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592:(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.
159:
400:
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
396:
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
369:
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
556:
the regular antiprisms) and numerous nonconvex solids. Later, Coxeter (1973) would quote Gosset's definition without comment, thus accepting it by implication.
835:
588:
Peter
Cromwell (1997) writes in a footnote to Page 149 that, "in current terminology, 'semiregular polyhedra' refers to the Archimedean and
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A further source of confusion lies in the way that the
Archimedean solids are defined, again with different interpretations appearing.
715:
792:
519:
with congruent rhombic faces, would also qualify as semiregular, though Kepler did not mention it specifically.
252:
401:
probably the most common failing. Coxeter, Cromwell, and Cundy & Rollett are all guilty of such slips.
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figures based on Gosset, and their vertically-regular (or versi-regular) and facially-transitive duals.
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Yet others have taken the opposite path, categorising more polyhedra as semiregular. These include:
631:
552:, a definition encompassing only six of the Archimedean solids, as well as the regular prisms (but
304:: a listing of the faces by number of sides, in order as they occur around a vertex. For example:
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Encyclopaedia of
Mathematics: Semi-regular polyhedra, uniform polyhedra, Archimedean solids
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The definition here does not exclude the case of all faces being congruent, but the
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as a semiregular polygon (being equilateral and alternating two angles) as well as
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which meet Gosset's definition, analogous to the three convex sets listed above.
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585:(overlapping with the cube as a prism and regular octahedron as an antiprism).
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including a twist has been argued for inclusion as a 14th
Archimedean solid by
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Gosset's definition of semiregular includes figures of higher symmetry: the
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The
Geometrical Foundation of Natural Structure: A Source Book of Design
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351:, with only a quite restricted subset classified as semiregular.
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On the
Regular and Semi-Regular Figures in Space of n Dimensions
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in different ways in relation to higher dimensional polytopes.
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found too artificial. Coxeter himself dubbed Gosset's figures
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We can distinguish between the facially-regular and
236:'s 1900 definition of the more general semiregular
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577:– including the Archimedean solids, the uniform
287:(their semiregular nature was first observed by
228:; today, this is more commonly referred to as a
831:George Hart: Archimedean Semi-regular Polyhedra
335:Since Gosset, other authors have used the term
841:polyhedra.mathmos.net: Semi-Regular Polyhedron
770:are not included in the article's enumeration.
255:, also called a pseudo-rhombicuboctahedron, a
328:. These polyhedra are sometimes described as
661:The Semiregular Polytopes of the Hyperspaces
462:coined the category semiregular in his book
530:. For example, Cundy & Rollett (1961).
408:
200:) is used variously by different authors.
18:
719:, 2nd Edn. Oxford University Press (1961)
779:
410:Rhombic semiregular polyhedra (Kepler)
853:
836:David Darling: semi-regular polyhedron
680:and Miller, J.C.P. Uniform Polyhedra,
812:
750:
16:Variously-defined concept in geometry
663:, Groningen: University of Groningen
658:
566:and others use the term to mean the
208:In its original definition, it is a
704:, Cambridge University Press (1977)
544:to classify uniform polyhedra with
540:Coxeter et al. (1954) use the term
377:, and the convex antidipyramids or
13:
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14:
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320:around each vertex. In contrast:
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738:Encyclopædia Britannica Online
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203:
1:
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283:An infinite series of convex
276:An infinite series of convex
253:elongated square gyrobicupola
713:Cundy H.M and Rollett, A.P.
413:
343:provided a definition which
300:can be fully specified by a
240:). These polyhedra include:
7:
787:. Dover Publications, Inc.
595:
480:on regular bases), and two
10:
877:
728:"Archimedes". (2006). In
691:, subscription required).
526:is used as a synonym for
472:, two infinite families (
468:(1619), including the 13
817:"Semiregular polyhedron"
755:"Semiregular polyhedron"
740:(subscription required).
632:Messenger of Mathematics
730:Encyclopædia Britannica
649:, 3rd Edn, Dover (1973)
495:. He also considered a
493:rhombic triacontahedron
450:Rhombic triacontahedron
312:, which alternates two
799:(Chapter 3: Polyhedra)
687:(1954), pp. 401-450. (
524:semiregular polyhedron
513:trigonal trapezohedron
422:Trigonal trapezohedron
194:semiregular polyhedron
20:Semiregular polyhedra:
678:Longuet-Higgins, M.S.
542:semiregular polyhedra
659:Elte, E. L. (1912),
602:Semiregular polytope
489:rhombic dodecahedron
435:Rhombic dodecahedron
326:pentagonal antiprism
302:vertex configuration
263:3.4.4.4, but is not
198:semiregular polytope
716:Mathematical models
573:excluding the five
411:
232:(this follows from
33:
814:Weisstein, Eric W.
752:Weisstein, Eric W.
607:Regular polyhedron
581:, and the uniform
470:Archimedean solids
409:
298:semiregular solids
246:Archimedean solids
230:uniform polyhedron
23:Archimedean solids
19:
647:Regular polytopes
634:, Macmillan, 1900
575:regular polyhedra
571:uniform polyhedra
535:vertex-transitive
528:Archimedean solid
507:which he used in
457:
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330:vertex-transitive
310:icosidodecahedron
265:vertex-transitive
214:regular polygonal
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781:Williams, Robert
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515:, a topological
505:isotoxal figures
465:Harmonices Mundi
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259:, has identical
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768:Platonic solids
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674:Coxeter, H.S.M.
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644:Coxeter, H.S.M.
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564:Robert Williams
482:edge-transitive
460:Johannes Kepler
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405:General remarks
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308:represents the
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269:Branko Grünbaum
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806:External links
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625:Thorold Gosset
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560:Eric Weisstein
546:Wythoff symbol
522:In many works
509:planar tilings
485:Catalan solids
455:
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371:Catalan solids
363:
360:star polyhedra
358:Three sets of
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261:vertex figures
234:Thorold Gosset
218:symmetry group
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732:. Retrieved
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689:JSTOR archive
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397:confusions.
394:quasiregular
387:
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379:trapezohedra
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336:
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207:
197:
193:
187:
736:2006, from
337:semiregular
204:Definitions
192:, the term
613:References
583:antiprisms
478:antiprisms
375:dipyramids
341:E. L. Elte
285:antiprisms
222:transitive
210:polyhedron
31:antiprisms
861:Polyhedra
822:MathWorld
760:MathWorld
702:Polyhedra
425:(V(3.3))
318:pentagons
314:triangles
220:which is
855:Category
783:(1979).
596:See also
316:and two
238:polytope
226:vertices
190:geometry
590:Catalan
550:p q | r
497:rhombus
453:V(3.5)
390:regular
349:uniform
345:Coxeter
322:3.3.3.5
306:3.5.3.5
224:on its
791:
734:19 Dec
579:prisms
568:convex
511:. The
487:, the
474:prisms
439:V(3.4)
296:These
289:Kepler
278:prisms
29:, and
27:prisms
685:246 A
367:duals
324:is a
212:with
789:ISBN
517:cube
491:and
476:and
392:and
365:The
251:The
196:(or
554:not
188:In
857::
819:.
757:.
676:,
630:,
562:,
332:.
291:).
248:.
25:,
825:.
797:.
763:.
280:.
271:.
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