Talk:Abstract polytope/Archive 6

Hasse diagrams

Although most recent edits are improving things, a couple of remarks on the Hasse diagram have been added that I feel should be either removed or referenced.

• "So in a very real sense, the Hasse diagram is the polytope." This is contentious to say the least, unless it can be found in the respectable literature. A Hasse diagram is no more an AP than a graph or a physical model is. Also, "real" has a specific meaning in this context and should be avoided elsewhere.

Thanks for compliment and feedback.

It is (I believe) universally accepted throughout modern mathematics that isomorphic objects are effectively one and the same. Hasse diagrams are indeed isomorphic to their corresponding posets, and the simplest formal definition of a Hasse diagram would be as a directed graph, which would also be isomorphic to the poset. The whole concept of AP is precisely the equating of (combinatorially) isomorphic objects. It think it is clear from the context that my use of "real" was only English. SteveWoolf (talk) 19:36, 10 July 2010 (UTC)

The graph of a polytope can never be isomorphic to the polytope - it doesn't have a null element, and even tweaking the graph concept to include one would still leave faces of rank > 1 undefined.

In formal logic, two objects are one and the same if and only if there is no way to tell them apart. Isomorphism is not necessarily identity, nor is equality. These are deep issues and I would rather avoid bringing them to the surface here. I think it better to let the reader impose their own understanding of these issues. The acid test: does ARP, or equally reputable source, make this claim for the Hasse diagram of a poset? I will add a citation tag, and perhaps we shall see. — Cheers, Steelpillow (Talk) 20:35, 11 July 2010 (UTC)

However, in practice, Steve is correct - we treat isomorphic objects as identical, even if they are not. Eg, we say "the" cube, as if there's only one - and up to isomorphism, that's true. We say "2+2=4", as if "2" and "4" were single numbers, rather than an equivalence classes of sets. mike40033 (talk) 05:16, 18 July 2010 (UTC)

• "The astute reader may notice that the Hasse diagram shown is itself the graph of a polyhedron with 8 square faces." This may be true but is confusing to the newcomer, and best left out of the introductory material. If enough on the relationships between graphs and Hasse diagrams can be found in the literature, then a separate section further on might be appropriate.

Yes, I also thought that too, so I'll remove this observation (which I hope you also find interesting!) at least until I am able to prove that all polytope Hasse's are the graph of an (n+1) polytope. In actual fact, I do know how to generate the complete face set of this "Hassotope" - it is the set of all sections of the original, with a new null face added in. An n-simplex gives an (n+1)-hypercube. All polytopes give Hassotopes with square faces because of the diamond property. SteveWoolf (talk) 19:36, 10 July 2010 (UTC)

This idea rings a bell - I'm sure I've seen it somewhere before. However, because of 'no original research', it should probably stay out of the article until someone finds that article and reads it (or writes it!) mike40033 (talk) 05:16, 18 July 2010 (UTC)

— Cheers, Steelpillow (Talk) 10:16, 10 July 2010 (UTC)

Created a Hasse for an irregular polytope as requested, using it to illustrate sections. I chose the triangular prism over the square pyramid, since the former is not self-dual - it's dual is the triangular bipyramid.

PS - My pictures are coming out a bit fuzzy - at least on my screen. Any suggestions? Is .SVG a better format? SteveWoolf (talk) 06:29, 11 July 2010 (UTC)

Image rendering has got a new engine, which does not do anti-aliasing well - svg is suffering too as for the time being that gets converted to png before serving. Your file looks fine on the commons, which presumably still uses the old engine. Technically svg is the preferred format, but fonts can cause trouble if the server does not recognise them. I wouldn't lose sleep over it. — Cheers, Steelpillow (Talk) 14:47, 11 July 2010 (UTC)

Thanks Guy SteveWoolf (talk) 17:13, 11 July 2010 (UTC)

Could Steelpillow first state his position on my statement that

So in a very real sense, the Hasse diagram is the polytope.

Does he still disagree with the statement, and/or does he genuinely believe it needs referencing?

I took time out from article editing (again) to try to explain the concepts involved (see above). But instead of doing me the courtesy of waiting while we continue the discussion about the mathematics, or waiting for an opinion from Mike or others who actually have knowledge in this field, I now find myself wondering whether I will have to reference every line in the article. At least a discussion about the need for references would have been civil.

Mike has already undertaken to oversee article development, and I doubt if Mike had written the statement in question that Steelpillow would have behaved thus. In my view this is disruptive editing by someone quite unqualified in this field - abstract polytope theory is NOT traditional polytope theory any more than the foundations of mathematics are primary school arithmetic.

Meanwhile, another potentially productive day of article improvement has been wasted. SteveWoolf (talk) 15:38, 12 July 2010 (UTC)

When challenging another user's edits, "Any editor has the right to challenge unsourced material by opening a discussion on the talk page or by tagging it." I will not waste words arguing my case further, unless the exchange can be carried out with some decorum. — Cheers, Steelpillow (Talk) 19:32, 13 July 2010 (UTC)

my 2.5c (inflation strikes even here)

I've reworded the contentious phrase. I don't believe citation is needed in the text as I've left it.

NB - the Hasse diagram is not isomorphic to the polytope. However, Hasse diagrams and polytopesposets are in one to one correspondence, so that isomorphic polytopes/posets give rise to isomorphic Hasse diagrams. If you wanted to, you could define a relation 'isomorphic' (§) on the set of Hasse diagrams that represent abstract polytopes (H), and another relation 'isomorphic' (°) on the set of abstract polytopes (P). Then, there is an invertible map from the set of polytopes to the set of Hasse diagrams, and this map preserves isomorphism. Therefore, (H,§) and (P,°) are isomorphic. However, you can't say that an individual polytopes is isomorphic to its Hasse diagram.

Please, guys, abstract polytopes are just an esoteric branch of mathematics. Please, no flame wars, no edit wars! mike40033 (talk) 00:59, 14 July 2010 (UTC)

Thanks Mike. — Cheers, Steelpillow (Talk) 21:26, 15 July 2010 (UTC)

A = B and A ≠ B ?

I don't get it - How can you say that "(H,§) and (P,°) are isomorphic" and then say "an individual polytope is not isomorphic to its Hasse diagram" ?

because it's true. mike40033 (talk) 03:41, 18 July 2010 (UTC)

So they are isomorphic AND they're not?

"(H,§) and (P,°) are isomorphic" means "The collection of all polytopal hasse diagrams, equipped with the equivalence relation § (being ihasse diagram isomorphism), is isomorphic to the collection of all abstract polytopes, equipped with the equivalence relation ° (being abstract polytope isomorphism)". This is very different from saying an individual hasse diagram is isomorphic to an individual abstract polytope. mike40033 (talk) 05:21, 18 July 2010 (UTC)

You don't have to define "§" - the Hasse diagram requires and already has it. As a drawing on a piece of paper, the relation is simply the pairs (F, G) where F is above G and joined to it. A Hasse diagram is a set of nodes with this relation - i.e. a poset.

No. A Hasse diagram is a picture. More formally, it's a directed graph. The relation 'there is an edge from A to B in the Hasse diagram' is not a partial order, since edges from A to B and from B to C are not accompanied by an edge from A to C - that would make the Hasse diagram too messy mike40033 (talk) 03:41, 18 July 2010 (UTC)

There is a simple 1-1 correspondence between (finite-rank) posets defined by means of any of the 3 relations <, ≤, or <: where "<:" means "immediately precedes", i.e. < with no in-between elements. In any case, the Hasse diagram does have the "<" relation: a < b if you can go from a < b in a series of upward edges.

Consider a set of 4 chickens {Lee, Art, Bert, Gretel} with the relation "pecks", where Gretel pecks everyone else, Art and Bert peck only Lee, and Lee is only pecked. According to our polytope definition, this IS the polytope of a line segment, to give an easy example.

Yes, but in the Hasse diagram, there's no edge drawn from Lee to Gretel. Please note that I'm being very pedantic here - it is technically not correct to say that the Hasse diagram is isomorphic to the polytope. However, there is an invertible function mapping polytopes to Hasse diagrams that preserves equivalence classes under polytope or Hasse diagram isomorphism. The Hasse diagram is a way to represent a polytope - but that doesn't make them isomorphic as algebraic structures. mike40033 (talk) 03:41, 18 July 2010 (UTC)

"Yes, but in the Hasse diagram, there's no edge drawn from Lee to Gretel". I gave my chicken set as an example of a polytope, not of a Hasse diagram. It is a set with a relation <. Nothing in the formal definition requires the drawing of any edges - only that the diamond property is satisfied. It is.

"it is technically not correct to say that the Hasse diagram is isomorphic to the polytope". In that case, what is your understanding if "isomorphic"? For I can most certainly give you a bijection from a polytope to its Hasse.

how so? You can construct the polytope from the Hasse diagram, yes, and vice-versa. So you can construct a bijection from P to H, and this bijection preserves equivalence classes of isomorphic objects, So (H,§) and (P,°) are isomorphic. However, you can't construct a structure-preserving bijection from the hasse diagram to the polytope. Try it on the digon, and you'll see. The polytope has Gretel related to Lee, the hasse diagram has no such edge.

A Hasse diagram is (usually and most simply) defined mathematically as a digraph, which has to be acyclic. And a graph is (usually and most simply) defined as a point set with an adjacency relation, which defines the "edges" (Not to be confused with flag adjacency). Acyclic digraphs are partial orders (the relation being given by the "direction", so our graph is a poset.

no, an acyclic digraph defines a poset in a unique way. That doesn't mean it is a poset. You might as well say I am the text "mike40033". Please be assured, I am not a string of 9 characters! mike40033 (talk) 03:41, 18 July 2010 (UTC)

A poset is a set with an order. A directed graph is a set of points, with a relation < where a < b if there is a directed edge from a to b (you can define it the opposite way round if you prefer - it's the same math). So a directed graph IS a poset.

a directed graph defines a relation on the set of all vertices : V1 ~ V2 if there is an edge from V1 to V2. This relation may or may not be a partial order. In the case of a Hasse diagram, the relation defined in this way from the Hasse diagram is not a poset. However, because it is an acyclic digraph, you could define a poset from it by saying V1 < V2 if there is a path from V1 to V2". this poset S=(V,<) would be isomorphic to the polytope, but that's not the same as saying that the original digraph is isomorphic to the polytope. mike40033 (talk) 05:34, 18 July 2010 (UTC)

I am a bit surprised by your 'I am the text "mike40033"' joke. No-one believes that the London Underground (the railway) is the same as its name. But everyone knows that the map of lines is isomorphic to the railway itself.

By definition an abstract polytope is a poset. No amount of 1984 truspeak is ever going to let you escape from one very simple mathematical fact: that anything isomorphic to a poset is a poset. In geometry, a square is not an irregular quad. In AP, they are the same polytope - or at least, two isomorphic pols, to be formal. And it is usual in AP theory to consider isomorphic pols as "essentially" the same polytope.

I was always led to believe that "isomorphic" meant "there exists a isomorphism (a structure-preserving bijection)". Now it may be 'obvious' that the map of the London Underground is isomorphic to the actual train system, but I don't see it. Where is the structure-preserving bijection? I always taught my students "if something is obvious, prove it". Rather, I'd say that the map captures certain important information about the real thing, but that doesn't mean they are isomorphic. mike40033 (talk) 05:34, 18 July 2010 (UTC)

I am no AP expert, but I am very qualified in the Foundations of Maths (no intention to brag - not my style) - so I hope you will do me the respect of discussing this - and also with honesty and civility. Regards Steve SteveWoolf (talk) 05:09, 18 July 2010 (UTC)

We have

• A Hasse Diagram is a digraph
• A digraph is a poset
• A poset is a polytope (if the polytope axioms are satisfied)

and in mathematics, A=B and B=C, and C=D implies A=D.

There are many issues here:

1) Is it true that a Hasse diagram IS a polytope?

2) If it is true, does it need a citation?

3) Can a mathematical proof be invalidated by a democratic vote?

4) Is it good policy to suppress a mathematical statement, even if true, to make people happy?

1) No. 2) Not necessarily. 3) No. 4) Sometimes. After all, is it not good policy to always express all possible true mathematical statements, irrespective of people's happiness. mike40033 (talk) 03:41, 18 July 2010 (UTC)

To claim (2) only because you don't believe (1) is intellectual dishonesty. I challenge each of you to take a stand on each of the above 4 issues - and to discuss each with sincerity. SteveWoolf (talk) 06:19, 15 July 2010 (UTC)

An editor needs to feel the environment is conducive to be willing to work hard with a good feeling. My requirements are simply that (a) We do sound mathematics and (b) We reach consensus and do not act unilaterally.

If, however, we are going to deny that the Earth goes round the sun, or that man evolved from apes, because it offends the church, rather than for scientific reasons, then I have no place here. I am not demanding instant agreement with my statements, only that they be discussed in a logical and respectful manner.

If, on the other hand, we are going to use Wikipedia technicalities to suppress statements we don't agree with, or remove statements without even attempting to achieve (mathematical) consensus, then someone else will have to take over my work.

Saccheri tried to prove Euclid's parallel postulate by assuming its opposite and finding a contradiction. Unable to find one, he nevertheless concluded that the theorems he thus proved were "repugnant to the nature of straight lines". Today, his results are theorems of hyperbolic geometry.

In Abstract Polytope theory, we have made a radical new definition of a polytope. As mathematicians, we cannot pick and choose its consequences according to personal taste. (We can however, question the worth of the axioms themselves - which I have often. But that is not what is going on here). SteveWoolf (talk) 06:56, 14 July 2010 (UTC)

However, in the context of editing a wikipedia article on Abstract Polytopes, we cannot even question the worth of the axioms. The axioms are as they are, and their worth is well established according to wikipedia's notability guidelines. Therefore there is no question that they are worth talking about in this article. People who prefer a different set of axioms need to move to a different wikipedia article, or just swallow the axioms as given.

Note also that although we can't pick and choose the consequences of the axioms according to personal taste, for the purposes of wikipedia, we need to pick and choose them according to wikipedia's guidelines - notability and 'no original research'. That is, anything that goes into the article ought to be published already elsewhere, or 'well known' amongst practioners in the field. Not everything needs a citation, though. mike40033 (talk) 04:46, 18 July 2010 (UTC)

Yes, accepted. But as a mathematician, I can still wonder why standard trad pol properties such as the 2 "nice" properties were "dropped". A different term could have been used - as was done with pre-polytopes - and the polytope term kept more consistent with trad concepts. Still, I have faithfully stuck to the status-quo axioms - only tried to make them as lucid and independent possible.

My hope is also that as mathematicians, we will always be interested in ideas primarily. After all, today's "original research" is more-than-likely already discovered anyway, and only awaits someone to reference it. For example, I am sure that the Cartesian product of two pols is a pol - and I have already worked out how to generate the face-set of the product and its < relation. But I couldn't find any references, so I shelved it. And many, many other ideas too. SteveWoolf (talk) 06:07, 18 July 2010 (UTC)

When you ask "why were they dropped", I guess you mean "why were they dropped and not retained". The truth of the matter is that they were both dropped and retained. If you drop them, you get abstract polytopes. If you retain them, you get something else. I've seen the assumptions of atomicity and latticehood of a poset applied from time to time, but since I was more interested in abstract polytopes, I didn't pay much attention. I'm sorry, I can't even recall the name given to these structures, let alone give a reference. However, my point is that the "nice" axioms were retained, and were dropped, by different groups of mathematicians at different times (that's as it should be) and as the concepts were developed they were given different collections of terminology and nobody read the other guys' articles (that's less ideal by far).

you should write up some of your ideas, and make sure the terminology you use fits in with some "standard", eg, the standard for abstract polytopes. Even if it turns out that it's all been done before, you shouldn't turn down the chance that there might be something new - or that as you write, your ideas evolve into something new... mike40033 (talk) 06:59, 19 July 2010 (UTC)

My 2c

Something to note about Abstract Polytopes :

Abstract polytopes is a very very very specialized field of pure mathematics. It is a field of combinatorics inspired by geometry. It is important to keep in mind that there are numerous other topics that are very very specialized fields combinatorics inspired by geometry. To name a few, there are lattice polytopes, eulerian posets, buildings, coset geometries, and more.

Each of these have their own set of axioms that define what they are. Unfortunately, they also tend to have their own set of terminology. Sometimes the terminology contradicts, for example, when an abstract polytopician says 'chain', he or she means something different from what a building theorist means.

Therefore in arguments about terminology, it is very, very, very important to refer back to the published sources on Abstract Polytopes. It's no use presenting options and voting, or even arguing for what we like "best". For any concept that should go into the article, there is a "correct" term and definition - and this correctness is a matter of historical fact, as recorded in the peer-reviewed publications on Abstract Polytopes. Any discussion and voting on terminology takes place by private communication between researchers, or by fiat publication of a journal article, long before the concept is a candidate for appearance on wikipedia.

It's likewise not much use (for the purposes of improving this article) appealing to these other fields of combinatorics, claiming that certain elements seem "better" or gaining inspiration from them, if the inspiration we gain contradicts the axioms and accepted terminology of abstract polytopes.

I promise to monitor the article over the next few months, and correct anything that is wrong. Will this promise of mine be enough to solve the arguments and discussions?

And in the meantime, please let's all remember WP:FAITH

mike40033 (talk) 12:25, 23 June 2010 (UTC)

Thanks Mike, Gabe and Tom for your constructive response.

As a gesture of good faith on my side, I retract my accusation of maliciousness, and shall assume Guy is genuinely interested in bringing the article forwards.

I shall implement all article recommendations, including re the Hasse Diagram section, shortly.

I entirely accept the suggestion that Mike, and perhaps also Gabe, being the most qualified in this field, will oversee development. I hope this is agreeable with Guy. I remain willing to consider suggestions from all parties, discuss them as neutrally as possible, though subject to reasonable time limitations, and to explain any concepts that are unclear. SteveWoolf (talk) 05:59, 24 June 2010 (UTC)

For what it's worth..... so far, all the edits I've seen on the article are fine, and are indeed improving the article... mike40033 (talk) 01:02, 6 July 2010 (UTC)

Thanks for that SteveWoolf (talk) 07:46, 6 July 2010 (UTC)

Intentional harassment

Mike, Gabe and others:

My patience is exhausted now. I have tried reasoning with Steelpillow ad nauseam, and frequently taken time out to explain some of the mathematics to him in order to build consensus, which you'd hope he might appreciate - but all to no avail; he ignores it and acts unilaterally.

I "went along" with your requests to give him the benefit of the doubt, but will do so no more. Are we so naive that we cannot admit that there people who vandalise, but who do it such a way as to conceal their real motivation, by pretending to be genuinely interested in improving Wikipedia when in fact they only wish to raise their own importance? Please read Wikipedia:Disruptive editing.

Steelpillow does not have the mathematical understanding necessary to contribute to this article, nor to judge the contributions of others.

I am not prepared to give my time and soul to a project to have it harassed and sabotaged in this manner. So either you people will have to find a way to discourage him, or the progress that everyone says they like will be paralysed until someone else qualified finds the time.

Thankyou all for your support and kind words, but there are times when justice needs more than that.

SteveWoolf (talk) 21:20, 13 July 2010 (UTC)

Notation

There has been an ongoing discussion as to whether this article should use a vertex-combinatorial notation for j-faces such as abc for a triangle, or a neutral notation such as F. Much is scattered through the archives so I will try to summarise, even expand a little. abc was, and remains to some extent, popular among combinatorialists. McMullen & Schulte have preferred F.

It is common to treat an abstract polytope (AP) as a vertex-combinatorial structure, in which case the abc notation seems apt. However this is just one interpretation, or application, of an AP. There is no a priori need for vertex combinations, indeed they raise difficulties in certain situations such as tilings of the projective plane, digons and so forth. Johnson [1] points out a more fundamental danger of treating a face as a vertex set - it can become indistinguishable from the associated polytope: is {φ,a,b,c,ab,bc,ca,abc} just the face element abc written in longhand, or is it the whole set of elements incident below that face (what Johnson calls the "facial")? As AP theory evolved, it became clear that treating F as an atomic element in its own right provides the most consistent, and most abstracted, approach. It has been remarked in our discussions that AP theory has shed the last vestiges of its geometrical origins - I would suggest that vertex sets are just such a vestige that needed to be shed. Even Grünbaum, long-time and continuing champion of vertex sets, now resorts to the F notation when formalising abstract polyhedra [2].

Whatever the merits of the two approaches, agreement was reached that we should be guided by McMullen and Schulte, and specifically by their book Abstract regular polytopes. Twice over the years (as I recall — Cheers, Steelpillow (Talk) 14:51, 17 July 2010 (UTC)), further agreement has been painfully thrashed out that this means using their F notation. Recent edits have made many worthy improvements to the article but sadly some, including a new Hasse diagram of the triangular prism, have ignored this agreement. I propose that we correct this trend before more damage is done, and rework the necessary content accordingly. The vertex-oriented approach remains important and deserves a well-thought-out section of its own. Given the turbulent history, I would earnestly ask the more serious editors to seek a clear consensus before proceeding.

— Cheers, Steelpillow (Talk) 10:30, 17 July 2010 (UTC)

Since no-one including you has the respect and courage to answer the 4 questions I presented as a challenge to intellectual honesty, my interest in this site has been killed forever. I have no doubt you will do whatever you want regardless of any pretense at consensus. Why don't you just replace the whole article with your vastly impressive website articles.

To be consistent with recent "decisions" you will need a 5th axiom: "The poset consisting of the nodes of a Hasse diagram (of a polytope), together with its relation "is above and joined to", is not a polytope".

If anyone wants to do any serious work I'll be happy to correspond outside Wikipedia.

SteveWoolf (talk) 16:43, 17 July 2010 (UTC)

(For context, I was asked to weigh in by Guy (Steelpillow) here here; archive, in a neutral message. As Guy states it, this is a difference over whether to use vertex set notation, particularly between Guy and Steve; see Hasse diagrams, My 2c, and Intentional harassment (above), and User talk:SteveWoolf: (section) for reference.)

I am not an expert in AP, but have extensive experience in math and Wikipedia. Regarding notation, following the literature is simply the rule. Polytopes have varying notation over the centuries, but McMullen & Schulte, esp. ARP, seems the current authoritative reference, and hence (as Guy suggests), should be followed – i.e., use F instead of abc.

Turning to exposition – using a single letter for atomic objects (sets, etc.) is ubiquitous in mathematics, and hence seems clearest for novices, which is the intended audience of Wikipedia. Also as mentioned, identifying polytopes and faces with vertex sets is fraught – the digon with vertices a and b has two faces (both ab), which are not identical with the whole polytope. There is certainly some use of referring to faces of “nice” polytopes by vertex sets, but in general, using vertices to refer to polytopes and their faces is limiting and confusing, and the point of APs is exactly this generality. (Compare with finite topological spaces: the point is the added structure.)

So, with the caveat that there may be some good use of using vertex sets in naming of nice polytopes, I’d agree that atomic names should be used in referring to APs, both following literature and for clear exposition.

The current revision of the page, AFAICT, initially uses vertex sets (to refer to faces), then generalizes to more abstract cases where this isn’t possible; currently the section “The digon” explains this issue pretty well. Is the question whether vertex sets should not be used at all, or whether they should only be used in the intro, and then dispensed with once generalized to polytopes where they’re not helpful?

My feeling is that abstract notation (E, F) should be preferred – and used to name the polytopes themselves – though some limited use in naming faces (of “nice” polytopes) seems ok. Hope this helps! —Nils von Barth (nbarth) (talk) 19:15, 17 July 2010 (UTC)

I'm not sure that I understand what this debate is about. Clearly, vertex notation is no good when discussinf general abstract polytopes. Equally clearly, it's fine when discussing particular cases. I have no objection to naming, eg, the faces of a prism or pyramid after their vertex sets. After all, the relation 'subset' on the sets of vertex sets of faces of a polytope is an abstract polytope. mike40033 (talk) 03:29, 18 July 2010 (UTC)

Thanks to those who have responded so far. This note is an attempt to clarify a few things in reply. The current issue is as much about establishing a clear consensus to try and end a "virtual edit war" as about what that consensus actually is. For example, currently the vertex notation appears at various places (but not in others) in the introductory discussions and associated Hasse diagrams, and in the first incidence matrix. Once we have a clear policy, we can judge which (if any) of these is inappropriate. My proposal is that "atomic" notation should be the default, and especially so throughout the introductory section, while the vertex notation should be used only where the atomic notation is unsatisfactory for some specific reason, such as in the (proposed) section on vertex sets or when specifically addressing mike40033's observation that "The relation 'subset' on the sets of vertex sets of faces of a polytope is an abstract polytope".

It occurs to me that vertex notations comes in two flavors: abc for algebraic combinatorialists, and ABC for elementary geometers. If people feel that elementary geometric notation is useful at some given point in the introductory material, then I would at least hope for capital letters and some brief rationale as to why "vestiges of traditional geometry" are retained. I could live with that.

— Cheers, Steelpillow (Talk) 09:19, 18 July 2010 (UTC)

We can't use vertex notation for non-atomistic polytopes. Okay fine. But for atomistic polytopes (such as the square pyramid or triangular prism) - there is one very clear advantage in v-notation - you can see immediately what is incident with what. This makes it much easier to work with. So what is wrong with using v-notation when we can, and face names when we can't?

We have previously agreed to use ARP as our touchstone. I have some papers by M&S that do not use vertex notation at all. Does ARP use is where appropriate? If it does not, are we now reconsidering that agreement? I am not implying that we should not, just trying to clarify whether we are. — Cheers, Steelpillow (Talk) 19:53, 22 July 2010 (UTC)

For the likes of many of us - including myself - ARP is a very difficult book to get through. It is just not aimed at those without advanced math skills. That's not a criticism of it - I wouldn't expect to understand Shakespeare in Chinese either. Wikipedia, on the other hand, is intended to be accessible to common folk. If we are going to copy ARP, there is little point in having an article. So I would say let us write the best possible article without flagrantly violating Wikipedia rules.

If you accept that, then we should discuss the (mathematical) merit of topics, including this one, at least as much as Wiki policy. With that in mind, is the article really improved for having the square example I provided in F notation, which I did reluctantly?

So - given Mike's latest remarks - can we now use vertex notation where possible, and face names where necessary? If we can agree on that, I would be willing to use as many "nasty" examples of polytopes as is consistent with a good article, i.e. not always use "nice" examples - despite my personal view that "nice" polytopes are more interesting. I can, for example, modify the hemicube picture - although it doesn't use vertex notation anyway.

There is quite a good reconciliation of the problem: the two edges of, for example, the digon ab could be labelled ab1 and ab2. I'd vote for that - would others care to comment? SteveWoolf (talk) 05:59, 23 July 2010 (UTC)

Mike's recent remarks include, "There are pedagogical (sp?) reasons not to" and "I think the standard notation is capitals (often F and G) for particular elements of the poset". Nils has also said that "My feeling is that abstract notation (E, F) should be preferred". I see no justification here for using vertex notation wherever possible, indeed I see a consensus for the reverse - using the F notation unless there is a good reason why not. I do happen to think that the square example is improved by using the F notation. I also think that the situation is if anything made more confusing by using ab1 and ab2 for the digon - I do not think that say {&phi, A, B, c} (where c is the line AB) is beyond the capable novice: indeed, I think it would help to get across the way these structures work. We already have lines in the Hasse diagram showing the incidences, there is in general no need to labour the point with the vertex notation. So as I read it, we have a 3-to-1 consensus that we should use face names where possible, and the vertex notation where necessary. Are we in a position to agree this? — Cheers, Steelpillow (Talk) 18:43, 25 July 2010 (UTC)

Sorry, for the digon I should of course have written something more like "{φ,A,B,F,G,M} where F and G are two distinct lines AB." It had been a long day. — Cheers, Steelpillow (Talk) 07:49, 26 July 2010 (UTC)

Although every face has a "corresponding" section of which it is the greatest face (tho' not all sections are faces), we should be a little careful. If P is the polytope, which is the set of all faces, it is not also the greatest face. True, M/S in ARP state we can informally identify faces with "their" sections, but we should remain aware of the difference, and use different symbols for the polytope (eg P or Pi) and the greatest face (eg G or M). SteveWoolf (talk) 15:17, 26 July 2010 (UTC)

Makes sense. Have amended my digon example above. — Cheers, Steelpillow (Talk) 21:03, 26 July 2010 (UTC)

It seems clear to me, from the context, that Mike was saying that capitals were standard for naming faces as opposed to small case, curly or Greek letters, and not as opposed to vertex notation.

When I asked

So can anyone give a mathematical reason why we should not use v-notation where possible that would outweigh it's clear advantage?

Mike's reply was

not me. As far as I can see there isn't one

- which seems to me to come down decisively in favour of vertex notation where possible. That alters the vote to 2 major contributors vs 2 occasional editors.

No! If you'd asked Can anyone give a mathematical reason why we should use v-notation where possible I would also have answered the same way. Although I, personally, prefer {ø, a, ab, abc} over {ø, a, E, G}, this is not for mathematical reasons. It's just easier (I think) to understand the former. However, if the former is used in this article, we must be careful to avoid giving the (false) impression that faces "are" or "contain" sets of vertices/subfaces. They don't (except by accident, for particular instances of certain polytopes). In particular, I'd strongly recommend not using ab1 and ab2 for the edges of the digon. mike40033 (talk) 07:19, 26 July 2010 (UTC)

Since Guy and I are unable to agree on this matter, which is having the effect of eroding a constructive environment for article progress, a clear statement by you (Mike) on which approach leads to the best article would resolve the matter, assuming Guy and I are both willing to accept the verdict!

Provided no other serious editor objects (and I don't expect anyone to), I will go along. However I have some sympathy for Mike if he chooses not to play God as definitively as one might ask. — Cheers, Steelpillow (Talk) 21:03, 26 July 2010 (UTC)

Thanks for that. I hope you are satisfied that the article makes the issues re vertex/face notation more than clear. And, yes, I do think Mike will be much happier if we can resolve things by intelligent discussion and not have to promote him from sage to court judge.SteveWoolf (talk) 07:30, 27 July 2010 (UTC)

Although I will accept your recommendation not to use "ab1" notation, I am curious as to why you are so averse to it. SteveWoolf (talk) 15:17, 26 July 2010 (UTC)

It just seems messy to me... :-) Being accused of being an aristocratic sage (the herb, I suppose, not the wise man) court judge, I just want to expose my irrational innate humanness for the world to see mike40033 (talk) 06:58, 28 July 2010 (UTC)

Ok, seriously folks... it's just that eventually you'll have to drop vertex notation - the hemicube, after all, has 3 faces with the same vertex set - and there are flat polytopes with more faces than that. After a while, appending numbers to the lists of vertices just doesn't seem so great. Do you really want to call the faces of the flat {4,infinity} with four vertices m, i, k and e mike1, mike2, .... mike40033, .... mike2883994724417895995.... ? Might as well call them F1, F2, ... mike40033 (talk) 07:04, 28 July 2010 (UTC)

(The following written purely humorously!) Yes, F2883994724417895995 is an order of magnitude shorter than mike2883994724417895995. Actually, I suppose F1,..., Fn would use less characters on average, tho' the maximum number required as a suffix will be greater. I am not so sure that there will be many examples of polytopes with 2883994724417895995 faces in the article, tho' with article growth and expanded Wikipedia article size limits made possible by advancing technology, there might be as many as 2883994724417895994 in a worst-case scenario.

Anyway, we had agreed (give or take the odd growl) on vertex notation where possible, so I'm happy to live without "vvvn" notation (eg "ab1"). Still, Guy hasn't commented on this latter "option" - so I'd be interested to know his opinion on it - i.e. its merits, regardless of actual practice. Not that I'm pushing it! SteveWoolf (talk) 17:35, 28 July 2010 (UTC)

Well, I suppose I could settle for binary - F_{001110101001001011011011010101011111101011101101001111} has a certain poetry, does it not? — Cheers, Steelpillow (Talk) 19:39, 28 July 2010 (UTC)

I find binary rather too concise. In Foundations of Mathematics, we define 0 is a number, then 1 is the successor of 0, written eg 0'. 2 is thus 0". There are of course axioms, eg equal numbers have equal successors.

So 16 would be 0"""""""'. I feel that this notation would improve the article, and that binary or decimal numbers assume too much prior cultural background. Also, we should translate the text into this notation, so the alphabet could be A, A', A" and so on. To avoid a rather dry text, we could add images, as a bitmap list naturally, with of course each pixel designated by its colour being a number, say, 0 to 2²⁴-1 (24 bit). SteveWoolf (talk) 23:26, 28 July 2010 (UTC)

That's if you use the Peano axioms. I believe Zermelo Fraenkel set theory is more widely accepted foundation for mathematics (see http://en.wikipedia.org/wiki/Natural_number#A_standard_construction) in which case you get F_{}, F_{{}}, F_{{},{{}}}, F_{{{},{{}}, {{},{{}}}}}, F_{{{},{{}}, {{},{{}}}, {{},{{}}, {{},{{}}}}}}, etc. In the interests of accuracy, I propose we change all numbers referred to in the article with their Zermelo-Fraenkel equivalents, and replace ambiguous terms like "cube" with much clearer terms like "the universal { {{},{{}}, {{},{{}}}, {{},{{}}, {{},{{}}}}} , {{},{{}}, {{},{{}}}} }". mike{{},{{}}, {{},{{}}}, {{},{{}}, {{},{{}}}}}{}{}{{},{{}}, {{},{{}}}}{{},{{}}, {{},{{}}}} (talk) 04:28, 29 July 2010 (UTC)

While democracy and consensus are important, I think that deferring to established professionals such as Mike should override. The fact that other professionals may use face names does not mean they insist on it, especially in Wikipedia.

Given that I have done most of the work in expanding and clarifying this article (and Mike the rest), and that Mike has already undertaken to oversee it, can you not do the gracious thing and accept the compromise already offered, so I can get on with further article development?

SteveWoolf (talk) 04:56, 26 July 2010 (UTC)

Given that I have held back at Steve's request, I find that illogical. The history shows that I started active editing a month before Steve, and I also should like to return to that. — Cheers, Steelpillow (Talk) 21:03, 26 July 2010 (UTC)

Okay, but.... A good mathematics article should not be an essay. It should be as free as possible from undefined terms, unless their meaning is quite clear and unambiguous. I have, for example, always adopted the convention of putting undefined "conversational" terms in quotes, as you will see many times in the article. And informal prose should be present only when it's meaning is clear and it supports the concepts under discussion.

Boldness has its place, but, particularly in mathematics, we need to be pretty sure what we are talking about before we put it in an article. And that it also adds something worthwhile. Often less is more. One clear crisp statement (or even none at all) is better than a paragraph of vagueness and repetition; eg much better than

The domesticated feline unilaterally decided to adopt a position of repose upon the woven textile floor-covering

is

The cat sat on the mat

I shall now take my leave of thee, sir(s), and bid you good health and cheer until our next meeting. SteveWoolf (talk) 07:30, 27 July 2010 (UTC)

The article already makes it abundantly clear when we can and when we can't (use v-notation) - tho' if anyone thinks it's not clear enough, I'd be happy to try making it clearer still.

Much of the current article would be more difficult to read without using vertex notation, such as the statement

{ø, a, ab, abc} is a flag in the triangle abc.

which appears in the Flags section. To replace this with

{ø, a, E, G} is a flag in the triangle P

forces you to tediously define all the < relations of P, and the reader to check them - more than enough to discourage many from bothering, and therefore from understanding.

We would transform a lucid, accessible article into a dry maths textbook; the article should be a bridge to more technical works, not a copy of them.

So can anyone give a mathematical reason why we should not use v-notation where possible that would outweigh it's clear advantage?

not me. As far as I can see there isn't one. There are pedagogical (sp?) reasons not to - we don't want to give people the impression that it can always be used. On the other hand, to never use it has its own pedagogical disadvantages - it makes polytopes which could use it somewhat more confusing. mike40033 (talk) 08:19, 22 July 2010 (UTC)

Then, if it really is true that all the aristocracy only use face names - and I don't know either way - let us at least be honest enough to make a fair decision on the mathematical merits or otherwise of v-notation. Ditto all other topics.

The logical consequence of Mike's remarks is that pedagogical criteria are the best we have. It seems to me that we are all in agreement that each notations is appropriate in certain situations (Steve's flag example is a case in point), but that there is no clear consensus yet on the default treatment where neither has an obvious difficulty - and these are the areas where issues continue to arise. — Cheers, Steelpillow (Talk) 19:53, 22 July 2010 (UTC)

yes, notation is about understanding (ie, pedagogy), not about the mathematics itself. As far as what practicing abstract polytopists use in their research, well, we usually jump straight into C-groups and their subgroups. It's easier figure stuff out that way. Actual faces etc will almost always be in very introductory texts (ie, section 1 of an article, or the first chapter or two of M&S's ARP)mike40033 (talk) 07:24, 26 July 2010 (UTC)

I don't think it's so critical whether we use upper or lower case for vertex letters. I use lower case to show their lower "level" to other faces F, G.... Similarly I prefer Greek capitals (eg capital Pi) for higher entities such as the Polytope itself, tho' out of laziness I have used P. Other authors use curly letters, but these are hard to read.

I think the standard notation is capitals (often F and G) for particular elements of the poset, curly letters (usually P, or Q for a quotient, or K and L for facets and vertex figures) for polytopes, capital Gamma (or sometimes W) for the associated C-group, capital N for a semisparse subgroup. mike40033 (talk) 08:19, 22 July 2010 (UTC)

On the human relations front, I will just say that when Guy seeks consensus before making edits that are likely to upset, and desists from using words such as "sadly" and "damage" in reference to my work, then a more pleasant and constructive - and productive - team will emerge. SteveWoolf (talk) 21:40, 20 July 2010 (UTC)

My mistake

Yes I was wrong! The Hasse with its "edges" representing the relation is NOT isomorphic to the polytope it represents. And I appreciate Mike's answers to my questions - and patience with my errors.

But it is quite subtle. In fact, just as a topology can be defined either as a set of open sets OR as a set of closed sets (with slightly different axioms), so to we could define a partial order not in terms of the normal < (Less Than) relation, but also by means of <: (the successor or covering relation). Then, if we do that, I am pretty sure that a pol and its Hasse will be isomorphic. (Actually not all partial orders can be redefined in this way - only those that have successors and predecesors. The real numbers wouldn't work - there is no next real after zero).

Equally, if we were to add to the Hasse ALL the incidences and not just the "successor" ones, we would have our isomorphism. The modified Hasse diagram would then be a mess of spaghetti (tho' perhaps pleasing to Italians!), and it would no longer have the most pleasing property of being the graph of an (n+1)-pol (well that's my conjecture).

I believe, however, that the undirected graph corresponding to the digraph of the poset which is the rank 1 polytope is, in fact, the graph of an irregular polytope of rank 3 with 3 faces. mike40033 (talk) 08:22, 22 July 2010 (UTC)

I am not suggesting we should do either one! But I do still say that the Hasse can effectively be thought of as the polytope itself - much more so than the graph, which is all you usually see in a polytope "picture".

SteveWoolf (talk) 19:53, 20 July 2010 (UTC)