Talk:Abstract polytope/Archive 5

"Nice" and "Nasty" terminology
Throughout the Discussion, I shall remove my use of the terms "Proper" and "Improper" in favour of "Nice" and "Nasty", since ARP uses Proper to refer to the least and greatest faces - and we now need this term in the article. In any case, I don't think anyone besides myself used proper/improper anyway - we seem to have de facto agreed on nice/nasty. Of course, my coining nice/nasty was tongue-in-cheek and we could consider other possibilities. But we all seem to have got used to them in the Discussion, so my feeling is, leave it as is - we have better things to do with our time. In the Article, however, it would probably be best if we just stick to standard terminology, i.e. dually atomistic and meet/join lattice.

I am not clear whether the polytope properties of being dually atomistic and a meet/join lattice are actually independent of each other. Note that lattice has nothing to do with all vertices have integer value co-ordinates - this is a geometric concept with no meaning in AP theory. SteveWoolf (talk) 03:14, 22 June 2010 (UTC)

Is Axiom 2 Independent?
I'm not sure. First, the fact that any flag can be changed into any other already implies that all flags must be of equal size. Then, we can ensure that flags always have n+2 elements by defining an n-polytope as one whose maximal element has rank n. (talk) 12:37, 14 November 2008 (UTC)


 * Technically, the statement that P is strongly connected is different from the statement that P is strongly flag-connected. In particular, McMullen and Schulte show that if P satisfies properties 1 and 2, then strong connectedness is the same as strong flag-connectedness. In general, the connectedness requirement is the following: every section of P of rank 2 or more must have the property that given any two proper faces F and G, there is a finite sequence of proper faces F = H_0, H_1, ..., H_{k-1}, H_k = G such that each pair of adjacent faces is incident. At any rate, it's not clear to me whether these two versions of property 3 are equivalent if you remove property 2. -CunningGabe (talk) 01:34, 17 November 2008 (UTC)


 * Very much appreciate the feedback - I will consider these matters with extreme care before meddling further with our Great Definition (- much better to get egg on one's face in the Talk than a 7-course meal in the Article).SteveWoolf (talk) 07:02, 17 November 2008 (UTC)

I have been studying the definition and the relevant pages of ARP at great length, and I can now clarify this topic - but not 100%.

Our current axiom 3 (strong flag-connectedness), given our rank definition, clearly implies axiom 2. We should, I think, therefore do what ARP does, i.e. state axiom 3 in its weaker "incidence sequence" form, as given by CunningGabe above; then state the current strong flag-connectedness as a provable theorem.

So far, so good. But given this "original" set of 4 ARP axioms (ie with the weaker axiom 3), I have been able to prove, by examples, the independence of axioms 1,3, and 4. But I have not been able to construct any poset that satisfies all axioms except 2. Nor can I prove it from the other 3. So I am left wondering again - is Axiom 2 independent (when we use the weaker axiom 3)? If M-S use axiom 2 in their proof of strong flag-connectedness, that does not imply it was necessary - there may be a proof without it.

So, a challenge for you all: Can you either prove axiom 2 independent of the other 3, or give an example of a poset which only satisfies the other 3? I may write to Schulte if we can't crack it, but I feel I should save communications for more advanced topics.

I shall provide my examples proving independence of the other in due course. SteveWoolf (talk) 18:55, 21 June 2010 (UTC)

This excellent article would benefit from a coherent and early definition of "universal"
I just noticed this article, and find it to be excellent.

It could, however, benefit from an early and formal definition of "universal", since it refers to the 11-cell and the 57-cell and one other ARP as "universal" without, defining the term anywhere before that in the article:

"''The 11-cell, discovered independently by H. S. M. Coxeter and Branko Grünbaum, is an abstract 4-polytope. Its facets are hemi-icosahedra. Since its facets are, topologically, projective planes instead of spheres, the 11-cell is not a tessellation of any manifold in the usual sense. Instead, the 11-cell is a locally projective polytope. The 11-cell is not only beautiful in the mathematical sense, it is also historically important as one of the first non-traditional abstract polytopes discovered. It is self-dual and universal: it is the only polytope with hemi-icosahedral facets and hemi-dodecahedral vertex figures. The 57-cell is also self-dual, with hemi-dodecahedral facets. It was discovered by H. S. M. Coxeter shortly after the discovery of the 11-cell. Like the 11-cell, it is also universal, being the only polytope with hemi-dodecahedral facets and hemi-icosahedral vertex figures. On the other hand, there are many other polytopes with hemi-dodecahedral facets and Schläfli type {5,3,5}. The universal polytope with hemi-dodecahedral facets and icosahedral (not hemi-icosahedral) vertex figures is finite, but very large, with 10006920 facets and half as many vertices.''"

When it is finally defined, many paragraphs later, it is not defined clearly, but rather in a conversational mode. I strongly suggest that

a) the term "universal" be defined *before* it is used as though the term were understood, and

b) when it is defined, it be defined a bit more coherently rather than conversationally.Daqu (talk) 01:27, 9 May 2010 (UTC)


 * Done my best, but now we need a definition of "covering". Hope someone who knows combinatorics better can make improvements. 83.104.46.71 (talk) 15:45, 9 May 2010 (UTC)
 * Sorry I don't think this really qualifies as a definition. First, it isn't clear or precise. Second, you can't just use terminology like "cover" without defining it. In any case, the term "cover" does have a precise meaning in a poset, namely that a covers b if a>b and no x satisfies a>x>b. Eg, in the set of integers, 4 covers 3 but 4 does NOT cover 2. But I don't think that is the meaning you intended.
 * May I suggest relegating your ideas to the talk page until a consensus is reached, preferably with some more expert help. Much of my stuff is here waiting for such help and references!SteveWoolf (talk) 11:31, 5 June 2010 (UTC)
 * I just tried to make it more succinct. I cannot see anything imprecise about it, apart from definitions of set-theoretical terms such as "maximal" or "covering". For that, suitable definitions probably already exist on the wiki and can be linked to from here, I see little point in repeating a definition that has a more appropriate home elsewhere. However, set theory is not my strong point so I will leave it to someone else provide such links. Meanwhile there is the eternal dilemma - to post less-than-perfect content or to post nothing; WP:BOLD seems appropriate here - "be bold but not too bold". &mdash; Cheers, Steelpillow (Talk) 13:24, 5 June 2010 (UTC)
 * I defined "covers" for you above - it's a very simple concept, and just means "is the immediate successor of". So in what sense does the cube cover the hemicube? Being bold is fine, but when the bold idea doesn't withstand scrutiny, then one needs to be humble. I don't have much free time, so if you want to prevent legitimate advancement of this article, I and others will stay out and exchange ideas elsewhere. Can I suggest that you do as I have and put your "bold" ideas in the talk section and await consensus before you give them "official"status. SteveWoolf (talk) 15:19, 5 June 2010 (UTC)
 * ??? I have no wish to "prevent legitimate advancement of this article". Whatever gave you that idea? I just tried to fix the bits that I felt I could fix, and remarked on some issues I felt unqualified to fix. I still feel unqualified to write up covering. If you know how to do things better, then it's probably quicker to do it yourself than to try and educate me. :-( &mdash; Cheers, Steelpillow (Talk) 16:04, 5 June 2010 (UTC)

Article structure etc
My original section "Formal definition" (singular) was the culmination of the preceding paragraphs, which started with an gentle, informal approach to the AP concept. It is not helpful to lump in with this other definitions. In any case, Regularity is defined later in the Properties section - where it belongs. The Article is "Abstract Polytopes" and not "Abstract Regular Polytopes". Regularity is merely a property of the general AP. The fact that the main book ARP chooses to limit its scope to regulars is not relevant here. The Universal defn in any case is incorrect, but should also be in the Properties section - or its own.

ARP does not, I think, use the term "Body" anywhere. I can only reiterate that abstract polytopes, by definition, consist of no more than a set of faces, with some of them "contained" in others, where "contained" is the order relation. The idea that a cube, for example, has an "interior" or internal space or points is a complete misconception of the AP concept. In traditional polytopes, yes - depending on your point of view. I therefore think it is best to use the term maximal face or n-face when talking about the faces (ie without their subfaces), or "whole" polytope, maximal section, or n-section when including the subfaces. I for one feel "Body" implies an interior space - and that is non-existent in AP theory. SteveWoolf (talk) 16:23, 5 June 2010 (UTC)


 * Yes, I can see the benefit of moving all the regular/universal stuff to its own section. Meanwhile I am still not clear why you think the definition of "universal" is wrong. "Body" seems appropriate for the graphical introduction, but I agree that this needs to be explained as the maximal face as soon as is sensible, and removed from the subsequent discussion. &mdash; Cheers, Steelpillow (Talk) 21:39, 5 June 2010 (UTC)


 * Appreciate your constructive response, and sorry I was a bit impatient, it's just I have so little time for this yet would dearly like the article to move ahead - it is so far behind the active research.


 * In fact, I have just noticed that the topic of Universal polytopes is covered reasonably well in Mike's Amalgamation Problem section. There is a minor problem, however, in the ambiguous use of the term "covers" - since it is also a well-established concept in order theory. In the latter sense, in the triangle abc, the edge ab would cover the vertex a since a < ab and no other face is "between" them; whereas the maximal face abc does not cover a, since there are two intermediate faces ab and ac. However, I think we can at least defer this issue for now by agreeing to use "covers" in Mike's sense, i.e. that P covers Q if Q is a quotient of P. We are not necessarily obliged to import all order-theoretic terms into AP theory, and in any case this article has not so far needed the order-theoretic term. I myself prefer the term "is an immediate superior of" (dually, inferior) which is very clear if rather long-winded, but I use the symbols >> and << which are very concise. I shall shortly integrate all this into the article, which should resolve all the issues very neatly to everyone's satisfaction.


 * Given that abstract polytopes have no "interior", I would prefer we avoid the term "Body" - unless you can point to its general use in contemporary recognised publications. SteveWoolf (talk) 23:55, 5 June 2010 (UTC)


 * Well, I made some changes to bring the regularity stuff together. I notice that universality also applies to noble polytopes (having isometries of facets and vertices but not necessaily of intermediate elements), so have explicitly kept that with the amalgamation problem and deleted my incorrect remark. I can't comment sensibly on covering, though it concerns me that set theorists and order theorists may be working at cross-purposes here - an unfortunately common phenomenon with modern polytope theories. &mdash; Cheers, Steelpillow (Talk) 09:27, 6 June 2010 (UTC)

(discussion of flags moved to its own section below) &mdash; Cheers, Steelpillow (Talk) 10:05, 6 June 2010 (UTC)

Geometric analogs
Besides "body" we also have terms such as "edge" and "vertex". The two MucMullen and Schulte papers I have to hand are careful to only use these words in the context of realization. Johnson's explanation in Polytopes - abstract and real uses the terms "body", edge" and "vertex" in an abstract context. His book remains in preparation and the web-published note was edited by myself (and approved by him) so I cannot possibly be the judge of its acceptability here. Should we go with M&S or allow ourselves the same latitude as Johnson? &mdash; Cheers, Steelpillow (Talk) 09:27, 6 June 2010 (UTC)

Flags
The following definition of flag WILL NOT DO:
 * "A flag is a maximal chain of faces, i.e. comprising one face from each rank such that each face is connected to the faces above and/or below it. For example, {ø, A, d, Π} is a flag in the triangle ABC having edges d,e,f and body Π."

First, flag elements are NOT in general connected to each other. Connectedness is a well defined term in AP theory and only applies to faces of equal rank, or to sections; the term you meant was "incident". Second, the minimal and maximal faces are NOT incident to any lower/higher face respectively. Finally, this is an introductory section and designed to present AP concepts in easy steps. The "vertex" format {ø, a, ab, abc} is much clearer here because it immediately shows the incidences. The current format does not even make clear that A < d.

I shall update the article accordingly. If you object, then please reach a consensus in the talk page where it can pass scrutiny - or otherwise.SteveWoolf (talk) 09:39, 6 June 2010 (UTC)


 * Definition in terms of vertices again WILL NOT DO. Each element of an AP is an abstract object in its own right. To say that the abc notation helps explain the incidences is oxymoronic - it is the incidences - to the next level down - and is not the element. The kind of confusion inherent in this notation is just the kind of nonsense that Johnson has been trying to clear up. I do not believe in dumbing-down the introductory material to the point of being misleading if not downright incorrect. Happy to change "connected" to "incident". &mdash; Cheers, Steelpillow (Talk) 10:05, 6 June 2010 (UTC)
 * Worse, your latest edit does not explain what "contains" means, falling foul of your own concerns about unexplained terms - and in the introductory section to boot. I'd be grateful if you follow your own advice and discuss such things here before making edits that other folk might consider bad. Will revert until then. &mdash; Cheers, Steelpillow (Talk) 10:09, 6 June 2010 (UTC)
 * Here is my proposal:
 * A flag is a maximal chain of faces, i.e. comprising one face from each rank such that each face is adjacent to the faces above and/or below it. For example, {ø, A, d, Π} is a flag in the triangle ABC having edges d,e,f and body Π.
 * I have done no more than replace "connected to" by "adjacent with". Note that the grammatical use of "and/or" is actually consistent with your concern that "the minimal and maximal faces are NOT incident to any lower/higher face respectively". Do you think this can be made clearer without getting longwinded?
 * [updated] Re. the vertex notation, there is already a remark on the page that it is not appropriate for certain polytopes, such as the hemicube. Further, it does not show the upwards incidences or containments, hence it does not dualise sensibly - a clear warning that something is awry.
 * &mdash; Cheers, Steelpillow (Talk) 10:18, 6 June 2010 (UTC)

There are now 3 problems with the Flag defn, namely

(a) The incidence defn needs to precede it - I will do that now.

(b) My previous edit stated
 * Each face, other than the minimal face, contains the one of preceding rank

whereas the current version has redundant information - i.e more conditions than the defn requires; this is not good form in mathematics. It suffices that each face contains its predecessor (if any). The converse, that each face is contained in its successor, is implied by that. It is therefore a consequence of the defn and not part of it.

(c) In your example, A is not necessarily contained in d - only if d is either AB or AC, but not BC. SteveWoolf (talk) 10:54, 7 June 2010 (UTC)


 * I take your point in (c). (b) depends in part on the definition of containment (see new section below). Re. (a) your definition of incidence conflicts with long-established usage in synthetic geometries and especially projective geometry, hence as a definition it is new to me. If taken merely as a statement, "Two given faces are said to be incident if one contains the other," this is consistent with traditional usage. I do not have M&S to hand (and the papers I do have make no reference), so cannot confirm their intent either way. Are you sure their usage here constitutes a definition rather than merely a particular case? &mdash; Cheers, Steelpillow (Talk) 19:26, 7 June 2010 (UTC)


 * Only just noticed this, sorry. Will answer now.


 * Incidence is defined clearly and unambiguously exactly as now in the article, on page 9 of ARP.


 * In the triangle abc, edges ab and ac are NOT incident (in the ARP sense), but are they not in the traditional sense, since they intersect at a vertex? Perhaps you can answer that for us.


 * Mathematics is replete with conflicting definitions both within and between its various branches. You could solve the Middle East problems as easily. Regrettable, of course, and we should always strive for clarity and consistency. But in the end, we have to accept the status quo (except when factually incorrect) so we can move on and develop the actual content, rather than 'ARPing on about terminology interminably. SteveWoolf (talk) 07:16, 12 June 2010 (UTC)
 * Groan! Meanwhile, digging into "the traditional" meaning of incidence - as one might suspect, it varies somewhat. Synthetic treatments are apt to talk of "axioms of incidence" (such as a point as the meet of two lines and a line as the join of two points) without formally defining incidence. This is understandable, since all terms used in axiomatisation are of necessity primitives (having no a priori meaning). However some authors do give definitions close to that given here. It is therefore hard to justify any statement that some given definition is "different from traditional theory". Might it be worth amending the article along the lines of "Note that traditional geometers sometimes use the term in a broader sense."? &mdash; Cheers, Steelpillow (Talk) 12:42, 12 June 2010 (UTC)


 * Thanks - will do now. SteveWoolf (talk) 18:33, 12 June 2010 (UTC)


 * Mm - the Wikipedia Incidence (geometry) article seems very definitely to say that two lines can be incident, ie share a vertex, which would not be consistent with our AP defn. Plus other hard-to-understand meanings which seem VERY different. So I modified the caution a bit, but you can amend it further if you feel it's necessary. I don't think it's such a big deal, I think most mathematicians expect definitions to apply only to the current project - except for the truly mainstream concepts eg integer, real number, partial order, or topology. SteveWoolf (talk) 22:33, 12 June 2010 (UTC)

Defininition still not right!
Our current flag definition


 * A flag is a set of faces {F&minus;1, F0, ..., Fn}, i.e. containing just one face of each rank, and such that F&minus;1 <  F0 < ... < Fn.

- which I copied from ARP, is still not quite right!

The problem: This definition assumes Axiom 2 of the later formal AP defn, which states that Every flag has exactly n + 2 faces.

Informally, a flag is a sequence of faces, each contained in the next, ranging from the least to the greatest face, and which is not a subset of any flag with more elements, i.e. there can be no "gaps". However, posets in general can have flags of varying size. For example, the poset


 * {ø, a, b, bc, abc}

(which is NOT a polytope!) has two flags of unequal size, namely (ø < a < abc} and (ø < b < bc < abc).

We cannot avoid this problem by defining flag after the formal AP defn, because the defn needs the flag concept.

The solution: It's not a serious problem, and I shall shortly present a formal correct definition. SteveWoolf (talk) 09:49, 12 June 2010 (UTC)


 * Okay done - I think it's okay now! SteveWoolf (talk) 15:38, 12 June 2010 (UTC)

Vertex representation and containment
The vertex representation does not uniquely represent every element of an AP, as noted in the article. The question, "does the set of incident vertices DEFINE a given face?" is thereby answered in the negative. For some 2-face P having adjacent vertices A,B,C, the representation ABC can thus be no more than that. Many combinatorial studies of APs treat ABC as being shorthand for {A,B,C}. This is fine if one is dealing with combinatorics, but is not true of APs in general, and is particularly unhelpful when dealing with tilings of projective spaces.

Johnson highlights this issue by distinguishing between j-faces such a P and j-facials such as {A,B,C,d,e,f,P} where d,e,f are the sides of the AP. In realising these elements as point sets, no point belongs to more than one element, so for example the 2-face P is an open region of the plane ("open" in the geometrical sense, in that it does not include its boundary) while the 2-facial {A,B,C,d,e,f,P} is a bounded triangle and a proper polytope (ignoring the empty set). Thus most faces of an AP are not in themselves APs, only the facials are. The vertex notation ABC adequately describes neither, and indeed some authors seem unaware of the difference.

This gives rise to a concern I have about the exact definition and implications of containment. Does "P contains A" mean that A is a set and B is a member of that set? Or is the idea closer to that of incidence, in that if two elements are incident then the higher-ranked element is said BY DEFINITION to contain the other? Or is containment something else again? The issue at stake is whether or not it is always true to say that if two elements are incident then the higher-ranked element contains the other. For example, intuitively the face P does not contain vertex A while the 2-facial {A,B,C,d,e,f,P} does. &mdash; Cheers, Steelpillow (Talk) 19:26, 7 June 2010 (UTC)


 * There really is no problem with these concepts; everything is well-defined in ARP and elsewhere, but I will try to clarify.


 * Yes, vertex representation cannot be used for some polytopes, eg the digon ab, since both edges contain the same two vertices. In that case we are forced to label them, eg, e and f.


 * Avoiding vertex representation for now, the containment relation < (or >) is the defining relation in any order. A poset (partially ordered set) is quite simply any set WITH a relation <, such that some, none or all pairs of elements x,y satisfy x < y. Our beloved AP's are simply that - a poset. The relation < is provided "free" with the poset - not something you can define. Thus a digon is the poset {phi, a ,b, e,f, D} where e and f are the edges, D the max face and the relation < is given by {phi<a, a<e, a<f, e<D, etc.}. If you are looking for something else - you won't find it - it really is as simple and basic as that. You should not think of containment in this context as necessarily involving sets. If you like, think of it initially as an ordering only; then you could say, for example, that in triangle abc, that a < ab means a is an ancestor of ab. Note that neither a < b nor b < a is true - this is a partial order only, unlike integers or real numbers (decimals).


 * Ok, now let's consider "nice" polytopes like the triangle abc. Here, we can use vertex representation - tho' we are never forced to. If then you think of each face as a set of vertices, then containment has the set-theoretic meaning "is a subset of" but not "is a member of". I admit that in this respect, "containment" is perhaps inappropriate, since in set theory "is contained in" means "is a member of". I shall check ARP and amend our article if it uses a different term. But let's not sweat over terminolgy here, the important thing is the concept.


 * Thus the edge ab = {a,b} clearly "contains" the vertex a = {a} (in the subset sense). But we hardly want to write a cube 3-face as {a,b,c,d,e,f,g,h} so we use abcdefgh as a shorthand - it's just a notational convention.


 * One final point: there is a somewhat subtle difference between a face and "its" section. A face is a single element of the poset, eg ab. Whereas the set {phi, a, b, ab} is a section of triangle abc - and also a polytope itself. Section is standard terminology, what Johnson calls facial.


 * Hope that helps - but if you have further questions I shall try to answer them.


 * PS Be careful with the term "adjacent" - it has a specific defined meaning in AP and absolutely does not mean adjacent (in the English sense) in the Hasse diagram. Only flags or faces of equal rank can be adjacent. So incident and adjacent are mutually exclusive. If you want to express the idea of a>b "with nothing in between" the correct term is "covers" or "is a successor of" (dually, "is covered by" or "is a predecessor of") SteveWoolf (talk) 00:16, 8 June 2010 (UTC)


 * ARP seems to avoid the terminology "problem" by sticking to the symbols < ≤ > and ≥ without using any English term. Will research further.... SteveWoolf (talk) 00:58, 8 June 2010 (UTC)

Have rewritten the Polytopes as Posets to clarify these issues. While I don't think we should make the article quite so "gentle" everywhere (we'd never get done!), this section is the most fundamental to understanding just what an AP is. Guy, please don't confuse this by hitting novices with the subtleties of digons and hemicubes here, which can't use vertex rep. Nothing in this section states that vertex representation is essential, while any novice can easily see that, for example, a < abc. The article later makes clear the impossibility of vertex representation for digons &c. The sole purpose of this section is to ease novices into the sea-change paradigm shift from touchy-feely geometry into pretty strange abstract algebra. SteveWoolf (talk) 07:15, 8 June 2010 (UTC)

In general, I am trying to tidy and tighten up the various imprecisions and other loose ends before proceeding to higher topics. Generally I shall follow ARP terminolgy and content - unless there are compelling reasons not to (like if they made a mistake!). Hope you will support this effort so we can end up with an article that shines like a beacon and fills the enormous void that currently exists between high school geometry and the very inaccessible (in terms of comprehensibility to lesser mortals) professional world. SteveWoolf (talk) 08:30, 8 June 2010 (UTC)


 * Many thanks for your wonderfully clear explanations. I am so glad I didn't put you off for good, I'd probably have made as many mistakes as I corrected. I share your concerns about introducing complications too soon, and am happy to have the vertex representation used with greater caution as you are doing. Your remarks on containment tend to confirm my suspicions that some authors (and probably wiki contributors) use the term out of context, in the mistaken assumption that an element represented as abc is necessarily the combinatorial set {&phi;,a,b,c,ab,bc,ca,abc}. I may be sweating over terminology, but it is just these little differences across disciplines that create the inconsistent mess that is polytope theory today and I see a major opportunity for to set these differences in context. One small point of information - in Johnson's approach, the set {&phi;, a, b, ab} is not a facial of triangle abc but of the 1-face ab; if abc is a face then the associated facial is {&phi;,a,b,c,ab,bc,ca,abc}. I seem to recall that one key reason for the distinction between face and facial is specifically to help avoid the mistaken assumption about set-theoretic containment. &mdash; Cheers, Steelpillow (Talk) 19:01, 8 June 2010 (UTC)


 * Appreciate the positive sentiments....


 * I most definitely think terminology and its correct and consistent usage are very important. I just didn't want to get sidetracked then into a discussion about whether "is contained in" was the best term for < in the case a < ab, because the polytope theory works just as well with any other term, eg "is pecked by" or "is a prisoner of" or "jabberwocks" etc.


 * But we do need a term. Since in trad theory we most certainly would say that the vertex a is contained in the edge ab, I think we should stick to that terminology. Using terms that people are used to makes for much greater readability - imagine how difficult it would be to read a maths book in a country where the symbols for + and X are reversed (as is the case in Slovakia with, and . !).  One disadvantage with familiar terminology is that one is tempted to assume properties that DON'T carry over into the new environment.  For example, in group theory, you have to keep reminding yourself that a x b = b x a is not always true; likewise incidence is quite different in AP to TP - I myself would NOT have chosen it (and used to use "coaxial" since I called a flag an "axis"). Still, we are stuck with incidence, and nobody serious wants to devote their time to terminology wars that can't be won rather than to making mathematical progress.


 * If/when we want to formalise AP concepts 100% using set-theoretic methods, then the faces of abc become {},(a},{b}, ... {a,b,c }. Then we have to say {a} is a subset of {a,b} and not use "is contained in". But we DON'T discuss AP in these terms any more than we see pixels on a computer screen when we look at Marilyn Monroe's image in Some Like It Hot. When we switch on a light, we don't consciously insert a piece of metal between two others. So there is no problem with our use of contains - we only have to remember that it has the trad pol sense and not the set theory meaning. That being said, it is probably best only to use it in the a < ab sense in the introduction, as a guide to intuition, and thereafter to stick to the math symbols < ≤ > and ≥.  However, we may still need "contains" in other situations, eg a is contained in the flag {ø, a, ab, abc}. This is the set theory meaning. I guess it might be best to write "is a member of" or "is an element of" instead - so I will generally try to avoid the word "contain" - unless readability demands it.  (Readability is very important - you only have to glance at ARP to see that - and compared to other stuff out there, ARP reads like a large print illustrated children's book).


 * Hope you find the above, and its conclusions, agreeable.SteveWoolf (talk) 05:06, 9 June 2010 (UTC)


 * No problem. If I still have any concerns later on I will raise them here. &mdash; Cheers, Steelpillow (Talk) 10:27, 9 June 2010 (UTC)

My short-term program for Article improvement:

(1) Review Egon Schulte's letter and ensure the article conforms to his suggestions

(2) Ensure that the article conforms to ARP terminology etc.

(Except where we have any compelling reason otherwise - eg clarity.) SteveWoolf (talk) 05:06, 9 June 2010 (UTC)

Meaning of Adjacency
Adjacency is a standard term in AP theory (see ARP) so its use in the flag definition needs replacing with the correct term, i.e. incident (with). 178.41.110.144 (talk) 16:24, 6 June 2010 (UTC)
 * Oops, sorry. It's been a long day. Guess I need a break too. &mdash; Cheers, Steelpillow (Talk) 19:08, 6 June 2010 (UTC)

Done SteveWoolf (talk) 14:58, 3 July 2010 (UTC)

Twins vs mirrors
Does "mirror" have any customary use in set theory? Otherwise, to me it suggests geometric reflection of some kind, which is inappropriate. Might it be better to omit the attempt at analogy and just write, "Every abstract polytope has a dual polytope, in which the partial order is reversed: ..."? &mdash; Cheers, Steelpillow (Talk) 09:59, 10 June 2010 (UTC)


 * No, there is no formal term "mirror" that I know of. We could just omit it - but surely the inverted Hasse diagram IS a mirror version (and a reflection) of the original, is it not? Admittedly a cube is not a reflection of an octahedron - but an abstract polytope is defined as a poset, so any geometric drawing or realisation of it is an aid to intuition only! Of course, by the same argument, the Hasse diagram is not the poset either, but it is in the sense of is isomorphic to, which is the pretty much the mathematical meaning of "is the same as".


 * I did put "mirror" in quotes, and I feel it assists intution, tho' I could live with its omission. (But we really do need to fix the Flag defn - see my recent comments). SteveWoolf (talk) 04:51, 11 June 2010 (UTC)
 * Without ranking labels, a Hasse diagram is unclear, for example as to which is the maximal element, so it needs the labels. When we "reflect" a Hasse diagram to dualise, we must leave the ranking labels unaffected. Alternatively, we may simply reverse the ranking labels and leave the rest of the diagram untouched: real mirrors do not work this way. Further, if one of the two diagrams has some elements rearranged (preserving incidences) it is no longer a physical mirror of the other but is still its dual. For these reasons I find the "mirror" analogy misleading. In the traditional arena, the chiral regular compounds of five tetrahedra cause enough trouble as it is, being both geometric duals and mirror twins. I will leave the definition of "flag" to you, having said my bit. &mdash; Cheers, Steelpillow (Talk) 09:37, 11 June 2010 (UTC)


 * A Hasse diagram does not actually require "ranking labels" - these are implied by the face position in the diagram, given the universal convention that we always put the higher faces on higher levels.


 * This leads to an interesting and non-trivial mathematics question: If you take a (valid) Hasse Diagram of some polytope, ie just the graph (only its vertices and edges with no directional or rank information), can you choose a different greatest face, i.e. so that the same graph is still a valid polytope. The answer is yes - for example an edge, whose Hasse is a square - any vertex can be selected as the greatest face. The interesting question is - are there any cases where the new polytope is isomorphic to neither the original nor its dual? Formally, we are asking if


 * Polytopes exist that have isomorphic Hasse diagrams (as undirected graphs) yet are neither isomorphic nor dual to each other (as abstract polytopes).


 * My guess is NO - but I'd love to see a proof or counterexample....


 * Again, the use of "mirror" was a conceptual aid to the novice and not a precise mathematical term.


 * Will fix the flag defn shortly, tho' I will make some further general comments here first. SteveWoolf (talk) 13:04, 11 June 2010 (UTC)

I am a little wary of using "mirror" here for two reasons. The first is that I'm afraid it will mislead people to thinking that the dual of a polytope really corresponds to some geometric reflection. The second is that abstract chiral polytopes have two mirrored forms (where the mirror does, roughly speaking, correspond to a geometric reflection), so that there's some danger of confusion of terms. (One of these days, I'm hoping to expand the article on Chiral Polytopes!) I think that a better way to guide intuition is to make the connection with duality of geometric polytopes by inversion through a sphere. --CunningGabe (talk) 18:34, 25 June 2010 (UTC)


 * Okay, will remove the "mirror" metaphor - but I am very reluctant to start talking about "inversion through a sphere", which seems to be wholly in the realm of Euclidean or traditional geometry. SteveWoolf (talk) 18:57, 25 June 2010 (UTC)

Ideas for improving the Article
First, I don't think we should have a new section in the Talk page for each minor point....

Introductory vs Rigorous Style
I think it is fair to say that for those not well aquainted with modern abstract mathematics, developing a real understanding of abstract polytope concepts is not easy. Many with even quite advanced knowledge of traditional polytope theory would probably consider the new world quite alien.

When we teach our kids arithemetic, we start with whole numbers. We teach them that 2 follows 1. We often don't start with zero. Have we misled them? In fact, 1 has no successor (in the set of decimal numbers, of course). Have we lied? Later we are taught that 0.5 = 1/2. But only much later do we learn that there are no integers p,q such that π = p/q. Were you taught that -1 has no square root? It sure does, namely "i". i is not a number? Then how can you say "-1" is a number then; what is that hyphen before it? Is it really better to teach our 3-year olds that there are numbers between 1 and 2, but we'll get back to that later?

The introductory sections are for novices primarily - our AP children! Can we not get the primary concept across first - before confusing them with more advanced ideas. What is the "primary" concept? It is that polytopes are no longer patterns of vertices, lines, planes in Euclidean (or other) Space. Now they are simply elements in an ordered set (with certain rules) - and nothing more.

It is incidental (in the English sense), though not fundamental, that McMullen and Schulte have decided to broaden the definition of what is a polytope as well as redefining the primary concept. (I am not discussing the merits of this broadening here).

One way in which the polytope idea has been widened is relaxing of the "dually atomistic" rule - AP polytopes are allowed to have distinct faces with the same vertices; the simplest case being the digon whose two edges both have the same 2 vertices.

Given all the above, I would very much prefer if we could NOT bring up the complication of "exceptional" polytopes (digon, hemicube &c) in the introduction. I would be entirely happy with writing a very clear explanation of these, with examples, very soon after the formal definition.

It is important not to imply, even in the introduction, that all polytopes can be represented by vertex sets.

I really hope that anyone weighing this topic will carefully consider the above points; to summarise, I really would like to get the main street AP idea across first, then address some of the strange side-roads. SteveWoolf (talk) 16:07, 11 June 2010 (UTC)


 * Agree with all salient points. Such broader ideas of polytopes have been around a long time - there has never been a "foundational" definition consistent with all modern streams of thought, and the original motivation for abstract theory was to develop just such a universal theoretical framework. As the theory got fleshed out it began to take on a life of its own, enveloping some modern ideas and excluding others. Meanwhile a sensible theory of realization remains elusive, limiting the value of AP in their original intent of clarifying traditional theory - for the most part it has just moved the goalposts. &mdash; Cheers, Steelpillow (Talk) 13:01, 12 June 2010 (UTC)


 * Thanks Guy. I'll write the promised explanation of "nasty" polytopes as my next thing to do. In any case, I will go back to Egon Schulte from time to time for his comments - hopefully he'll be willing. But first I'll try to get everything as tidy, accurate and clear as possible. SteveWoolf (talk) 18:29, 12 June 2010 (UTC)


 * Re. Rank, am I right in thinking that the least face should be denoted by F&minus;1 ? &mdash; Cheers, Steelpillow (Talk) 19:41, 12 June 2010 (UTC)
 * Silly me - fixed and thanks! SteveWoolf (talk) 13:54, 13 June 2010 (UTC)

Okay I have now written a section The digon which - hopefully - makes the "vertex notation" issue very clear, and explains why digons, hemicubes etc. can't use it, and shows a digon and its Hasse suitably labelled.

So, given that you agree with my "salient points" above - may I amend the Hasse Diagram section in the introduction so that it only illustrates what a Hasse diagram is without also burdening the novice with the subtleties of "nasty" polytopes? I would be very happy if we could go back to original "Vertex-notational" Hasse, given that the digon section now illustrates the exceptions. SteveWoolf (talk) 12:45, 16 June 2010 (UTC)

Use and abuse of "Contains"
On further reflection, I think we most definitely should never use the term "contains" except when it clearly means "is an element of [a set]"; for example, when a flag contains a face. We should not use it to mean "<", as in vertex a < edge ab.

This is what ARP does, and it will clearly avoid confusion. I have not as yet seen any English equivalent for "<" in ARP - but I think we should have one. I feel that "includes" is probably the best term, as in set theory "X includes Y" means "Y is a subset of X", and clearly a is included in ab in the English sense. I think "X includes Y" also reads better than "Y belongs to X".


 * Personally, I use 'is incident to', as do many published articles. The sole disadvantage of 'incident' is that it's a reflexive term - edges are incident to vertices as well as the other way round. However, anything like 'includes' or 'contains' smacks against the abstract nature of these objects. mike40033 (talk) 02:53, 20 June 2010 (UTC)


 * Yes, incident is an established term, and I shall use it when F < F' and or F' < F. But we also need a "one-way" term to mean <;  Guy and I have both agreed on is a subface of, as its meaning is self-explanatory. This is already implemented, and includes has been purged. I shall only use contains for set membership, eg Flag &psi; contains faces F, F'. Hope you concur. Guy and I are now leaning towards geometric rather than traditional for non-AP's - what do you now think?  Good to see you back. SteveWoolf (talk) 07:54, 20 June 2010 (UTC)


 * "when F < F' and F' < F" : I think you mean 'or', not 'and'. Yes, 'incident' is a two-way term. I don't recall 'subface' being used anywhere in the abstract polytope literature, but I can't think of anything better for now... mike40033 (talk) 01:10, 25 June 2010 (UTC)
 * Corrected above - ta. Defined correctly in the article. SteveWoolf (talk) 13:28, 25 June 2010 (UTC)


 * Re. the use of "subface", an example already referenced below is: Edelsbrunner, H.; Algorithms in combinatorial geometry, p.141. For more, google on for example polytope subface. &mdash; Cheers, Steelpillow (Talk) 16:10, 27 June 2010 (UTC)

In the absence of any dissent, and in the interests of consistency and clarity, I shall implement this very soon. In any case, it would be better than the present ambiguity, and we can still continue to haggle over terminlogy afterwards.

This will affect our oft-updated flag defn! But I shall not flag in my quest for mathematical perfection. SteveWoolf (talk) 08:16, 14 June 2010 (UTC)


 * Done - together with some readability improvements. A much clearer exposition now, I think. SteveWoolf (talk) 09:20, 15 June 2010 (UTC)


 * I am concerned that includes is no improvement on contains. Informally it still has connotations of one element inside another, and more strictly if ab represents an edge element then in general vertex element a is not a subset of ab (only &phi; and ab are) and therefore is not included in ab. Also, from what you say I assume that inclusion is not used in ARP. Perhaps it might be best to simply use the symbol "<" on these occasions, as ARP does. &mdash; Cheers, Steelpillow (Talk) 18:28, 16 June 2010 (UTC)


 * If we formalise completely, then our faces are sets of vertices; eg the vertex a is the set {a}, the edge ab is the set {a,b}. With this definition, it is true that a is a subset of ab, or ab includes a. Of course, we can't formalise all the time - it probably takes 50 lines to prove in arithmetic that a + b = b + a. The purpose of formalising is just to be sure we can be precise when necessary. So in practice we always abbreviate {a, b} to ab. Is contained in is definitely inconsistent, because it means "is an element of", so includes is an improvement, and I swapped over precisely because of your correct observation that we had been using contain ambiguously.


 * This hark back to the "element abc of some polytope P is not {&phi;, a, b, c, ab, bc, ca, abc}" issue. Likewise there is no a priori need for abc to be the set {&phi;, a, b, c}. In general abc is merely an element and is not per se a set. The "abc is a set" idea is just one interpretation of the abstract structure; it can be a useful simplification in combinatorics - which is where AP theory began - but otherwise it blurs the distinction between the element and the associated subsets of which abc is also an element. This is key to Johnson's formalism, which he says is entirely equivalent to ARP, so we need to respect it. To paraphrase your comment, if we formalise completely then our faces are mere elements and NOT sets of vertices; eg the vertex a is NOT the set {a}, the edge ab is NOT the set {a,b}, and it is NOT true that a is a subset of ab. &mdash; Cheers, Steelpillow (Talk) 09:38, 17 June 2010 (UTC)


 * 1. You are right in saying that in AP theory, it is not required that faces are defined as vertex sets; an AP can be completely formalised as a partial order of elements (faces) which are not further divisible. Therefore, my argument in favour of the "includes" term is weakened (tho' not destroyed).


 * 2. However, you yourself, if I recollect correctly, were confused/dissatisfied, as I should have been, by the previously ambiguous use of "Contains". Flags are sets of faces, and do therefore contain faces (as elements). If we also use "contains" in the sense edge ab contains vertex a, we are now using it to mean "<", i.e. our partial order relation. It is definitely NOT a good idea within the same project to use the same term with different mathematical meanings. I have stated that we should use the symbol most of the time. But we also want a readable article. For example, in the article I used the term inclusion pairs which I think reads very nicely.  I really can't think of a better term. "Belongs to" rather implies set membership. "Is a subface of" doesn't appeal to me. I used to use the terms "is [an] inferior/superior of/to" - so we could use those. (I'm too chicken to use "pecks", and Alice might revert "jabberwocks"). Any other candidates, anyone?


 * 3. Hmm... you could actually formalise in such a way that, eg, a &isin; ab &isin; abc, by defining each face as the set of its subfaces plus its name symbol. To do this, define &phi;, a, ab, abc respectively as &phi;, {&phi;, a}, {&phi;, a, b, ab} and {&phi;, a, b, c, ab, ac, bc, abc}. (Keep a clear head here... a is not the same as a, which is {&phi;, a}. This also works for naughty polytopes like the digon, because the name of each face is included (oops... excuse my English) in its definition. Interesting, huh? I only just thought of it.


 * Mathematics is full of such esoteric constructions, but usually, having formalised, we drop our formalisation like yesterday's fad. We only go back to it when the informal approach gets fuzzy - as has happened here.


 * So - by this method, we could correctly say that each face contains its subfaces. But I am still queasy about it, because it depends on this formalisation, which is NOT a necessary requirement of AP theory. Yes, we could simply define "contains" between faces to mean "<". But wouldn't it make a not-particularly-easy subject easier to use a different term? More later... SteveWoolf (talk) 07:43, 18 June 2010 (UTC)


 * You are right to be queasy. Such containing formalisms are typical of early approaches to AP theory, and remain one interpretation - or perhaps I should say application - of it. However, they tend to blur the distinction between face and facial, and so have limited value. Modern AP theory is careful to step away from them. &mdash; Cheers, Steelpillow (Talk) 09:04, 19 June 2010 (UTC)


 * Yes, it's important not to think of faces being "contained" in higher rank faces - this will lead us to mentally identify a face with a subset of its containing face, which will lead to the study of subset lattices. Nothing wrong with this, of course, unless we still call our objects abstract polytopes. There are abstract polytopes which can't be written as subset lattices - for example, the digon or the hemicube (and all the other 'flat' polytopes too). mike40033 (talk) 03:01, 20 June 2010 (UTC)
 * I think having an English term as well as a symbol such as < makes for better readability. I don't think we wish to make the article a copy of ARP, even if it were legal! I haven't and don't plan to use includes much anyway, mostly the symbol - but mostly where I have used it, replacing with the symbol < would read a bit clumsily. SteveWoolf (talk) 22:51, 16 June 2010 (UTC)
 * Well, if we can find a turn of phrase that is not misleading, that's OK. How about saying "F is ordered below G" for "F<G"? &mdash; Cheers, Steelpillow (Talk) 09:38, 17 June 2010 (UTC)
 * Don't care for that - might as well just use less than or greater than which I also don't like. But on 92nd thoughts. subface is pretty good: F is a subface of G, G has subfaces F', F". Crystal clear if not as beautiful as crystal. SteveWoolf (talk) 14:16, 18 June 2010 (UTC)
 * D'oh! I knew that! :-/ Yes, subface is the exact correct term. See for example Edelsbrunner, H.; Algorithms in combinatorial geometry, p.141. &mdash; Cheers, Steelpillow (Talk) 09:04, 19 June 2010 (UTC)
 * Will update SteveWoolf (talk) 13:43, 19 June 2010 (UTC)

Restructuring
I have moved things round a bit to make for a more logical and readable article, and expanded on some topics - especially the issue of when vertex notation can be used.

Lots of thought and time went into this - please do likewise before updating, or discuss here. Thanks SteveWoolf (talk) 01:45, 16 June 2010 (UTC)


 * Great work, Steve. Much better. Just one suggestion, I think it might be sensible to bring the section on abstract vs. traditional under the introductory concepts.


 * The Introductory concepts section is the beginning of our formal theory, where precise mathematical terms are defined. Everything prior to it is about motivation and orientation - no symbols or equations here! So logically they should be separate to keep them separate.


 * There is no Stamp Duty payable on Sections (forgive my irreverent wit - hope you won't feel a duty to stamp it out) - I feel it's confusing to try to force topics together to economise on Sections. For example, Duality is not a polytope property, but a relation between polytopes. (I will fix this and other things up in due course). SteveWoolf (talk) 23:27, 16 June 2010 (UTC)


 * I also notice a few places where minor improvements could be made to the text, for example for continuing consistency with your recent work, or where assumptions are not stated (such as when non-spherical polytopes are introduced or a restricted view of "traditional" definitions is taken). I think I understand your POV sufficiently now that most of these tweaks can be made safely without discussion, but would you prefer me to raise them individually here? &mdash; Cheers, Steelpillow (Talk) 18:45, 16 June 2010 (UTC)


 * Well I suppose when we are pretty sure we are improving the article, then we should be bold - other parties can always politely revert and justify. But I think when we are not so sure, then we should first defer to a discussion on the talk page. Reasonable? SteveWoolf (talk) 23:04, 16 June 2010 (UTC)
 * Yes, thanks. &mdash; Cheers, Steelpillow (Talk) 09:52, 17 June 2010 (UTC)

Misc
Thanks for corrections.

I personally feel that "The connections on a railway map or electrical circuit...", while technically more accurate, is a bit verbose and unnecessarily technical in an informal sentence. However, I can live with it if you don't have second thoughts.
 * I'll sleep on it. I'm not entirely happy with the examples chosen since each has an inner hierarchy that might be analogous to rank, e.g. tube stations where you can or can't change lines or the different kinds of electrical component. &mdash; Cheers, Steelpillow (Talk) 10:24, 19 June 2010 (UTC)
 * The point of the examples is to illustrate only that the map or circuit diagram captures the connectivity and that physical positions are irrelevant. For this they achieve the purpose very satisfactorily. I spent my childhood absorbed with the London Tube map - what an education. And when at 10 or 12 years old I applied the same idea to geometrical figures, the school master ridiculed my efforts! SteveWoolf (talk) 13:43, 19 June 2010 (UTC)

Null face "Virtually" essential. Try convincing a certain user of the essentiality in the simplex and hypercube articles. User Sephia Karta and I argued at length for the inclusion of the null face in the otherwise elegant tables - but he stubbornly ignored our arguments in favour of trivial and easily refuted ones - till me and Sephia gave up. I guess this is just one of those Wiki problems that we're stuck with - it's a mirror of the less-than-perfect world. Anyway, agree with your edit. Read the fascinating brawl at Talk:Simplex.
 * Have responded there. In the present context of APs, identifying the empty set with the null face is fundamental to consistent treatment. &mdash; Cheers, Steelpillow (Talk) 10:24, 19 June 2010 (UTC)
 * Of course, those are not AP articles, tho' I think the article would benefit from a less negative view of -1.

Will respond to "faces as sets" comments shortly. SteveWoolf (talk) 05:29, 18 June 2010 (UTC)

Will remove "The terminology used in abstract theory also differs in many ways for traditional usage, with terms such as vertex figure and section having different meanings in each field." This is a repeat of what has already been said, and the details are given later also. Is vertex figure different? I don't think so - just defined in AP terms, that's all. SteveWoolf (talk) 05:25, 19 June 2010 (UTC)
 * FYI: In a faithful realisation of some abstract polytope, the abstract vertex figure is realised as a vertex star - the vertex point together with every element of which it is a subface. The geometric vertex figure is obtained by slicing through the polytope close to the vertex. In descriptive geometry we call this a section but it is not an abstract section. The vertex star and sliced surface are both realisations of the same abstract vertex figure - which is a section of the parent AP - but only one is faithful. The vertex star is a faithful realisation of the abstract vertex figure in situ within the parent AP, while the sliced surface is a faithful realisation of the sectioned figure in isolation as an AP in its own right. Neither is a faithful realisation of the other, because rank and dimension do not correspond. Thus, the geometric vertex figure is not a faithful realisation of the abstract vertex figure. I avoid discussing the term vertex section due to concern over the finite number of electrons available on . Note however that in both geometric and abstract theories the statement, "A vertex figure is a section of the polytope at a vertex" is true, though in each case it means something quite different. Confused? You should be! (thanks to Dan Rowan - or should that be Dick Martin? Oh dear, I'm showing my age...) &mdash; Cheers, Steelpillow (Talk) 10:24, 19 June 2010 (UTC)


 * Mm.. I'll get back to that SteveWoolf (talk) 13:43, 19 June 2010 (UTC)

I now think "Geometric" would be better than "Traditional" - M-S use it, it's shorter and sounds better, and the -metric implies measuring which is mainly what differs the trads from the APs. If no strong objections within a short while, I'll change it. SteveWoolf (talk) 05:40, 19 June 2010 (UTC)
 * Happy, with caveat. I am beginning to think of the intermediate step between geometry and APs as "traditional combinatorics". There may be moments where "traditional" encompasses the broken combinatorial models that were the direct ancestors of APs theory. &mdash; Cheers, Steelpillow (Talk) 10:27, 19 June 2010 (UTC)
 * Don't follow - can you be more concrete? SteveWoolf (talk) 13:43, 19 June 2010 (UTC)


 * Having re-read the article with this in mind, I can find no instance where the caveat applies, so let's go for a quiet life and ignore it. Consider me happy. &mdash; Cheers, Steelpillow (Talk) 18:43, 20 June 2010 (UTC)

Hasse diagram
I am most unhappy with recent changes to this section. If I make a stick, ball and panel model of a polyhedron, a face is clearly a panel, and not just the handful of incident vertex balls. If I draw a line l and mark on it two points A and B I do not regard the segment AB as just its endpoints - indeed in projective geometry the two points do not uniquely bound a single segment. Coming from such understandings, the present explanation is well-nigh impenetrable. Further, the abc notation does not transfer easily to the algebraic treatment of generalised AP theory, while it is obvious enough to relate a panel to F2. Attempting to bridge from geometry to APs via the abc notation is horribly convoluted if not downright misleading (Well, if someone is coming from a knowledge of combinatorial polytope theory the present version may be a gentle lead in, though they will find little that is new). Some time ago I made extensive edits and revised the diagram, treating j-faces as atomic elements in their own right and not being combinatorial objects, specifically in order to create a more direct and intuitive bridge - and also to help combinatorialists understand how AP theory differs from their own traditions. Reverting to the earlier failed approach is truly a backwards step. I am also disappointed that such a fundamental change was not discussed beforehand. I would very much like to revert wholesale, and then discuss individual changes before updating. &mdash; Cheers, Steelpillow (Talk) 11:37, 19 June 2010 (UTC)


 * We had better discuss first, else I will feel that it is impossible to improve the article.
 * 1) I already explained my arguments at length, and you "agreed with my salient points"
 * 2) It's only your opinion that this "fails". In what way?
 * 3) I was equally unhappy when you altered my original, that's partly why I took a year off - I was spending all my time arguing and little of it improving, and here we are again.
 * 4) If you believe an edge is anything other than a set of elements of a poset, then fine. But then stick to traditional geometry and leave the abstract side to those who can and wish to develop it. This could be a major article - if it's allowed to be.
 * 5) I will say for one final time: the purpose of the Hasse diagram section is to introduce the Hasse diagram, and NOT to point out the subtleties of abstract pols. I took the time to address your points in detail later in the Digon section, but if that doesn't suffice, we can give the hemicube as another example. Why do have to ram difficult ideas home before we cover the easy ones?
 * If you want to help me improve this article, then please let me get on with it. If you want to own it, then I guess I must bow out. I am perfectly willing to discuss topics, but when logic doesn't win out, then other priorities will become more worthwhile. My wife does not like the time I spend on this, and I am neglecting my music and my business for it. I can only do that if the time is well spent. SteveWoolf (talk) 13:43, 19 June 2010 (UTC)


 * What if, instead of giving the Hasse diagram for a pyramid, we gave it for a polygon? Then we could label the vertices and edges of the polygon separately so that in our notation, we are treating the vertices and edges as atomic, while still showing that convex polytopes have a natural realization as a Hasse diagram? I feel that the problem with the pyramid is that it is very difficult to convey all the information in the Hasse diagram unless you essentially label each face by the vertices it contains. --CunningGabe (talk) 12:22, 23 June 2010 (UTC)


 * ... or maybe, have a picture of the pyramid with each vertex, edge and face labeled? mike40033 (talk) 01:06, 25 June 2010 (UTC)
 * But how do you prefer to see them labelled - in "vertex notation" or with separate names? I still feel we should only introduce the Hasse Diagram concept here, leaving non-atomistic polytopes (digon &c.) to be introduced further on, as I have done. But I will defer to your decision here as the deciding vote. SteveWoolf (talk) 19:50, 25 June 2010 (UTC)

Here are the previous diagram and associated table:

This seems to be along the lines that Mike2003 is suggesting. The risk with labelling faces and edges by their vertices is that readers might be led to think that these elements are necessarily vertex sets, which was a common enough principle in the combinatorial days before the arrival of formal APs, but is not in fact so (it remains the preferred interpretation by authors such as Gr&uuml;nbaum, so definitely needs a place in the article, but I feel srongly that the intrdouctory section is the wrong place - just as Steve realised that it was also the wrong place for what he calls "nasty" polytopes). I believe very strongly that this previous version is far better - which is unfortunately at the heart of the dispute between Steve and myself.&mdash; Cheers, Steelpillow (Talk) 20:22, 26 June 2010 (UTC)


 * Suggest we await Mike's clear preference, which I asked him to indicate. I thought you had agreed we were right in teaching our 3-year-olds that 1 is followed by 2 - we don't worry them at first with 1.5. Then we correct our "misinformation" later, when they have reached a suitable age. If you agree with that, why doesn't the same apply here? The digon example follows very shortly afterwards and clarifies the point, and which I wrote precisely to address your concern - and was happy to do, the article is the better for it.


 * I gave my reasons why at the start of this sub-topic. There is a difference between being economical with the truth which can later be expanded on (as with rational numbers and the digon), and giving downright misinformation which later has to be corrected (as with vertex sets). &mdash; Cheers, Steelpillow (Talk) 13:52, 27 June 2010 (UTC)


 * Reviewing recent edits, the article is now moving in the right direction. Thanks, Steve. Much appreciated. &mdash; Cheers, Steelpillow (Talk) 13:58, 27 June 2010 (UTC)


 * Not aware of any misinformation, only of not providing an overload of it. I remain convinced that it was better to stick to familiar concepts in the introduction, and so far only you are clearly on the side of your "right direction". But I decided it wasn't worth the time wasted that could be used advancing the article. SteveWoolf (talk) 20:27, 27 June 2010 (UTC)


 * Guy - I think I have an idea that will not only improve the article by making the AP concept even clearer, but will entirely address your current concern right at the start. Give me 24hrs to do this - I'll start now but I'll have to break off if the rolling pin starts being brandished!SteveWoolf (talk) 05:34, 27 June 2010 (UTC)


 * PS Can I suggest that "supervise" rather than "police", "complaint" rather than "outcry" would be more neutral - and likely to lead to constructive discussion. Reminds me of a bible-thumper I knew who always asked his rebellious kids if they thought it was right to sin - the terminology presupposes the answer! SteveWoolf (talk) 20:58, 26 June 2010 (UTC)
 * Likewise I trust that no more accusations of disruption, insanity, need of restraint and so forth will come my way. If I fail to achieve neutrality, it is because I am only human. :-/ &mdash; Cheers, Steelpillow (Talk) 13:52, 27 June 2010 (UTC)


 * A good strategy for not being branded disruptive (etc.) is not to be disruptive; and if you ask forgiveness for your human weaknesses, then you must show some willingness to sincerely reform: I will start to trust your intentions again when demonstrate, over time, your desire to cooperate. A clear statement of your intent would immediately create better feelings. I gave you your olive tree, no hard feelings - shall we get back to content now? SteveWoolf (talk) 20:46, 27 June 2010 (UTC)


 * Steve, with the greatest respect, will you just let sleeping dogs lie. If you must have the last word, make it a polite one. That way you will have your wish and there will be no hard feelings. Deal? &mdash; Cheers, Steelpillow (Talk) 19:07, 28 June 2010 (UTC)


 * the main advantages of the two diagrams are as follows :
 * the square is simpler, and the 'style' is more in line with the rest of the article. However, for some, the square is too simple - eg, you can't 'see' the hasse diagrams of facets and vertex figures in the square anything like as well as you can in the pyramid.
 * the pyramid is not regular, so might help dispose any misconception that abstract polytopes have to be regular. It also gives more detail, allowing amateurs who want a deeper understanding to think about it more deeply
 * I suppose it's too messy to put both diagrams in the article. Perhaps, if the square is chosen, the pyramid one could be moved to the article on pyramids or on the Hasse Diagram (if there is one) ? —Preceding unsigned comment added by Mike40033 (talk • contribs) 01:57, 29 June 2010 (UTC)


 * Perhaps the pyramid one could be put into an expandable somehow....? mike40033 (talk) 01:59, 29 June 2010 (UTC)

Like this?

&mdash; Cheers, Steelpillow (Talk) 08:14, 1 July 2010 (UTC)

Square Rools, ok?
(For the non-British readers, "xxx rules, ok?" is a typical bit of graffiti to be found painted on walls in the UK, expressing a preference or admiration of xxx" - eg a football team. Often misplet "rools", probably intentionally)

Guys, gals... We're going to go round and round forever here. For a section that is designed to merely state the very simple concept of a Hasse diagram, it might as well be a section about which religion is the true one for all the controversy it has stirred!


 * Someone (Gabe?) said the pyramid was too complex to see the incidences in non-vertex notation, so we should use a simpler polytope. That I agree with.
 * Steelpillow wants non-vertex notation, okay, I can live with it, given the previous statement.


 * Now the square is "naughty" because it's regular and self dual, not to mention its number of vertices and edges is a perfect square and the first non-composite (oops!) non-prime (oops!) composite number. Okay, we could use my example of an amorphic polyhedron, but then we're back to rather complex and unintuitive diagrams.


 * Of course, since as the medieval scholars knew, 1 is not a number. mike40033 (talk) 08:17, 8 July 2010 (UTC)


 * Ah... we live and learn (sometimes). 1 is "no longer" (since the 19th Century) considered prime, so that the unique factorisation theorem works! You 1 that won! SteveWoolf (talk) 18:50, 9 July 2010 (UTC)


 * Frankly I really don't care for clickable popups. If it's necessary for the concept, it should be there all the time. If it's not, why clutter the article?

We don't have to (and nor should we wish to) teach all there is to know about AP's in a single section, and in fact Queen Elizabeth looks likely to become Britain's longest-reigning monarch. Less is more - a section should stay on-message. I will in due course expand the regularity section by giving examples of irregulars. So can we move on to other topics? I'll be soooo happy if anyone can answer any of my "Unanswered questions". But thanks for all the suggestions! SteveWoolf (talk) 20:16, 3 July 2010 (UTC)


 * Pretty much agree. Looking at my pop-up example, this is not really what pop-ups are for (collapsing a pop-up also requires javascript, which some users block for security reasons). I'd like to see a less regular example later in the article, just to illustrate that they are not all symmetrical. How about one for the hemicube, to go alongside its graph? &mdash; Cheers, Steelpillow (Talk) 10:00, 4 July 2010 (UTC)


 * ok, I concur... no popup :-) and no pyramid :-( mike40033 (talk) 00:59, 6 July 2010 (UTC)


 * hmm... well, the hemicube is just as regular as any other regular polytope, but I have no objection otherwise... mike40033 (talk) 01:00, 6 July 2010 (UTC)


 * Yes, we definitely should have an irregular example - I actually feel irregulars deserve much more consideration than they generally receive, so I will put my "amorphic", i.e. totally irregular, polyhedron into the article - for now without its Hasse, which would be fairly large - quite a bit more so than the pyramid.


 * But should we include the Hasse for this? The article should explain concepts, using examples to illustrate. While Mike's Atlas of Regular Polytopes is great, we wouldn't want to embed it in the article (though we should link to it) - to give a sledgehammer example. But if y'all'd like one, Steve is here to serve - I'll create one. SteveWoolf (talk) 07:41, 6 July 2010 (UTC)


 * Since the hemicube I suggested is of course regular, I'd like to see a suitable alternative. The pyramid drawing already exists, or one for Steve's example would also do. &mdash; Cheers, Steelpillow (Talk) 18:49, 7 July 2010 (UTC)


 * I have been planning to use a Hasse diagram to illustrate the "Sections" section by red-highlighting a section of an irregular polytope. Then we can mention it as another example in the Hasse diagram section. I'll think about the best pol to use - we don't want to get overly complex. Will archive soon. Regards SteveWoolf (talk) 04:33, 8 July 2010 (UTC)

Name of the 1-polytope
Let us try to reach a correct consensus here.

We are talking about "standalone" 1-pols, not sections here. 3 choices: Edge, Line, Line Segment.

1) Edge. Personally, I like it. Short and identical with the 1-section use (eg the edge of cube).
 * A 1-polytope is only an edge when it is a subpolytope. On its own, it is not an edge of anything. &mdash; Cheers, Steelpillow (Talk) 10:29, 20 June 2010 (UTC)

2) Line segment. Verbose, and I never liked it, but I thought that was the standard term; Mike left it in his recent edit, which seems to suggest he thinks so - would he care to confirm/deny?
 * "segment" is a geometrical notion, not an abstract one. It is a consequence of realisation. &mdash; Cheers, Steelpillow (Talk) 10:29, 20 June 2010 (UTC)

3) Line. Short and sweet - but maybe not so short. Lines in geometry can be unbounded, and our 1-pols are very, very bounded - only a point is shorter (well, ok, the (-1)-pol also, is it's length -1? I suppose....). More to the 0-pol, I have never seen line used to mean 1-polytope. One certainly can think of the apeirogon as an infinite unbounded line (informally).
 * Axiomatically, "Line" is a primitive and hence wholly abstract - it has no context except by interpretation. This is what we need to imply.
 * Sez who? I can say the same for "edge". SteveWoolf (talk) 12:12, 20 June 2010 (UTC)
 * See the article on Axiomatic geometry. &mdash; Cheers, Steelpillow (Talk) 20:22, 22 June 2010 (UTC)


 * Historically, geometers often treated a polygon as a connected sequence of points and lines (e.g. Lachlan, R. An elementary treatise on modern geometry, 1893). I recall this still being the case during my schooling in the 1960's. Coxeter drew close parallels between polytopes and configurations, to the point that in some places his treatment in Regular polytopes ceases to apply to the former, while for his "Regular complex polytopes" all faces above 1 dimension are unbounded. An AP may be realised in many different ways, and a geometric polytope is only one of them.
 * In general and apeirogon is not a line but a piecewise curve - even the regular variety may be skewed in higher dimensions as a zigzag or a piecewise spiral. &mdash; Cheers, Steelpillow (Talk) 10:29, 20 June 2010 (UTC)

I will leave the recent edit to "line" as is pending further comments. SteveWoolf (talk) 09:27, 20 June 2010 (UTC)


 * Mm - the article now has inconsistent usage of "line" and "line segment". I would prefer to revert to "line segment" pending consensus, but will leave that to Guy to decide. Hopefully Mike will settle this, tho' other contributors can also chime in.... SteveWoolf (talk) 09:42, 20 June 2010 (UTC)


 * From McMullen and Schulte, "Regular polytopes in ordinary space", Discrete & computational geometry 17 (1997) &mdash; "A 0-polytope can only be realised as a point, and a 1-polytope only non-trivially as a (line) segment" (their brackets). "Edge" is used only in the context of a section. &mdash; Cheers, Steelpillow (Talk) 10:39, 20 June 2010 (UTC)


 * In that case we should use segment not line - the latter is bracketed. And what are they using in 2010? 13 years is long ago in this field. Suggest we wait for Mike to comment, this debate is becoming less than constructive - again. SteveWoolf (talk) 12:12, 20 June 2010 (UTC)


 * In all my reading, I've never seen it referred to anything except a 'line segment' (or as 'the rank 0 polytope'). I guess I didn't notice the bracket in M&S. mike40033 (talk) 06:43, 21 June 2010 (UTC)


 * In truth, the abstract 1-polytope has no other accepted name. I think it best to follow established usage, e.g. M&S, and stick to 1-polytope" for the AP and line segment (or segment where context allows) for the geometric polytope. (FYI I have suggested the term ditelon from the Classical Greek, meaning "two-ended thing". Johnson prefers ditel. Neither has yet gained any traction in the literature). &mdash; Cheers, Steelpillow (Talk) 20:14, 22 June 2010 (UTC)


 * As you say, it's best to stick to '1-polytope' or 'line segment' mike40033 (talk) 01:35, 29 June 2010 (UTC)

I would probably use line segment. If I'm being precise, I would just call it the 1-polytope but it's useful to have another term that people are already familiar with, and that for most intents and purposes is the same as the 1-polytope. The fact that the only nontrivial realization of the 1-polytope is a line segment means that we may as well identify the 1-polytope with its geometric realization. --CunningGabe (talk) 12:17, 23 June 2010 (UTC)

Disruptive Editing - again.
I am again most upset - to find "agreement on my salient points" to be now replaced by threats to "revert wholesale". I have really worked hard to advance this article, but now I have to spend more time trying to protect it than on improving it.

You give me a Barnstar for diligence, and admit I was "driven to distraction". But still you don't let me advance the topic and forever nitpick as a substitute for logic and comprehension.

Extracts from "Wikipedia:Disruptive editing"

Disruptive editing is a pattern of edits, which may extend over a considerable period of time or number of articles, that has the effect of disrupting progress toward improving an article or ....

Wikipedia ... openness sometimes attracts people who seek to exploit the site as a platform for pushing a single point of view.

Sometimes a Wikipedia editor creates long-term problems by persistently editing a page or set of pages with information which is not verifiable through reliable sources or insisting on giving undue weight to a minority view.

...disruptive editors harm Wikipedia by degrading its reliability as a reference source and by exhausting the patience of productive editors who may quit the project in frustration when a disruptive editor continues with impunity.

...a series of edits over time may form a pattern that seriously disrupts the project.

Disruptive editors may seek to disguise their behavior as productive editing, yet distinctive traits separate them from productive editors.

SteveWoolf (talk) 12:19, 20 June 2010 (UTC)
 * Steve, I have responded on your talk page. &mdash; Cheers, Steelpillow (Talk) 19:47, 20 June 2010 (UTC)

Final Appeal for Sanity
"I see that you are not happy with Mike40033's removal of your disruptive editing remarks." I removed it, not Mike, but put it back again exasperated by your response to my neutral attempt to resolve the 1-polytope name dispute.

It has been a mere 2 weeks since I decided to resume work on the article after a long break. I think it is much better than it was two weeks ago. But I have spent more time having to deal with you (Steelpillow) than in advancing the article. Had you supported my efforts instead of harrassing them, it would already have advanced much further. Here I am again, wasting time instead of doing mathematics - because you leave me no choice.

Abstract polytope theory requires knowledge of abstract mathematics, which you don't have. Were you to accept this limitation and confine your contribution to constructive suggestions, there would be no problem. I took your point about misuse of "contains" and fixed the problem. Fine.

But when you make edits to the article that are mathematically incorrect, or that are unnecessarily repetititive, or unsuitable in other ways, I have to spend time fixing them. Worse, I have to spend time endlessly arguing with you about it. Intelligent discussion about disputes is fine, stonewalling is not. The result is that I feel unable to continue my work, because every sentence added to or improved in the article requires an essay to justify it to you.

So I am only willing to continue work on the article if you have more respect for my mathematical knowledge, and are willing to accept the lack of your own. Specifically, I am willing to continue if you leave the article alone, and limit yourself to comments and suggestions on the talk page. I genuinely do want these, and as I did with the "contains" issue. You have my word that I will address any concerns that have merit, as objectively and neutrally as possible.

If you prefer the article as it was 2 years ago, before I entered the scene, then of course you will be happy to see me go. But do you really want to own the article, when you know you can't advance it very far? You've got your website, and numerous non-abstract polytope sites - can't you be satisfied with those, and let me make this article really worthwhile?

You have constantly lamented "the sorry state of polytope theory today" - it's fuzziness, imprecisions, lack of solid foundation. That is precisely what Abstract Polytope theory successfully addresses, though admittedly only a subset of traditional theory, and with extensions into new areas. Our Wikipedia article may not rival ARP (yet) in its content, but it does make it accessible to people who would otherwise have no possibility to make any headway into this domain - such as yourself. Yet instead of helping the revolution, you obstruct it.

I cannot squander any more of my time. If, by the Grace of God, you make a personal decision to stop "driving me to distraction" as you yourself put it, and instead, genuinely support my efforts, you would be improving Wikipedia, your own knowledge, your own self-respect, and the respect of others. Before responding, please think on this. Could you at least try backing off for, say, 3 months - I think you would be amazed and pleased by how far I could take the article...? SteveWoolf (talk) 04:23, 21 June 2010 (UTC)


 * I think we would both welcome some sanity in this dispute. For the benefit of others, here are my salient remarks copied from Steve's talk page, and which Steve has not addressed:
 * "'I remain genuinely puzzled that at times you seem to acknowledge that "in AP theory, it is not required that faces are defined as vertex sets" while at other times you seem to insist that "If we formalise completely, then our faces are sets of vertices". We have agreed that we should be guided by M&S, and especially by ARP. The papers by M&S that I have to hand make no mention of vertex sets and also avoid the abc notation, using say F instead. Does ARP describe j-faces as vertex sets and/or use the abc notation, and if not then what is your reference point?'"


 * The issue is discussed above in several places, for example here, here and here. I do not wish Steve to waste three months taking the article in the wrong direction. Can anybody else offer enlightenment? &mdash; Cheers, Steelpillow (Talk) 20:05, 22 June 2010 (UTC)


 * I think the best approach is to let Steve work on the article, and I'll watch it and tweak anything that needs tweaking. I'll soon find out if he's wasting time, and let him know. mike40033 (talk) 12:03, 23 June 2010 (UTC)


 * I would hope that you can also police this talk page too? Steve's most recent outcry was over remarks I made here, not in the article itself. &mdash; Cheers, Steelpillow (Talk) 20:31, 26 June 2010 (UTC)

Mathematics, common sense, and camraderie above all else
I regretfully have to resign from my work here from now, since I feel Steelpillow is intentionally disrupting my attempts to advance the article. I therefore appeal to you to make it clear that his behaviour is unacceptable, to avoid my having to waste a lot of time taking the matter to the Wikipedia authorities.

I would ask you both to review the various "discussions" in the Talk section, to understand my conclusion that I cannot work in an environment where I am constantly being sniped at.

Some points:

1) Recently, I suggested, in a friendly manner, a neutral discussion on the name for a 1-polytope. His response was to arrogantly advance his particular preference with opinions stated as though they were facts, and arguments for his choice that would apply equally to the other choices.

2) In Talk:Abstract_polytope I conceded:
 * "You are right in saying that in AP theory, it is not required that faces are defined as vertex sets; an AP can be completely formalised as a partial order of elements (faces) which are not further divisible."

That seems clear enough, yet he has twice claimed subsequently that I am insisting on defining faces as vertex sets, which I elsewhere made clear was not always possible. True, I did go on to suggest a method of defining faces in a way that would make one face a subset of a greater face - but I presented that merely as an interesting bit of mathematics; I stated again that it was unnecessary to AP, and that I felt "queasy" about adopting it anyway.

Either he doesn't have enough respect to take the trouble to read and understand discussion points, or he has insufficient math abilities, in which case he should not be undermining the efforts of those who do have them.

3) Steelpillow wrote yesterday "I do not wish Steve to waste three months taking the article in the wrong direction.".

At best, this is unbelievably insulting. Does Steelpillow think the article was better before I worked on it? Then let him say so. Does anyone else (with any credentials) share this opinion? Steelpillow himself awarded me a Barnstar for diligence - is his opinion of my work so easily reversed?

At worst, it is a veiled threat to revert any work I do. No mathematician of any worth is going to devote his time only to see it maliciously vandalised.

Would anyone who considers Wikipedia, the study of Abstract Polytopes, and its accessible presentation to be worthwhile please make your feelings known or do whatever you can to remedy this shameful situation, and reflect on the consequences of inaction. Thankyou, Stephen Woolf. SteveWoolf (talk) 06:38, 23 June 2010 (UTC)


 * I think you are reading more malicious intent than Steelpillow is writing, though it is of course always hard to tell in a written medium. I have left a couple of comments on the various recent debates; hopefully, having someone else chime in will help break the stalemates. I am very pleased that Wikipedia has such a fleshed-out Abstract Polytopes article, and even more pleased that so much attention is being paid to making it accessible -- many other math articles would benefit from the same treatment. Keep up the good work, and I'll stick around to give my input on proposed changes. --CunningGabe (talk) 12:33, 23 June 2010 (UTC)


 * I'm SURE there's no maliciousness here. Steelpillow is conscientious editor (challenges me too!), and ONLY interested in improving the article and his own knowledge through collaboration. If he's wrong, or doesn't understand I'm sure he'll admit this, but I'm sure its hard to be impatient with improvements when constantly challenged. Tom Ruen (talk) 22:49, 23 June 2010 (UTC)


 * Mike, Gabe and others... Thanks to Steelpillow's recent remarks, I don't feel like contributing here further at this time. If and when you are able to change the situation, I shall return. Meanwhile, I do thank you for your support. It is a sad day when ignorance and emotion triumph over the pursuit of knowledge. SteveWoolf (talk) 20:23, 28 June 2010 (UTC)


 * Steve, when I read your exchanges with Steelpillow, it seems to me that he is making every effort to be reasonable, while you take every piece of constructive criticism personally. Nobody is attacking you; we are all trying to work together to make the article better. It is natural to be hurt when someone doesn't agree with your idea, but resist the urge to fire back an angry response. Just take a deep breath, and either let this one idea go, or try explaining again in a different way why it would be good to use such-and-such term or explanation. --CunningGabe (talk) 22:15, 28 June 2010 (UTC)


 * Are you sure you've read all the recent exchanges carefully? How about his words: "I do not wish Steve to waste three months taking the article in the wrong direction"? The entire article up to and including the Digon section was virtually all my work, made at great personal sacrifice to other priorities. Is this the thanks I have a right to expect? Even you said "many other math articles would benefit from the same treatment", and other encouraging words.


 * That is just one example of many. Steelpillow in his own words admitted I was "driven [by him] to distraction" and apologised more than once for his words. I can only ask you (and others) to reflect again on how the resolution of these issues, or failure to resolve them, will impact on the article's future. SteveWoolf (talk) 03:46, 29 June 2010 (UTC)


 * Steve - we all have a right to draw our lines in the sand to defend our own needs. No one can "restrain" Steelpillow/Guy except himself. He knows he has hurt you, and also has a vision to defend, and ideally that wouldn't be a conflict, but a chance to learn. But you seem to need him to "completely back off" to feel safe to continue. Maybe you could suggest to him how you are able to collaborate now, and he can tone back his criticism, constructive or not. I know more about polytope than abstract ones, but contents look greatly improves since you started. Thank you! Good luck. Tom Ruen (talk) 05:02, 29 June 2010 (UTC)


 * Thanks Tom for the very kind words, and I really feel much better for having read them. You are right that differences of opinion can be a chance to learn and grow. I guess I could try to be less sensitive, and respond only to constructive remarks, ignoring others. I would be willing to try again if Guy could at least give some credit for what he thinks I have done right, instead of only saying what he doesn't like, and take more care to understand and think through issues before responding. SteveWoolf (talk) 01:14, 30 June 2010 (UTC)

Many thanks for the constructive intervention by so many of you. You have said much that I was not in a position to say. I hope Steve will take a positive view of the context for my remarks about being driven to distraction - I was awarding him a well-deserved barnstar. Keep at it Steve, Wikipedia needs you. &mdash; Cheers, Steelpillow (Talk) 08:22, 30 June 2010 (UTC)


 * Thankyou, Guy. And I would like to keep at it, as I have lots of good ideas for the article. I will archive the recent disputes as soon as I have time, and replace them with a summary of agreed AP content and any ongoing AP discussion. (I think I should do that, to avoid any suggestion of censorship falling on anyone else - even tho' the "material" quite likely doesn't fall within Wiki guidelines!). Again, may I thank everyone for putting mathematics, common sense, and camraderie above all else. SteveWoolf (talk) 13:14, 1 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)

Recent Archive
While there was clearly a need for the discussion to be selectively archived, why was it done in such an undemocratic and sledgehammer fashion? If any of the regular contributors to this article are responsible, or if any of the users MiszaBot, Kslotte and Airplaneman are their aliases, would they please show good faith by coming clean and so stating. SteveWoolf (talk) 01:53, 30 June 2010 (UTC)


 * I think it would be a good idea now to archive everything (except the sticky Standards & Defns section) and then each of us can copy back anything that they consider worthy. Any dissenters? I'll go ahead in a few days if not. SteveWoolf (talk) 20:28, 3 July 2010 (UTC)