 | Uniform Polytopes (inactive) | |||
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Definition
editThe definition given here, requiring that a chiral figure must have a certain symmetry and that no finite polyhedron (3-polytope) can be chiral, flies in the face of many published works. Even mathworld, referenced in support of this definition, is clear that polyhedra such as the snub cube are said to be chiral, e.g. http://mathworld.wolfram.com/Chiral.html and http://demonstrations.wolfram.com/ChiralPolyhedra/
So - is there some specialist area of abstract mathematics that has its own incompatible definition of the term "chiral", or is the current content incorrect? Either way, it is not telling the full story.
— Cheers, Steelpillow (Talk) 12:28, 1 September 2012 (UTC)
- I took a stab at what I think is going on. Feel free to revert/correct as necessary. — Cheers, Steelpillow (Talk) 13:01, 1 September 2012 (UTC)
- I disagree — I think the strict definition can apply in all cases, and is not particularly abstract (it applies perfectly well to convex polytopes in geometric spaces, not just abstract ones, it merely happens to be the case that there aren't any with this weird kind of symmetry until you get to higher dimensions). I will try writing to make this more clear and to reflect the fact that the term has been applied less strictly in some sources. —David Eppstein (talk) 15:53, 1 September 2012 (UTC)
- While I can sympathise with this as a point of view, I find the emphasis unusual. In forty years of reading about polyhedra I have found the definition, requiring only a lack of mirror symmetry, to be universal and have never before come across the idea that for example the snub cube is not chiral (though Cromwell spells it "cheiral", and some authors such as Coxetere tend to prefer "enantiomorphic"). Indeed, on Wikipedia itself we find this definition apparent in the pages you link to on chirality (mathematics) and on the snub cube. It is also given the Mathworld page given in the external links. I find it hard to maintain the view that these are merely "some sources" or that their usage is not "strict", or that this article should differ in definition from that given at chirality (mathematics). Rather, the definition you prefer seems to have arisen only in the last few years, as a consequence of abstraction away from concrete space. Yes it may be applied to real convex figures, but the result runs contrary to the usual understanding, e.g. for the snub cube. We do at least agree that the abstract treatment is of significant value and needs to be properly explaine here, but I do think that the more long-standing and widely used definition should be given first. — Cheers, Steelpillow (Talk) 18:30, 1 September 2012 (UTC)
- So your position is that e.g. every irregular polytope (having a trivial symmetry group) is also chiral? —David Eppstein (talk) 18:32, 1 September 2012 (UTC)
- Yes, in exactly the same way that chirality (mathematics) gives a shoe as an example of a chiral object. However I am aware that this is not helpful for the study of abstract polytopes. — Cheers, Steelpillow (Talk) 18:35, 1 September 2012 (UTC)
- Ok, but I don't see the point of having an article for that definition of "chiral polytope", any more than I need to have a separate article for "red polytope": we have an article on the color red, and we have an article on poytopes, but there are no special properties of red polytopes that do not follow immediately from the separate properties of redness and polytopes. —David Eppstein (talk) 18:38, 1 September 2012 (UTC)
- Agreed. But given that the article title is inclusive of it, I think it requires a brief mention. Since it is the broader and more widely-used definition, best to get it out of the way first. — Cheers, Steelpillow (Talk) 19:00, 1 September 2012 (UTC)
- Ok, but I don't see the point of having an article for that definition of "chiral polytope", any more than I need to have a separate article for "red polytope": we have an article on the color red, and we have an article on poytopes, but there are no special properties of red polytopes that do not follow immediately from the separate properties of redness and polytopes. —David Eppstein (talk) 18:38, 1 September 2012 (UTC)
- Yes, in exactly the same way that chirality (mathematics) gives a shoe as an example of a chiral object. However I am aware that this is not helpful for the study of abstract polytopes. — Cheers, Steelpillow (Talk) 18:35, 1 September 2012 (UTC)
- So your position is that e.g. every irregular polytope (having a trivial symmetry group) is also chiral? —David Eppstein (talk) 18:32, 1 September 2012 (UTC)
- While I can sympathise with this as a point of view, I find the emphasis unusual. In forty years of reading about polyhedra I have found the definition, requiring only a lack of mirror symmetry, to be universal and have never before come across the idea that for example the snub cube is not chiral (though Cromwell spells it "cheiral", and some authors such as Coxetere tend to prefer "enantiomorphic"). Indeed, on Wikipedia itself we find this definition apparent in the pages you link to on chirality (mathematics) and on the snub cube. It is also given the Mathworld page given in the external links. I find it hard to maintain the view that these are merely "some sources" or that their usage is not "strict", or that this article should differ in definition from that given at chirality (mathematics). Rather, the definition you prefer seems to have arisen only in the last few years, as a consequence of abstraction away from concrete space. Yes it may be applied to real convex figures, but the result runs contrary to the usual understanding, e.g. for the snub cube. We do at least agree that the abstract treatment is of significant value and needs to be properly explaine here, but I do think that the more long-standing and widely used definition should be given first. — Cheers, Steelpillow (Talk) 18:30, 1 September 2012 (UTC)
- I disagree — I think the strict definition can apply in all cases, and is not particularly abstract (it applies perfectly well to convex polytopes in geometric spaces, not just abstract ones, it merely happens to be the case that there aren't any with this weird kind of symmetry until you get to higher dimensions). I will try writing to make this more clear and to reflect the fact that the term has been applied less strictly in some sources. —David Eppstein (talk) 15:53, 1 September 2012 (UTC)