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Pagal originalų apibrėžimą, '''pustaisyklingis briaunainis''' yra toks [[briaunainis]], kurio sienos yra taisyklingi daugiakampiai, o viršūnės yra tranzityvios (pagal briaunaniui būdingą simetrijos grupę). Pustaisyklingiai briaunainiai pagal tranzityvumą yra paprasčiausi [[tolygusis briaunainis|tolygieji briaunainiai]], nes kiti pasižymi įvairesniu tranzityvumu ([[kvazitaisyklingasis briaunainis|kvazitaisyklingieji]] – be viršūnių, dar yra tranzityvios briaunos, bet sienos netranzityvios; [[taisyklingasis briaunanisbriaunainis|taisyklingieji]] – tranzityvūs visi trys elementai: viršūnės, sienos ir briaunos). Šis apibrėžimas remiasi labiau apibendrintu 1900 metais publikuotu matematiko ''Toroldo Goseto'' (Thorold Gosset) pustaisyklingių [[politopas|politopų]] apibrėžimu<ref>Thorold Gosset ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900</ref>, bei 1973 metų ''Kokseterio'' (H.S.M. Coxeter)<ref>Coxeter, H.S.M., ''Regular polytopes'', 3rd Edn, Dover (1973)</ref> apibrėžimu.
definition, it is a [[polyhedron]] with [[regular polygon|regular]] faces and a [[symmetry group]] which is [[Transitive action|transitive]] on its [[Vertex (geometry)|vertices]], which is more commonly referred to today as a [[uniform polyhedron]] (this follows from [[Thorold Gosset]]'s 1900 definition of the more general semiregular [[polytope]]).<ref>[[Thorold Gosset]] ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', [[Messenger of Mathematics]], Macmillan, 1900</ref><ref>[[Coxeter|Coxeter, H.S.M.]] ''Regular polytopes'', 3rd Edn, Dover (1973)</ref> These polyhedra include:
*The thirteen '''[[Archimedean solid]]s'''.
*An infinite series of convex '''[[Prism (geometry)|prism]]s'''.
*An infinite series of convex '''[[antiprism]]s''' (their semiregular nature was first observed by [[Kepler]]).
Pustaisyklingiais briaunainiais yra laikomi:
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]].
*Trylika '''[[Archimedo kūnas|Archimedo kūnų]]''';
*Begalinė aibė iškilų '''[[Prizmė|prizmių]]''';
*Begalinė aibė iškilų '''[[prizmė|antiprizmių]]''' (pastarųjų pustaisyklingę prigimtį aprašė dar [[Johannes Kepler|Johanas Kepleris]]).
'''Pustaisyklingį briaunainį''' visiškai nusako jo viršūnės konfigūracijos planas, kitaip tariant, į viršūnę sueinančių daugiakampių sąrašas. Pavyzdžiui, užrašymas ''3.5.3.5'' nusako [[dodekaedras#ikosidodekaedras|ikosidodekaedrą]], kurio kiekvienoje viršūnėje pakaitomis sueina po du taisyklingus trikampius ir penkiakampius; o ''3.3.3.5'' reiškia penkiakampę antiprizmę. Šie briaunainiai neretai dar vadinami tiesiog „briaunainiai tranzityviomis viršūnėmis“.
Since [[Thorold Gosset|Gosset]], other authors have used the term '''semiregular''' in different ways in relation to higher dimensional polytopes. [[E. L. Elte]] <ref>{{citation | last = Elte | first = E. L. | title = The Semiregular Polytopes of the Hyperspaces | publisher = University of Groningen | location = Groningen | year = 1912}}</ref> provided a definition which Coxeter found too artificial. Coxeter himself dubbed Gosset's figures '''[[Uniform polyhedron|uniform]]''', with only a quite restricted subset classified as semiregular.<ref>[[Coxeter|Coxeter, H.S.M.]], [[Michael S. Longuet-Higgins|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. ([http://links.jstor.org/sici?sici=0080-4614%2819540513%29246%3A916%3C401%3AUP%3E2.0.CO%3B2-4 JSTOR archive], subscription required).</ref>
Po to, kai ''Gosetas'' paskelbė '''pustaisyklingio''' briaunainio sąvoką ir apibrėžimą, šio termino taikymas nebuvo nuoseklus, ypač skirtingai jis buvo taikomas daugiamačių politopų teorijoje<ref>{{citation | last = Elte | first = E. L. | title = The Semiregular Polytopes of the Hyperspaces | publisher = University of Groningen | location = Groningen | year = 1912}}</ref>. ''Kokseteris'' perėmė ''Goseto'' apibrėžimą, bet jį pritaikė visai [[tolygusis briaunainis|tolygiųjų briaunainių]] klasei, o '''pustaisyklingiams''' liko tik mažiausiai simetriškų briaunainių poaibis, kurio figūroms būdingas tiktai viršūnės tranzityvumas.
Yet others have taken the opposite path, categorising more polyhedra as semiregular. These include:
*Three sets of '''[[star polyhedron|star polyhedra]]''' which meet Gosset's definition, analogous to the three convex sets listed above.
*The '''[[Dual polyhedron|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 solid]]s''', the '''convex [[dipyramid]]s''' and '''antidipyramids or [[trapezohedron|trapezohedra]]''', and their nonconvex analogues.
Nors įvedus apjungiančią [[tolygusis briaunainis|tolygiųjų briaunainių]] klasę ir buvo išspręsta didelė dalis įvairių su briaunainių klasifikavimu susijusių problemų, vis dar iškyla svarstymai, kaip skirstyti briaunainius į klases. Nepaisant visko, šiuo metu plačiausiai pripažįstama [[tolygusis briaunainis|tolygiųjų briaunainių]] klasė, kurią sudaro trys poklasiai: [[taisyklingasis briaunainis|taisyklingieji]] briaunainiai (jei yra tranzityvios viršūnės, sienos ir briaunos), [[kvazitaisyklingasis briaunainis|kvazitaisyklingieji]] (jei yra tranzityvios viršūnės ir briaunos, bet sienos netranzityvios) ir '''pustaisyklingiai''' (jei tranzityvios vien viršūnės, o sienos ir briaunos netranzityvios).
A further source of confusion lies in the way that the [[Archimedean solid]]s are defined, again with different interpretations appearing.
==Nuorodos==
Gosset's definition of semiregular includes figures of higher symmetry, the [[Platonic solid|regular]] and [[quasiregular polyhedron|quasiregular]] polyhedra. Some later authors prefer to say that these are not semiregular, because they are more regular than that - the [[uniform polyhedron|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.
===Išnašos===
In practice even the most eminent authorities can get themselves confused, defining a given set of polyhedra as semiregular and/or [[Archimedean solid|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<ref>Cromwell, P. ''Polyhedra'', Cambridge University Press (1977)</ref> and Cundy & Rollett<ref>Cundy H.M and Rollett, A.P. ''Mathematical models'', 2nd Edn. Oxford University Press (1961)</ref> are all guilty of such slips.
==General remarks==
In many works ''semiregular polyhedron'' is used as a synonym for [[Archimedean solid]].<ref>"Archimedes". (2006). In ''Encyclopædia Britannica''. Retrieved [[19 Dec]] 2006, from [http://www.search.eb.com/eb/article-21480 Encyclopædia Britannica Online] (subscription required).</ref> 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.
Coxeter et al. (1954) use the term ''semiregular polyhedra'' to classify uniform polyhedra with [[Wythoff construction|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 Gosset's definition without comment, thus accepting it by implication.
[[Eric Weisstein]], [[Robert Williams (geometer)|Robert Williams]] and others use the term to mean the [[Convex set|convex]] [[Uniform polyhedron|uniform polyhedra]] excluding the five [[regular polyhedron|regular polyhedra]] – including the Archimedean solids, the uniform [[prism (geometry)|prisms]], and the uniform [[antiprism]]s (overlapping with the cube as a prism and regular octahedron as an antiprism).<ref>{{MathWorld | urlname=SemiregularPolyhedron | title=Semiregular polyhedron}} The definition here does not exclude the case of all faces being congruent, but the [[Platonic solid]]s are not included in the article's enumeration.</ref><ref>{{The Geometrical Foundation of Natural Structure (book)}} (Chapter 3: Polyhedra)</ref>
Peter Cromwell (1997) writes in a footnote to Page 149 that, "in current terminology, 'semiregular polyhedra' refers to the Archimedean and [[Catalan solid|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.
==See also==
* [[Semiregular polytope]]
==References==
<references />
=== Išorinės nuorodos (anglų k.) ===
== External links ==
* {{MathWorld | urlname=[http://mathworld.wolfram.com/SemiregularPolyhedron.html | title=Semiregular polyhedron }}Polyhedron]
* [http://www.georgehart.com/virtual-polyhedra/archimedean-info.html George Hart: Archimedean Semi-regular Polyhedra]
* [http://www.daviddarling.info/encyclopedia/S/semi-regular_polyhedron.html David Darling: semi-regular polyhedron]
* [http://polyhedra.mathmos.net/entry/semiregularpolyhedron.html polyhedra.mathmos.net: Semi-Regular Polyhedron]
* [http://eom.springer.de/s/s084300.htm Encyclopaedia of Mathematics: Semi-regular polyhedra, uniform polyhedra, Archimedean solids]
[[Category:Polyhedra]]
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