uniform polyhedra are then said to include the regular, quasiregular and semiregular ones. This naming system works well, and reconciles many (but by no means all) of the confusions.
In practice even the most eminent authorities can get themselves confused, defining a given set of polyhedra as semiregular and/or Archimedean, and then assuming (or even stating) a different set in subsequent discussions. Assuming that one's stated definition applies only to convex polyhedra is probably the commonest failing. Coxeter, Cromwell[4] and Cundy & Rollett[5] are all guilty of such slips.
General remarks
In many works "semiregular polyhedron" is used as a synonym for Archimedean solid.[6] For example Cundy & Rollett (1961).
We can distinguish between the facially-regular and vertex-uniform figures based on Gosset, and their vertically-regular (or versi-regular) and facially-uniform duals.
Coxeter et. al. (1954) use the term "semiregular polyhedra" to classify uniform polyhedra with Wythoff symbol of the form p q | r, a definition encompassing only six of the Archimedean solids, as well as the regular prisms (but not the regular antiprisms) and numerous nonconvex solids. Later, Coxeter (1973) would quote Gossett's definition without comment, thus accepting it by implication.
Eric Weisstein, Robert Williams and others use the term to mean the convex uniform polyhedra excluding the five regular polyhedra-- including the Archimedean solids, the uniform prisms, and the uniform antiprisms (overlapping with the cube as a prism and regular octahedron as an antiprism).[7][8]
Peter Cromwell (1997) writes in a footnote to Page 149 that, "in current terminology, 'semiregular polyhedra' refers to the Archimedean and Catalan (Archimedean dual) solids". On Page 80 he describes the thirteen Archimedeans as semiregular, while on Pages 367 ff. he discusses the Catalans and their relationship to the 'semiregular' Archimedeans. By implication this treats the Catalans as not semiregular, thus effectively contradicting (or at least confusing) the definition he provided in the earlier footnote. He ignores nonconvex polyhedra.
References
- ^ Thorold Gosset On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
- ^ Coxeter, H.S.M. Regular polytopes, 3rd Edn, Dover (1973)
- ^ Coxeter, H.S.M. Longuet-Higgins, M.S. and Miller, J.C.P. Uniform Polyhedra, Philosophical Transactions of the Royal Society of London 246 A (1954), pp. 401-450. (JSTOR archive, subscription required).
- ^ Cromwell, P. Polyhedra, Cambridge University Press (1977)
- ^ Cundy H.M and Rollett, A.P. Mathematical models, 2nd Edn. Oxford University Press (1961)
- ^ "Archimedes". (2006). In Encyclopædia Britannica. Retrieved 19 Dec 2006, from Encyclopædia Britannica Online (subscription required).
- ^ Weisstein, Eric W. Semiregular polyhedron. (2006). Retrieved 19 Dec 2006, from MathWorld. The definition here does not exclude the case of all faces being congruent, but the Platonic solids are not included in the article's enumeration.
- ^ Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Chapter 3: Polyhedra)