| Semiregular Polyhedron |
Article Index for Semiregular |
Information AboutSemiregular Polyhedron |
| CATEGORIES ABOUT SEMIREGULAR POLYHEDRON | |
| polyhedra | |
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 Triangle s and two Pentagon s around each vertex. ''3.3.3.5'' in contrast is a Pentagonal Antiprism . These polyhedra are sometimes described as Vertex-transitive . Since Gosset, other authors have used the term semiregular in different ways. E. L. Elte provided a definition which Coxeter found too artificial. Coxeter himself dubbed Gosset's figures ''' Uniform ''', with only a quite restricted subset classified as semiregular. 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). Yet others have taken the opposite path, categorising more polyhedra as semiregular. These include:
A further source of confusion lies in the way that the Archimedean Solid s are defined, again with different interpretations appearing. Gosset'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, CromwellCromwell, P. ''Polyhedra'', Cambridge University Press (1977) and Cundy & RollettCundy H.M and Rollett, A.P. ''Mathematical models'', 2nd Edn. Oxford University Press (1961) are all guilty of such slips. GENERAL REMARKS In many works ''semiregular polyhedron'' is used as a synonym for Archimedean Solid ."Archimedes". (2006). In ''Encyclopædia Britannica''. Retrieved 19 Dec 2006, from Encyclopædia Britannica Online (subscription required). For example Cundy & Rollett (1961). We can distinguish between the facially-regular and Vertex-transitive figures based on Gosset, and their vertically-regular (or versi-regular) and facially-transitive duals. |
|
|