Talk:Abstract polytope/Archive 2

Improved Definition?
I offer a slightly more formal definition, using standard Order theory terms, that avoids the redundancy of requiring the rank function within the definition.

An abstract n-polytope is a partially-ordered set (poset) P, whose elements we call faces, satisfying:


 * (P1) P is bounded, i.e. it has a unique minimal face and a unique maximal face (usually, but not necessarily distinct)


 * (P2) Every maximal chain (which we call a flag) has exactly n + 2 faces.


 * (P3) It is strongly connected, that is, any flag can be "changed" into any other by "changing" just one face at a time.


 * (P4) If faces F, G, H satisfy F&raquo;G&raquo;H, then there is precisely one other face G'≠G such that F&raquo;G'&raquo;H, where &raquo; means "covers" or "succeeds": x covers z if x>z and no y satisfies x>y>z.

Then, we can define the rank or dimension of a face F as z - 2, where z is the number of elements in a "vertical" chain from the minimal element to F. A face of dimension k is a k-face.

Yes/No/Trivial ???SteveWoolf (talk) 07:23, 2 November 2008 (UTC)


 * Trivially equivalent (mathematically) to the "standard" definition(s), but it uses terminology that is unfamiliar to or conflicts with that used in the AP literature. (I particularly don't like "covers". mike40033 (talk) 05:12, 3 November 2008 (UTC)
 * Thanks for responding. I intentionally used standard order-theory terms. After all, the widely used poset concept belongs to that field. I never liked "covers" much also but it is standard in order theory (see Hasse diagram and Covering relation), and less of a mouthful than "is an [immediate] successor of". What about "succeeds"? Then my axiom P4 reads


 * (P4) If faces F, G, H satisfy F&raquo;G&raquo;H, then there is precisely one other face G'≠G such that F&raquo;G'&raquo;H, where &raquo; means "succeeds": x succeeds z if x>z and no y satisfies x>y>z.


 * Q1: Do you agree that defining rank within the AP defn is redundant?
 * so it seems. mike40033 (talk) 05:35, 5 November 2008 (UTC)
 * Q2: As a general principle, are we constrained to stick with published material regardless, right or wrong?
 * no mike40033 (talk) 05:35, 5 November 2008 (UTC)
 * Sadly, we are indeed constrained. See wp:or which is the Wikipedia policy which forbids original research. (hey, please don't shoot the messenger). -- Cheers, Steelpillow (Talk) 21:13, 5 November 2008 (UTC)
 * The ayes have it then, unless some precincts haven't reported. Or rather, they would have if WP was a democracy. But does WP want to leave patently false info even if referenced? Tho I admit this particular topic doesn't commit that sin.SteveWoolf (talk) 06:32, 6 November 2008 (UTC)


 * If Q1 Yes and Q2 No then surely it is better maths to define it independently?
 * as long as we don't do original research mike40033 (talk) 05:35, 5 November 2008 (UTC)

Are not all of us unreservedly united in the view that the definition should be as clear and concise as possible? And familiarity, while comfortable, can often be the antithesis of progress.
 * yes, well, me anyway. This doesn't mean we should use the terminology of Order Theory though. In some contexts, no set of mathematical jargon is any more clear intrinsically than any other. mike40033 (talk) 05:35, 5 November 2008 (UTC)

Looking foreward to the next thrilling episode....SteveWoolf (talk) 06:06, 3 November 2008 (UTC)

Serious error in the article! (Now fixed)
Well I'm gonna have egg on my face if I'm wrong about this, but the statement that the poset (of an AP) is a lattice is incorrect! "Nasty" pols such as the digon or hemicube do NOT satisfy the lattice requirement that every pair of faces has a meet and a join. Remember, a join must be a Least upper bound and therefore unique. If the two vertices of a digon are a,b then the two 1-faces and the 2-face are all Upper Bounds of a and b, but none of them is a least upper bound - there is no join, and poset is NOT a lattice.

Of course, many nice polytopes DO have posets that are lattices, but having defined AP we must remain consistent. SteveWoolf (talk) 06:55, 3 November 2008 (UTC)


 * Fixed! Feel free to express it better.... mike40033 (talk) 07:52, 3 November 2008 (UTC)
 * Done - took great care to preserve what was there, but present it more clearly - hope it pleases all. SteveWoolf (talk) 07:19, 4 November 2008 (UTC)

Rigorous definition of Nice
(I will relegate this item to my talk page shortly)

¶Defn 7.3.1. Nice is an intentionally vague word, not only completely unoriginal but also illegally plagiarised, self-referencing (see the "Rigorous definition of Nice" section outside these paretheses), both consistent and self-contradictory, which can be recursively defined to mean slightly more pleasing than slightly less than nice, to the precise extent that pleases everyone to the point that no-one would question it.

It is an acronym for the circular "Nice If Completely Elegant", and is found in ancient Wikipedia scripts dating back more than 3.14159 picoseconds.

A famous town in France was named after it, with the benefit of foresight and rewriting history, much to the chagrin of the EnglishLanguageophobes. Nasty is defined dually.

Lemma 7.3.2. It follows immediately from the definition that you cannot question it (and doing so might cause a universe-destroying paradox).

Note that a given polytope, and even some not given but misappropriated, might be nice or nasty at different times, depending on many factors, such as the campaign funds used in promoting them, the state of the economy, the quality of one's spectacles, the fashions of the time, which side of the bed you got up from, - but most of all the amount of tea, beer or other mind-destroying substances in the brain of the beholder.

Notable Nice Polytopes. (Please subscribe to Mathematics-For-Profit to read further).


 * ADVERTISEMENT

Mathematical truths are eternal; however, for a brown paper envelope containing used, non-consecutive banknotes from countries whose economic fundamentals are sound, you too can have your favourite really, really nasty polytope reclassified in my time-honoured, greatly esteemed definition as being nice. Complete discretion.

Ok, back to work.... SteveWoolf (talk) 14:17, 6 November 2008 (UTC)


 * This old crumbly suddenly recalls a term he used a few years ago to distinguish "tidy" polyhedra. Tidiness is defined as our intuitive ideas about what is allowed for real geometric polyhedra, and therefore has no rigorous mathematical formulation. This is pretty much equivalent to the above mathematical properties of niceness, with the exception of cause-and-effect time reversal. -- Cheers, Steelpillow (Talk) 08:53, 7 November 2008 (UTC)


 * I have been thinking about intuition vs cold precise logic. I think intuition gives a feeling of where to go, but then we must use careful precision to evaluate the place we reach. My intuitions led me with almost no outside influence into the AP world, and to many concrete results therein. So I trust intuition while always questioning it. Right now I'm asking myself if my negative intuitions about some polytopes are justified or simply a result of having been brainwashed, or rather, braindamaged at school by Euclidean propaganda. But I did like my maths teachers - all - probably why I am here now!

Speaking of cause and effect: they are now claiming a measured strong correlation bewteen watching sexy soaps and getting pregnant means the programmes erode the morals. While this might be true, it might also be that people with loose morals, or with time for love are more likely to watch sexy soaps. Or, much more likely, that advances in polytope theory cause both.

SteveWoolf (talk) 13:09, 7 November 2008 (UTC)

Re: Hemi-polyhedron Picture Style
The Hemicube picture style does not match those of the other hemi-polyhedrons in their separate articles, though I appreciate it attempts to explain its genesis from a cube. I have drawn an alternative which I think is clearer (poor resolution, I'm no graphics expert). Vertices are traditionally labelled with letters, edges with numbers, 2-faces (optionally) with Romans. I personally would prefer to see all the hemi pictures follow this standard, I'm not real crazy about the multiple chevrons, which get a bit unwieldy. Different coloured vertices help, but letters are clearest. This style makes it easy to deduce the full i-face set and its incidences.SteveWoolf (talk) 16:28, 7 November 2008 (UTC)


 * I made the simple images for the hemi-forms, glad if they can be replaced with a consistent standard. Just make sure to use lossless PNG, not JPEG. And I guess the value of the arrows mainly is clarity of orientation, but that can be shown by labeled vertices. Tom Ruen (talk) 18:37, 7 November 2008 (UTC)
 * [[Image:Hemioctahedron.png|100px]] [[Image:Hemi-dodecahedron.png|100px]] [[Image:Hemi-icosahedron.png|100px]]


 * Thanks for the feedback Tom. Can you enlighten me about the need for orientation? Is that the same concept as "orientable" polytope about which I know nothing - yet? Regards SteveWoolf (talk) 19:42, 7 November 2008 (UTC)

My Recent Changes
The article needs to be more accessible to Joe SixPack - and Sarah Palin, of course. I hope you will all agree that my latest edits are a step in that direction. There is a long way to go - I think we need a user-friendly Motivation section first, and that the Definition, crucially important tho' it is, should go after the reader has been "prepared" for the cataclysm of the definition's incomprehensibility without a 20-year orientation course. Makes Relativity seem like a children's story.

Comment away....SteveWoolf (talk) 08:37, 8 November 2008 (UTC)


 * I think you mean Joe ThePlumber? Not sure about a "motivation" section. Perhaps the Definition section, or maybe the article lead, could do with a gentle introduction to abstraction and partially-ordered sets before diving into the brain-tearing stuff. A History section could then discuss the motivation for, and evolution of, AP theory. Much recent work remains unpublished (Johnson is not the only example), and while I have cautioned against original research I do wonder whether any of the OR police would be capable of spotting a certain amount of creativity in making APs understandable (not as high profile as writing about aircraft {shudder}). -- Cheers, Steelpillow (Talk) 10:09, 8 November 2008 (UTC)


 * perhaps this plumber... mike40033 (talk) 01:52, 10 November 2008 (UTC)]
 * This sentence haz two errors.
 * Diplomacy is the art of telling someone to go to hell in such a way as they look foreward to the trip.

Incidences

 * Incidences - I'm not 100% sure I defined them correctly in my new article section. For example, in a triangle abc, by my definition, the two edges ab and bc are not incident because neither one contains the other. But clearly they intersect at a point and are incident in that sense. It seems incidence has a different meaning in combinatorics, though.


 * I am trying to clarify "incidence geometry" that the article mentioned. If one of you corrects/deletes this, please do so constructively. Reverting will remove other more recent good stuff!  Thanks SteveWoolf (talk) 17:53, 8 November 2008 (UTC)

Ok - I got it right. See Abstract Regular Polytopes p22 SteveWoolf (talk) 18:21, 13 November 2008 (UTC)

Latest Article Updates
I hope all my recent updates will meet with approval. I have tried to stay within the consensus reached in the discussion page. I have tried not to remove any worthwhile information, merely to organise it more logically and present it more understandably. It was a lot of work, and I'd greatly both appreciate both careful reading and discussion here before axing recklessly. I sincerely hope it will be received as a clearer and more complete introduction to our subject, but if it can be improved further nothing would please me more. SteveWoolf (talk) 18:44, 12 November 2008 (UTC)

'z' vs 's'
I saw, in the article, the word 'realization', and in the next paragraph 'realisation'. Much as I prefer Old World spelling, most research in abstract polytopes now takes place in the New World. So I changed the 's' to a 'z'. I propose we use American spelling as the standard for this article. Votes? mike40033 (talk) 00:48, 9 December 2008 (UTC)


 * Yes, absolutely, ARP uzez the Zed spelling, and that should probably clinch the argument, as adopting ARP as our de facto standard will save us from interminable mess of the "classical" "polytope" world, where it seems there iz no consensus. I personally don't think we have to worry unduly about non-technical words, though, e.g. centre, disc, summarise - but if any Puritan wants to Americanise those too, I'm sure Shakespeare won't mind. We don't have worry here, I think. SteveWoolf (talk) 05:35, 9 December 2008 (UTC)


 * Agreed. The Old World spelling is probably mine, oh dear - at least it may help the Wiki Police feel wanted. -- Cheers, Steelpillow (Talk) 14:03, 9 December 2008 (UTC)


 * Actually, I think it was mine. And should that be Wiki Polize? mike40033 (talk) 03:44, 10 December 2008 (UTC)

Term for Non-Abstract Polytopes
I am unhappy about the description "classical". It is not used in the literature, except perhaps to indicate those polyhedra known to Classical civilisations such as ancient Greece. I get the feeling that it is a neologism made up by one of our contributors - if so then it is also against Wikipedia policy. I suggest that we stick to the usual distinction made between "abstract" and "geometric". Some minor changes of argument will be needed, but nothing drastic. -- Cheers, Steelpillow (Talk) 12:48, 8 December 2008 (UTC)


 * Guy, you can read some of ARP at the site below, in fact, all of the relevant bits. If you search for "classical" you will see that this is M-S's pretty much standard terminology for "non-abstract". On the other hand, abstract polytopes are considered, at least by Wikipedia, to be a part of geometry, in my view correctly, and therefore the term "geometric" does not imply non-abstract (unless you also attach the word "traditional").


 * http://books.google.com/books?id=JfmlMYe6MJgC&pg=PA22&vq=classical&dq=%22abstract+regular+polytopes%22&output=html&source=gbs_search_s&cad=0


 * Regards SteveWoolf (talk) 13:28, 8 December 2008 (UTC)

Fair enough. However I think it would be useful to explain what a "classical" polyhedron is. -- Cheers, Steelpillow (Talk) 18:17, 8 December 2008 (UTC)

Actually I disagree. Abstract polytopes are a part of algebraic set theory. It is only in seeking realisations that we enter the realm of geometry. There is a historical similarity with the way that "symmetry" began as a property of polytopes, but moved on to embrace particle physics and other abstract things with no "metry" (metric) to be "sym" (similar), including partially-ordered sets. I don't know where Wikipedia took its lead from, maybe I'm just a lone rebel again. -- Cheers, Steelpillow (Talk) 18:28, 8 December 2008 (UTC)


 * Well, now we're into "What is geometry?"! The word means "Measurement of the Earth". But we have many modern "Geometries" that clearly do not conform to this narrow definition, so clearly that is no longer relevant. The fact that (geometric) concepts can be represented algebraically does not make the concepts non-geometric. Even traditional geometry can be 100% represented by algebra, i.e Cartesian geometry. with points in n-space becoming n-tuples (x,y,z,...). Abstract polytope theory, like topology, may look very algebraic, but it is fundamentally and primarily concerned with points (vertices), lines (edges), planes (faces), solids (cells) and so on.  I think these concepts are essentially what modern geometry is about. You can of course distinguish between classical and modern (or abstract) geometry, where "classical" of course means Euclidean.


 * So I really think that to describe non-abstract polytopes as geometric is a bit like calling the integers "Numeric" quantities to distinguish them from fractions and decimals, when they are all numbers. Or, a bit less trivially, like calling the Reals "Numeric" in order to exclude Complex Numbers.


 * In defense of "Classical": Sometimes this word does refer specifically to the Greek and Roman eras, especially in the context of Art, Architecture, and History. But as an English word, it more generally means just "old" or "traditional" as in "classical music", "the classical approach", or "classical theory". Since our Abstract theory is very new, I think "Classical" is the right antithesis.


 * Hope you will reflect on this; meanwhile, anyone else care to agree/ disagree? I do think we need to reach a consensus on what to call non-abstract pols, and please - not non-abstract!!! SteveWoolf (talk) 06:07, 9 December 2008 (UTC)

Any synthetic geometry can be formulated as a symbolic logic, i.e. as a separate branch of abstract logic from mathematical algebra (which is the most commonly studied branch). Simply because an idea has its roots in geometry, does not mean that it is merely a part of geometry - geometry may be a part of it, or they may just overlap a bit. Food in turn for your reflection, and maybe even rotation or inversion. -- Cheers, Steelpillow (Talk) 14:23, 9 December 2008 (UTC)

Well, no U-turn or conversion on my part, I still say abstract polytope theory is a branch of geometry. Our objects don't require a space to contain them; however, every polytope is a space in a meaningful sense. It is, for example, trivially easy to define the distance between two points (vertices), and therefore a "straight" line. A cubic honeycomb is very similar to Euclidean 3-space, the main difference being that you can't go continuously from one vertex to the next, only by quantum jumps. And if abstract polytopes are geometric objects, then the term "geometric polytopes" is no more helpful than "numeric numbers".

I really think we MUST reach a consensus here and soon, so that our articles make sense, i.e. on


 *  Whether "non-abstract" polytopes should, in abstract polytope articles, be called "Classical" or "Geometric". 

It seems clear to me that McMullen/Shulte in ARP prefer classical, and that Wikipedia considers abstract polytopes and other abstract objects as geometry. Would Mike and CunningGabe, or others, care to comment on this so that we can clean up? SteveWoolf (talk) 08:00, 10 December 2008 (UTC)


 * I would certainly prefer that we don't use "Geometric" to mean "non-abstract" in this sense. On the other hand, I see why it's kind of funny to use "Classical" for some of the weird higher-dimensional polytopes that are less than a century old. I'd suggest "Concrete", except that it's probably not our place in Wikipedia to introduce a new term like that. So I'll throw my weight behind "Classical". -CunningGabe (talk) 14:49, 10 December 2008 (UTC)

Review of alternatives
The idea of discontinuousness which SteveWoolf discusses leads to the well-established field of incidence geometry and the associated discrete geometries of embedded structures. It is certainly possible to construct polytopes in these spaces, but the spaces themselves are not usually regarded as polytopes. Rather, a polytope may be treated as a piecewise decomposition of some (discrete or continuous) space. But he is right in calling these combinatorial spaces geometries, even where no physical manifestation is being considered. So I agree that "geometric" is not a good contrast to "abstract".

For reasons such as those put forward by by CunningGabe or myself, and also recalling non-classical oddities such as Petrie polygons, spherical polyhedra and hollow-faced polyhedra, the term "Classical" is not a good choice either.

Unfortunately for "concrete", this is already used to denote geometries, and figures within them, which have an established metric, such as Euclidean geometry.

Various other promising terms, such as "plastic", also have established usages.

In my own unpublished OR I use the term "morphic" to describe geometries and figures which have some physical aspect of points, lines and planes, but no metric. One can then contrast abstract polytopes with morphic and concrete varieties. But OR is not allowed here.

In summary, there is no satisfactory choice, so we must make an unsatisfactory one. "Ordinary" and "traditional" are ordinary (sic) words, free of any mathematical niceties, and have both been used in other contexts. My choice would be "traditional".

-- Cheers, Steelpillow (Talk) 19:58, 10 December 2008 (UTC)


 * I, too, like the choice "traditional", better than either "classical" or "geometric". -CunningGabe (talk) 03:44, 11 December 2008 (UTC)


 * I second "traditional".&mdash;Tetracube (talk) 17:57, 11 December 2008 (UTC)

Appreciate your responses, guys. I would like to suggest that we wait at least for Mike to weigh in here, and TomRuen if he'd like to vote - and any other closet APists (Sarah Palin?) lurking in the wings! If CunningGabe could get an opinion from Egon Schulte that would be interesting. I have written to Peter McMullen, and if he finds time to respond I will copy his reply here.


 * Weighing in.... with a new section below.... maybe completely off-track though...... mike40033 (talk) 08:11, 12 December 2008 (UTC)

So now it is between "Classical" and "Traditional". But should we also consider "Euclidean"??? - it is nearer to SteelPillow's "Geometric" preference, and a lot more meaningful than either "Classical" or "Traditional".


 * Re: Euclidean unfortunately, Euclid's polytopes are now considered spherical by all good topologists. :-( mike40033 (talk) 08:14, 12 December 2008 (UTC)

Contrary to my previous assertions, I now find that ARP (Blessed be its Name) uses both - I haven't done a frequency count. On the other hand, a Google search on "Classical polytope" -Wikipedia produced 114 results, vs only 2 results for "Traditional polytope". However, I haven't read the google links to see what "Classical" is used to mean.

"Classical" has one less syllable! SteveWoolf (talk) 07:29, 12 December 2008 (UTC)

But... Why?
Publications on abstract polytopes tend to refer to "The classical theory" without defining clearly what that is. In fact, the entire history of "the classical theory" has been one of progressively widening the boundaries of what objects should be studied - from 3D platonics, to archimedian, to kepler-poinsot, to 4-D, to tesselations, to apeirotopes, petrie polygons, polytopes with non-planar faces, to simplicial complexes, ARP's, geometries and buildings, projective planes, lattices, eulerian posets and more. Now we are debating exactly what the "classical" theory "means". I'd like to ask a different question. What are we trying to accomplish here, exactly? Under what circumstances is it really important or necessary to pin down exactly what is meant by the "classical theory"? mike40033 (talk) 08:11, 12 December 2008 (UTC)


 * Hmm, it seems I was indeed off-track. Ok, I vote for classical, with traditional an acceptable alternative. mike40033 (talk) 08:15, 12 December 2008 (UTC)

Consensus???
Okay, we now have


 * CLASSICAL....... Steve, Mike, Quick GoogleSearch


 * TRADITIONAL..... CunningGabe, SteelPillow, TetraCube


 * SCHIZOPHRENIC... ARP (no disrespect intended, I just mean "variable"; but perhaps ARP's differing terms actually have different meanings)

Based on my "research" and writing this morning, does anyone wish to reconsider? We actually have THREE possibilities now, the third is SCHIZOPHRENIC, i.e. each of us can choose either CLASSICAL or TRADITIONAL according to whim. But I really feel ONE consistent term would be best. We could probably afford a few days for possible votes from ES/PM or other lesser mortals.

If the Traditionalists feel that they have won 3:2 and that I am stalling, by all means edit away, but since I wrote to McMullen I'd like to await the reply!

Makes the US election seem like a sideshow. SteveWoolf (talk) 08:57, 12 December 2008 (UTC)


 * To quote (more or less) from Lewis Carroll: "'When I use a word"', said Humpty Dumpty, 'It means whatever I want it to mean.'" Anyway, after examining those Google stats a bit more closely, many of the "classical" hits are scrapings from Wikipedia, leaving around 47 hits on genuine usage, while neither of the "traditional" ones refers to traditional polytopes as such. The score is therefore 47-0 in favor of "classical". This established usage forces me to U-turn and transfer my vote to "classical". Meanwhile, waiting for McMullen can do no harm. -- Cheers, Steelpillow (Talk) 15:01, 12 December 2008 (UTC)


 * I very much appreciate the time you took to follow the google links, and your willingness to do the right thing - but just when we seemed to have it settled, I received this from Peter McMullen:


 * Dear Steve,


 * In my recent usage, I have reserved "classical" for the regular polytopes of (for example) Coxeter's book "Regular Polytopes". I suppose that these are also the "traditional" regular polytopes.  More generally, I have been calling a realization of an abstract polytope a "geometric" polytope.  Thus the main contrast is between "abstract" and "geometric" polytopes.  Of course, the latter (regular, chiral and even more general objects such as incidence complexes) are often investigated in their own right, particularly in a dimension-by-dimension classification.


 * I hope that this clarifies my viewpoint. I imagine that Egon Schulte will have his opinions, but I would be surprised if they differ very much from mine.


 * With best regards -  Peter.

It seems, then, we should go with "Geometric", with the proviso that every significant article that uses this term should spell out that it means non-abstract. Actually, this logic seems to point to non-abstract as by far the most logical term, but we are aesthetic animals, not robots, so we must have elegant words!

Let us wait a day or two for any last minute views, after which I suggest we adopt "Geometric", mentioning that other authors use "Classical" and "Traditional". SteveWoolf (talk) 15:48, 12 December 2008 (UTC)


 * From what McMullen said, it seems that "classical" is the sense intended in this article in such places as where the reader is cautioned not to blindly carry over things from the classical realm, since they do not always match the abstract definition. If we are to follow McMullen's definitions (and I lean in that direction since most of our references are from him), then "geometric" should only be applied to realizations of abstract polytopes. To elaborate, a geometric polytope would include the classical polytopes, but they can possibly be a superset, since the definition of an abstract polytope allows for structures that were not included in the classical definition. I guess this makes me a schizophrenic. :-)&mdash;Tetracube (talk) 19:30, 12 December 2008 (UTC)


 * Googling "geometric polytope", it seems about as popular as "classical polytope", so I am happy with either. Trying to draw fine distinctions between "geometric" and "classical" would require defining both in mathematical terms, either of which we now know to be a fool's errand. What's the trigonal equivalent of a U-turn? For somewhat updated reasons, I too return to my original suggestion of "geometric", with a remark along the lines of, "geometric polytopes, also called traditional or classical". I had better not mention that Johnson uses the term "real" (implying a polytope in some real geometric space, as opposed to complex or abstract spaces) and this also scores quite well with Google. -- Cheers, Steelpillow (Talk) 17:31, 13 December 2008 (UTC)


 * I can live with Geometric although I don't like it. After all, I used it in my dissertation (to contrast them with Combinatorial polytopes, by which I mean Abstract). Then again, I did have a whole chapter explaining what a Geometric polytope was.
 * By the way, I haven't checked the google references, but the term real polytope will sometimes be used to mean a polytope with real-valued coordinates - to distinguish it from a rational polytope. mike40033 (talk) 00:41, 15 December 2008 (UTC)

"Abstract Regular Polytopes"
After searching (my incomplete) ARP, some surprises:


 * Geometric: I found not one single reference to Geometric Polytopes.


 * Traditional: There are a few references to Traditional polytopes and many, many to Traditional theory.


 * Classical: This term seems to be a bit more restrictive - I'm not sure - and might not encompass all non-abstract polytopes.

So - ARP itself doesn't seem to jive with Dr McMullen's letter to me!! (Is this internet fraud, with our Great Leader held in some ghastly dungeon somewhere until he recants his heresies?)


 * AIUI, McMullen has changed his habits since co-writing ARP, and now uses "geometric".
 * That would make sense, given his letter. Bye the way, I only just noticed this and all these other edits of yours, which I shall now digest and reflect upon. (Yes, I know, prepositions are not good to end sentences, like the one you're looking, with the attention it's worthy of, at, with.) - Stellpillow signed after the event, sorry.


 * Shouldn't you sign each edit when it's inserted into my text, so it's clear who wrote what? Also, I managed to figure out AIUI but SWWEITEW? (Shouldn't we write English in the English Wikipedia?) Arrivederci, au revoir SteveWoolf (talk) 13:44, 14 December 2008 (UTC)
 * Sorry I forgot to sign. Put it down to old age. AIUI is pretty commonplace IMHO. It just saves so much keyboard-bashing. -- Cheers, Steelpillow (Talk) 20:59, 14 December 2008 (UTC)


 * LOL mike40033 (talk) 00:43, 15 December 2008 (UTC)

Other Options

 * Real: The strongest objection to this is Coxeter's interest in Complex polytopes, so I suspect "Real" often means "Not Complex". Secondly, it has too many English connotations, and I don't think we want to view Abstract Polytopes as either "Unreal" or "Surreal" (or Fake!)
 * "Real" indeed refers to real-number coordinate spaces as opposed to complex coordinate spaces. Besides Euclidean these include hyperbolic, projective and other more exotic metric spaces. In the context of polytopes, its English connotations are precisely those which led to its choice in the established context of non-imaginary numbers, so this is not so much the "strongest objection" as the main selling point. I have no problem living with this aspect. On closer examination, "complex polytopes" turn out to be mathematical configurations of points, lines, planes and so on, and not at all the kind of closed surfaces that we usually understand as "polytopes" - it is something of a misnomer IMHO, unless one adopts the late-19th/early 20th century ability to ignore whether an edge is a line or a segment, a face is a plane or a bounded region. Anyway, the significant point here is that the abstract structures of complex polytopes are a much wider set than those of real polytopes, which they include as a more or less degenerate subset (some authorities do not even regard real polytopes structures as valid configurations, analogous to the way that some deny that digons and dihedra are valid real/geometric/classical/traditional polytopes). One may thus distinguish between "abstract real polytopes" and "abstract complex polytopes". All of which helps explain why I should never have mentioned "real" in the first place - the wooden spoon just kind of jumped into my hand, Sir. - Stellpillow signed after the event, sorry.


 * Didn't really follow all of that, but no matter, the conclusion is fine! Where does the clever wooden spoon quote come from? Couldn't find it on Google. SteveWoolf (talk) 14:01, 14 December 2008 (UTC)
 * The "it jumped into my hand" is an old Army or workman's excuse for picking up something they shouldn't - in my case it happened to be the traditional tool of the stirrer. -- Cheers, Steelpillow (Talk) 20:59, 14 December 2008 (UTC)


 * Cartesian: As far as I know, this term has never been used, which, like a virgin, probably puts it strictly off-limits! It does, however, seem to capture the essence of non-abstract polytopes with its "coordinates" (x, y, z, ...) view of things contrasting nicely with the AP combinatorial dogma. A most elegant term, in my view, though I am not sure if ALL non-abstract polytopes exist in spaces that have coordinates. I guess this sweet virgin can only be taken by a prince, and not us lesser AP mortals.
 * Even in Euclidean space one may wish to use polar coordinates or some other non-Cartesian system. Real projective spaces require the use of homogeneous coordinates rather than Cartesian, while affine space has only a partial metric (i.e. only of linear distance in a given direction, with no measurement of angles), and general projective space has no coordinate system at all. This truly is a non-starter - to mix metaphors, the lady is a tramp. - Stellpillow signed after the event, sorry.


 * Non-abstract: All this desperate soul-searching for an opposite of "Abstract" that is beyond misinterpretation, and we had it all the time! This is clearly the most logical term. But - it's lack of elegance might make the articles less readable. Worse, it will be less intelligible to a wider audience not familiar with Abstract Polytopes, and will inevitably provoke endless "What do you mean, exactly, by non-abstract?" questions which will then force us back to one of the other terms to explain our opacity.
 * To me, "non-abstract" implies that the properties of abstract polytopes do not apply. This is false. - Stellpillow signed after the event, sorry.


 * That can't be right. An irregular quadrilateral still has many properties in common with a square - 4 sides and angles, an angle sum of 360 degrees etc. So a non-abstract pol can still have many of the same properties, just not the defining one (i.e. being ONLY a special poset and nothing else). Still, for me "non-abstract" is out, tho' for different reasons. SteveWoolf (talk) 14:01, 14 December 2008 (UTC)

Conclusion
In view of the latest revelations, I am now once more against "Geometric" on the grounds that Abstract polytopes are also geometric objects (though of a different sort), which means that the term "Geometric Polytope" will always be unclear and will constantly need to be accompanied by a verbose definition of the form (i.e. "Traditional" or "Classical" or "In Euclidean space"). So we will end up with very cluttered articles. Furthermore, we occasionally need to contrast modern or abstract geometry with traditional geometry, as I did in Abstract polytope; I hardly think we want to talk about "Geometric" geometry!!!

I very definitely feel that Traditional is the clear winner, since:


 * ARP uses it extensively, and more than any other term


 * It is clear enough to be understandable by most readers


 * It is just vague enough not to cause us to make questionable statements when we use it


 * We can refer to "Traditional" geometry also.

I suggest allowing a few days more for any further opinions or research on this matter, before any extensive editing is undertaken.

I look foreward to all your responses, but silence implying consent is fine! Regards SteveWoolf (talk) 05:10, 14 December 2008 (UTC)


 * rather trivial
 * "One man's trivia is another man's ulimate truth". Will Global warming matter when the Andromeda galaxy collides with our Milky Way? (It will). You're right though, I would rather be finding out how many nasty polytopes there are in 27 dimensions.SteveWoolf (talk) 13:07, 14 December 2008 (UTC)


 * While the niceties of "abstract geometry" may cause heartache, one can mount an equally (in)valid counter-argument that abstract polytopes are strictly a branch of set theory, which is not geometry: we do not enter the realm of geometry until we seek to realize them. Either way, the tacit assumption in this article that "geometric" is to be taken as meaning "existing in some physical geometric space", would not confuse the reader any more than any other of the options we have been discussing. I remain happiest with "geometric", with "traditional" a close second. In fact, we could introduce them with a phrase along the lines of, "abstract polytopes capture the combinatorial properties of traditional geometric (or classical) polytopes in the form of a partially-ordered set blah, blah, ... ," and then use either "traditional" or "geometric" to taste, whichever is most readable at that point. -- Cheers, Steelpillow (Talk) 08:58, 14 December 2008 (UTC)


 * Have considered your words with microscopic care, hope you'll do the same!


 * You say again that Abstract Polytopes are not geometric objects. However, your arguments defending this view do not stand up to careful scrutiny:


 * (1) Virtually all mathematicians of stature today use set theory as a foundation, but often only when they need to formalise it for absolute rigour. It is entirely incorrect to conclude that this makes (e.g.) abstract polytopes a branch of set theory. Noone would say that Computer Science is a branch of electronics - it is NOT - or that Chemistry is a branch of Nuclear Physics. Related - yes.


 * (2) The "tacit assumption that geometric [means] existing in some physical geometric space" is a very shaky one.  In what sense are Euclidean 5-space, Riemannian and Hyperbolic spaces to name a few, a "physical geometric space"? We need a term that is the opposite of abstract, and excludes ALL non-abstract objects.


 * Claiming that Abstract Polytopes are not geometric objects is, at best, merely an opinion, and at worst, false. It seems absurd to me to say that an object is not geometric when it has vertices, edges, faces,... and which has properties such as spherical, toroidal, hyperbolic and many, many others. AP theory is just one more of many different kinds of modern abstract geometry.


 * In fact, AP's don't fit very well anywhere in the MSC 2000, but articles on AP usually get slotted into 51, 52B or 52C somewhere. mike40033 (talk) 07:13, 17 December 2008 (UTC)


 * If we have to explain our choice of term all over the place with a verbose phrase to define it, that seems to suggest (scream?) "not a good choice"! I feel using "Traditional" would almost never need explanation, or at most once at the top of the main article.


 * Using either according to whim? Don't we want to to have a nice precise subject and avoid the mess of non-abstract polytope theory? ONE term would be best, surely.


 * I really feel that the four arguments at the top of this section in favour of "Traditional" are very compelling, plus what I just added about our not needing to often define it. If, as you say, "Traditional" was a close 2nd for you, I hope these arguments might find a few undecided cells in your brain! Whereas for me, "Geometric" is most unsatisfactory, though I was prepared to accept it, until I found that ARP doesn't use it.


 * Hope we can resolve this before HyperAbstract polytopes are invented and AP's become classical. Season's Greetings. SteveWoolf (talk) 13:07, 14 December 2008 (UTC)


 * I will only repeat that I regard our respective cases as "equally (in)valid", and that McMullen nowadays prefers "geometric". -- Cheers, Steelpillow (Talk) 20:59, 14 December 2008 (UTC)

Mike "doesn't like Geometric but could live with it", which is pretty much how I feel also. Your statement about McMullen's recent preference for Geometric, which his letter confirms, is relevant. Not 100% of a clincher because I had hoped (as agreed) that ARP would be our standard, but if God keeps rewriting the Bible, where is our Rock, our Centre? Our castle will be on shifting sand. Well, okay, I guess nothing stands still forever. Let us give the rest of the gang time to reconsider, re-opinionate, and revote (or re-yawn) as they wish for a few days. Then perhaps we can do a final vote count again, implement the outcome, and get back to counting how many angels fit on a pinhead in 27 dimensions. SteveWoolf (talk) 04:24, 15 December 2008 (UTC)

Tradition

 * Who says a face must have a ring of edges,
 * Each pair of them a vertex,
 * slotted in between?
 * And who says the edges
 * always are coplanar,
 * and that we know what that must mean?


 * Tradition!


 * There is a book I read
 * By Imre Lakatos
 * I sighed and shook my head -
 * it left me at a loss..


 * Tradition!

mike40033 (talk) 06:58, 17 December 2008 (UTC), with apologies to whoever feels they need them.


 * Is that a poem that you wrote,
 * Or a third-party quote?
 * But Geo' or Tradition?
 * To choose is my mission
 * So is the latter your vote?


 * Because if so, I think "Traditional" seems to win for now at least, until the Gods publish a new Great Work to end all Works. SteveWoolf (talk) 12:04, 17 December 2008 (UTC)


 * Classical > Traditional > Non-abstract > Geometric. Where '>' is the partial order relation 'is preferred over, by me'. mike40033 (talk) 01:28, 18 December 2008 (UTC)

Tea Break
In an LA autorepair shop, I saw this myself:
 * Labor: $50/hr
 * If you wait: $60/hr
 * If you watch: $80/hr
 * If you help: $100/hr.


 * This one is very nice... :-) mike40033 (talk) 01:26, 18 December 2008 (UTC)

On a London Underground Poster depicting Henry VIII buying a round-trip ticket to Tower Hill: Graffitied in, "And a Single (One-Way) for the wife".SteveWoolf (talk) 12:04, 17 December 2008 (UTC)

Sequence of discussion
I have re-ordered and grouped various subsections, to try and improve the flow of the discussion and to make the table of contents easier to navigate. Feel free to disagree/modify/revert. Many thanks for the many improvements to date. -- Cheers, Steelpillow (Talk) 23:25, 22 November 2008 (UTC)
 * Assuming that was in part addressed to me, much appreciated. I think we have a great team going here. SteveWoolf (talk) 11:10, 18 December 2008 (UTC)
 * I will not be upset if you don't thank me for my thanks for your thanks. But just in case you do, or even think it, I may as well, while I'm here anyway, thank you for.... oops lost track. SteveWoolf (talk) 11:10, 18 December 2008 (UTC)

Some Outstanding Issues
(-1) Can the classical polytopes be identified purely within the abstract theory?

(0) Is there an abstract criterion equivalent to realisability?

(1) What precisely is meant by a "flat" polytope?

(2) Is the uniqueness property significant, i.e. whether or not each i-face has unique vertex and facet sets? (From which I think it follows that all j-face incidences are unique, for any j).

(3) Are there any other criteria we might want to consider for classifying polytopes as nice vs nasty?

(4) Could we clarify whether Johnson is offering an alternative definition of Abstract polytope? If so, what is this difference? What precisely is meant by "monal"? (Diadic, I believe, means the "diamond" property - my axiom P4 above)?

Meta-issue: our subject is poorly linked to related maths topics (... but maybe that's good, keeps it safe from the barbarians?)SteveWoolf (talk) 08:10, 2 November 2008 (UTC)

Some outstanding misconceptions, and some answers

 * Better etiquette to merely respond to perceived misconceptions without stating the shortcomings of others. Yes I have made many mistakes, but few of us have made none.SteveWoolf (talk) 22:22, 2 November 2008 (UTC)

(-1) and (0) are basically the same question: a classical polytope is just one which has been realised faithfully in Euclidean space. Any abstract polytope is realizable in any real space of any dimensionality - but not necessarily faithfully. I have amended (0) accordingly.
 * Sorry - I deleted your copy of my o/s issues at first thinking they were just that, accidentally. In any case, in the interests of conciseness, would it not be better just to respond to specific points?
 * (-1) is mike's and I think he meant something different.SteveWoolf (talk) 12:46, 2 November 2008 (UTC)

(1) A "flat" polytope is a realization which exists in some space of zero curvature (e.g. Euclidean space) of the same dimensionality, such that all its bounding facets are also flat. This is usually one of the criteria for faithfulness.


 * No! At least, not if you mean abstract polytope. A "flat" AP is one whose vertex set is the same as the vertex set of a facet. Nothing to do with topology. Sorry! mike40033 (talk) 05:16, 3 November 2008 (UTC)


 * Fair enough. I have no knowledge of "flat" in the context of set theory. -- Cheers, Steelpillow (Talk) 18:00, 3 November 2008 (UTC)

(2) and (4) The uniqueness property is effectively that which Johnson calls monal. Yes, it is highly significant, and is one of the defining properties for an abstract polytope. "Dyadic" (note the spelling) refers to the property of some n-dimensional polytope that each edge connects to just two vertices, each (n-2)-facet to just two (n-1)-faces, and so on. Johnson's formulation is mathematically equivalent to that of Danze, McMullen and Schulte, and quite deliberately so - the only difference being that it is shorter and more understandable. More on all this can be found in the article which I linked to earlier.


 * My uniqueness property does not hold for the digon or hemicube, and therefore cannot be part of an abstract polytope definition that includes these "nasties". We definitely need to be talking the same language here and resolve this.
 * The digon is not monal, so no disagreement there. The faces of the hemicube have different facet (edge) sets, so are monal - but does this not also make them unique? -- Cheers, Steelpillow (Talk) 18:00, 3 November 2008 (UTC)
 * Could you define monal, and give the simplest possible example and non-example. Does it have any meaning for an abstract polytope?SteveWoolf (talk) 12:03, 4 November 2008 (UTC)


 * Dyadic as you define is equivalent to the diamond property (axiom 4 of the definition).
 * It also requires edges to have only two vertices. Johnson describes it thus, "The dyadic property says that every 1-face is incident with just two 0-faces and every (n-2)-face is incident with just two (n-1)-faces." -- Cheers, Steelpillow (Talk) 18:00, 3 November 2008 (UTC)
 * This is provably equivalent to the diamond property (axiom 4 of the definition), and from this axiom it immediately follows that an edge always has two vertices.


 * Is Johnson's alternative really "shorter and more understandable"? If so, why not state it here for consideration. And how can that be consistent with your statement below, namely "I had to re-read Johnson's remarks several times before it all began to make sense, and when I edited them together I found I had still missed stuff"? SteveWoolf (talk) 22:31, 2 November 2008 (UTC)
 * That's easy - McMullen, Schulte and co. write in an abstruse higher mathematical jargon which I hardly understand at all. Is Johnson's shorter? Well, you're not going to take my word for it if you haven't already. You'll just have to compare his, as referenced earlier, with other published works. -- Cheers, Steelpillow (Talk) 18:00, 3 November 2008 (UTC)

(3) There is no "nice vs. nasty" criterion for abstract polytopes: a poset is either an abstract polytope or it is not. This is one of their fundamental attractions - the topology is stripped of all connotations about Euclidean spaces and faithfulness. As far as faithfulness goes, clearly a "nice" polytope can be realised faithfully in the chosen space (which might be Euclidean, hyperbolic, spherical, or something more exotic), while a "nasty" polytope cannot. However, what is nice in one space might be nasty in another. Also, faithfulness means different things to different people - some might allow geometrically coincident though abstractly distinct vertices, or overlapping faces, or whatever, while others might not. There is no single "right" answer. Johnson at least provides a reasonably consistent definition, though it is not to everyone's taste - for example he does not allow coincident vertices while Gr&uuml;nbaum generally does (they both agree on the underlying abstract polytope).


 * My point about nice and nasty polytopes is that, given that abstract polytope is already defined, there are probably some of these that we might consider nicer than others, e.g cube vs hemicube. I don't have a clear intuition right now just where we need to draw such distinctions, only that some pols are clearly better behaved than others. For example, a concrete question might be this: are all polytopes satisfying my unique property realisable? SteveWoolf (talk) 22:08, 2 November 2008 (UTC)
 * Much of the point of abstract polytopes is precisely in order to detach the theory from ideas of niceness or realisability. These ideas have been cluttering up the field for decades and causing nothing but trouble; they are highly subjective and often self-contradictory, and are subject to variations in different spaces. They are also highly sensitive to one's definition of "polytope" - something on which there remain fundamental mathematical disagreements. By all means play with these ideas (I like to), but do not be under any illusion that they are relevant to the theory of abstract polytopes. And if you ever do get "a clear intuition just where we need to draw such distinctions" as you put it, please tell us about it! -- Cheers, Steelpillow (Talk) 18:00, 3 November 2008 (UTC)


 * You seem to have completely missed my point again, which is define in a precise standard way certain properties of AP's so that they are NOT "subjective or often self-contradictory". For someone who devotes so much time (correctly) lamenting the past vagueness of the subject, you seem surprisingly resistant to attempts to formalise it more precisely and unwilling to define your own concepts. As far as I am aware, the AP definition in this article IS that of Mullen-Shulte, and that seems as clear and concise as possible, except that Rank is redundant. If you and Johnson can better it, let's see it here, because I saw no definition in his article based on defined terms, only a rather wooly and lengthy essay that did anything but draw a line between abstract and classical.
 * May I recommend you spend a bit of time on elementary set theory, its very easy and intuitive, will help you to tighten up your ideas, and you'll be glad you did. Virtually all mathematics these days uses this as a foundation. And please refrain from referring to the "misconceptions" and "illusions" of others, at least until you have reached Coxeter or Mullen status. Please, could we constructively discuss and develop ideas in a humble, respectful and non-competitive manner, otherwise Wikipedia's AP section risks being an irrelevant sideshow instead of a cutting-edge oasis of inspiration. Deal?SteveWoolf (talk) 03:49, 4 November 2008 (UTC)


 * No, I think it is you who miss my point. You say that you wish to define in a precise standard way certain properties of AP's so that they are NOT "subjective or often self-contradictory". My point is that the properties of "niceness" which you seek are based on our classical geometric ideas - and these ideas are indeed subjective and often self-contradictory, which is precisely the reason that abstract theory was developed - to distance itself from these ideas that you are trying to force back in. "Niceness" and "precise standard way" are impossible to reconcile, for reasons I have already explained: Johnson has one reasonably self-consistent formulation, Gr&uuml;nbaum another - neither is perfect, but doing better would be a real challenge. As far as Johnson's note goes, bear in mind that it is a mashup of many separate posts during a long conversation in which many of the refinements were still being worked out, and a more rigorous presentation must await his forthcoming book. He is a professional mathematician and polyhedronist of some note, so do not dismiss his ideas because of the presentation.
 * So far as my understanding goes I try to know my limits, to stay within them, and to acknowledge freely when I lose the plot. I am constantly astonished how few self-styled logical thinkers are capable of such self-criticism (And yes, I do have formal qualifications in logic and suchlike). This is not a reflection on your good self, merely a defence of mine.
 * Anyway, I think this ping-pong match has gone on long enough, so I am signing off. -- Cheers, Steelpillow (Talk) 10:23, 4 November 2008 (UTC)


 * What you say is demonstrably false: My definition of (vertex/facet) uniqueness, which you clearly have not understood, is 100% abstract. Whether it's useful, I can't say. Mike's definition of Flat is also purely abstract. Both of these concepts, and no doubt others, precisely subdivide polytopes into different groups, no subjectivity here. But you seem to oppose every attempt at defining terms precisely, while at the same time being unwilling to define your own (eg "monal").


 * If Johnson's article is such a still-being-worked-out mashup [awaiting] a more rigorous presentation (your words) then you cannot be surprised if it is received as such.


 * Progress would be rapid here, I think, if you would discuss ideas more thoughfully instead of discarding the unfamiliar without first understanding.SteveWoolf (talk) 12:34, 4 November 2008 (UTC)


 * Well, if you really think it will help, then I will repeat some things I have already said. Johnson's definition of "monal", and much else, may be found here. The idea of "uniqueness" appears to be much the same thing - I have no problem with it, whether it is the identical concept or a closely related one. I have acknowledged my ignorance of "flat" posets. What I do reject is the idea that faithfulness (aka "niceness") can be easily applied to APs, or that it yet has a unique definition. -- Cheers, Steelpillow (Talk) 17:42, 4 November 2008 (UTC)


 * Good news - have reread Johnson's article, understood (I hope) "monal" and resolved one of our issues - see the section below "Monal vs VF-Unique" issue resolved. SteveWoolf (talk) 05:42, 6 November 2008 (UTC)

The field really has been given some deep thought by serious mathematicians, and anybody hoping to find something new will need to do a great deal of careful spadework first. I had to re-read Johnson's remarks several times before it all began to make sense, and when I edited them together I found I had still missed stuff. But don't let that stop you trying.

Finally, shouldn't issue (-1) be the empty or null issue? -- Cheers, Steelpillow (Talk) 11:24, 2 November 2008 (UTC)
 * Good one, and let us always let civil words and knowledge rise above lesser human drives.SteveWoolf (talk) 22:31, 2 November 2008 (UTC)

Archimedean Solids
This reminds me of an interesting article I read, which may already be reflected in wikipedia... It revolves around the concept of an Archimedean Polyhedron. What is it, exactly, and how many are there? Please answer below.... mike40033 (talk) 05:39, 5 November 2008 (UTC)

I don't quite get your point... The article Archimedean solid exists, but seems to contradict itself. It defines Archimedean as having equivalent vertices and non-equivalent 2-faces, and the triangular prism is an example of this. But the article says prisms are not Archimedean! ??? I'll look into this further. SteveWoolf (talk) 18:22, 5 November 2008 (UTC)

Okay, right, seems there aren't yet universally agreed definitions for this, semi-regular and uniform. Mathworld says there are 13 Archimedean solids. So, point taken.

I never defined "nice" or "nasty" - only said that some polytopes "behave" better than others, and that we MUST define what words we do use rigorously in order to talk the same language. Vertex/facet uniqueness I have defined, as you have Flat. Realisabilty seems to be an outstanding problem. Steelpillow's Monal I haven't grasped yet. Then there are the topological categories....Once we have a common language, then we can meaningfully ask how these various properties relate to each other.SteveWoolf (talk) 19:44, 5 November 2008 (UTC)


 * The interesting point is that using Archimedes' definition, there are actually 14. The pseudorhombicuboctahedron is not as "nice" as the others, but fits perfectly well Archimedes' definition. But people - including serious research mathematicians - have had a great deal of trouble reconciling their intuitive desire to exclude the pseudorhombicuboctahedron withthe fact that the definition allows such an ugly duckling. mike40033 (talk) 06:32, 6 November 2008 (UTC)


 * Our understanding of Archimedes' polyhedra comes down to us through Pappus's brief summary of the topic, as translated by Thomas. Pappus enumerated 13 regular-faced figures which he attributed to Archimedes. It was not until the last century that JCP Miller recognised the significance of the pseudorhombicuboctahedron, when he was attempting a card model of the actual rhombicuboctahedron and made the pseudo- one by mistake. For this reason it is sometimes called "Miller's mistake". Besides this example, there are two more which meet (Thomas' translation of) Pappus' definition but are not in his list - these are similar pseudo- variants of the rhombicosidodecahedron, and like it they may be found among the Johnson solids. I do not know whether Thomas translated Pappus' definition correctly, but let us assume that he did. Unless some lost manuscript is found (perhaps another palimpsest), we will never know for certain whether Pappus' flawed definition accurately reflects Archimedes' understanding. As a result of the flaw, modern authorities differ as to exactly which polyhedra should, or should not, be considered to be Archimedean. The two bones of contention are of course the pseudo-uniform figures and the uniform prisms. -- Cheers, Steelpillow (Talk) 08:25, 7 November 2008 (UTC)

"Monal vs VF-Unique" issue resolved
A polytope is monal if no two faces have identical incidence sets, i.e. the set of faces that contain or contain them. (Have I understood Johnson correctly?)

An n-polytope is vertex/facet unique if no two distinct i-faces (-1 &le; i &le; n) have either identical vertex sets or identical facet sets. (Trivially true for i=-1 or i=n so we only have to worry about 0 &le; i &le; n-1)SteveWoolf (talk) 14:43, 7 November 2008 (UTC)

The second of these two is stronger, i.e.
 * P is vertex/facet unique => P is monal.

For example, the hemicube is monal (different 2-faces have different edge sets), but not VF-Unique (different 2-faces have the same vertex set).

I have no opinion right now how useful each of these two are. The hemidodecahedron is VF-Unique, as Steelpillow pointed out. I would guess the hemiicosahedron is also.SteveWoolf (talk) 06:01, 6 November 2008 (UTC)


 * If so, to require X-polytopes to be monal would then mean not every X-polytope has a dual. mike40033 (talk) 06:35, 6 November 2008 (UTC)


 * Johnson states that the monal property means that "no node can be the top (or bottom) of two different subdiagrams each representing a valid j-facial (or j-cofacial)", i.e. duality is indeed respected. I understand this to mean, in plain language, that it forbids not just abstractly coincident faces sharing the same boundary but also coincident vertices sharing the same vertex figure. -- Cheers, Steelpillow (Talk) 08:39, 7 November 2008 (UTC)


 * As S/p says, monality is "dualistic", so I would be surprised if you (Mike - or anyone) could find an example of a monal polytope without a monal dual - but go ahead, make my day!


 * What is true is that some abstract polytopes are clearly not monal, for example, the digon, a nasty if ever there was one. This means that any definition of polytope requiring monality cannot be equivalent to our current one. All is not lost, however. We will just say that some polytopes have the nice property of being monal. Anyone have any more non-monals in your cupboards? I expect "derivatives" of the digon aren't monal, like digonal prisms, antiprisms, products, pyramids and so on. SteveWoolf (talk) 15:01, 7 November 2008 (UTC)


 * Non-monal posets are not abstract polytopes. Johnson states that "Ludwig Danzer, Egon Schulte, and Peter McMullen define an abstract n-polytope as a partially ordered set of elements or "j-faces," subject to conditions that ensure that it is monal, so that no two j-faces coincide, that it is dyadic, so that a 1-face (edge) joins just two 0-faces (vertices) and just two (n-1)-faces (facets) meet at any (n-2)-face (ridge), and that it is properly connected, so that it does not split into a compound of two or more n-polytopes.", i.e. "Monal" is just a term coined by Johnson to indicate a certain property inherent in Danzer, Schulte, and McMullen's formulation. And again, "My definition of an abstract n-polytope is stated differently but is entirely equivalent to the definitions given by Danzer, Schulte, and McMullen." This means that any conflicting definitions are not correct.


 * I am more likely to be wrong in my interpretation. Perhaps digons are valid abstract polytopes, I am not sure - there is no direct answer to that question in the note. -- Cheers, Steelpillow (Talk) 15:50, 7 November 2008 (UTC)
 * A digon, being so little, is easily shown to satisfy our (current AP)definition. Unless you can prove me wrong in the above (quite possible), Johnson's AP's are not equivalent to the article's = McMullen-Shulte. Which is not say Johnson's might not have been better, but I think, Comrades, we are stuck with the status quo until the next, if any, revolution. SteveWoolf (talk) 17:59, 7 November 2008 (UTC)

The digon is not monal. Nor does it satisfy uniqueness as I understand it. That means it is not a valid abstract polytope. The edges ab are each the top of the same diamond (j-facial in Johnson's terminology), and so break the monal (and uniqueness?) property. If the present definition really does allow it, then the definition would appear to be wrong. But does it? (BTW, in the diagram, it would be more correct to use P, better still &Pi; (upper case pi), for the top element.) -- Cheers, Steelpillow (Talk) 11:57, 8 November 2008 (UTC)
 * As you correctly say, the digon is neither monal nor VFU. But the articles's defn, which is I think the same as McMullen-Shulte, DOES NOT require monality. However much you or Johnson (or Sarah) may disagree with the definition, I think we all have to live with it - remember, we are reporting the status quo, not rewriting it. I also am not comfortable with the un-American nasties that this "wide" definition admits into our heartland. BUT - only when we agree on the meanings of our terms can we have worthwhile discussion. But as I said previously, it is not a calamity. We can still shut out the nasties with other terms, eg your monal concept (which, by the way, I now like). If you don't like digons, then limit your research to monal polytopes, just as many polyphiles won't soil their hands with non-regular polytopes. And the people who like digons can go their separate way within a larger polytope community - you know, all polytopists are equal but monal polytopists are much more equal. Really Guy, I'm on your side here - but we can't keep fighting a lost battle. M-S rules, right or wrong.


 * My digon picture shows the vertex sets of the faces, not their names. Have a nice digon-free weekend SteveWoolf (talk) 12:50, 8 November 2008 (UTC)


 * digons most definately are AP's. If Johnson's topes exclude the digon, and if he also asserts his definition is equivalent to that of an AP, then he is mistaken on the latter point. mike40033 (talk) 00:34, 11 November 2008 (UTC)


 * Keep in mind that every face contains itself! Thus it is allowed for two faces to contain the same lower-dimensional faces, and to be contained in the same higher-dimensional faces. I think the difficulty in interpretation comes about because of the natural map between a j-face of a polytope (a node of the Hasse diagram), and what Johnson calls a j-facial (a sub-Hasse diagram). -CunningGabe (talk) 04:58, 11 November 2008 (UTC)


 * Guy (and others) - suggest you just call polytopes as defined by Johnson monal. Calling them Johnson polytopes would be inconsistent with Johnson solids. It would be probably be confusing to coin any more terms!
 * I think we share the opinion that the polytope net has been cast too wide, but probably differ about where the line should have been drawn. Me and the hemicube will never be friends! Still, ones's dislikes help us define what we like. I am asking myself just which objects I want to devote my years to. I think I consider as nice only pols that are VFU and for which two faces always intersect at a single face (possibly &oslash;), and ditto the dual. But I am wary of my intuitions, as they are oft little better than the results of brainwashing. Regards SteveWoolf (talk) 14:36, 12 November 2008 (UTC)


 * Not that it particularly matters, but I don't share the opinion that the polytope net has been cast too wide! I like all the nasties, and I've been spending some time discovering a bunch of interesting degenerate abstract polyhedra. For instance, there are a lot of dihedra: not only do you have dihedral n-gons, but also you have shapes like a pentagon divided into 3 triangles glued to another pentagon divided into a quadrilateral and a triangle. I'm actually working (in my free time) on enumerating how many abstract polyhedra there are with a given number of proper faces.
 * It is nevertheless useful to classify abstract polytopes - that helps me count them! That said, it is still not clear to me that we are interpreting Johnson's use of "monal" correctly - clearly either we are misinterpreting him or he makes an incorrect assertion. Your other two properties certainly do characterize some niceness, but as you say, it is difficult to know that they include only what you want and exclude what you don't want. A good exercise for starters is to make sure they exclude polytopes with only two facets, which I think you'd agree are degenerate. Then you can check if these properties also prevent each facet from being degenerate in the same way. I suspect that these will not be too difficult to check. -CunningGabe (talk) 17:31, 12 November 2008 (UTC)
 * See section below "Classifying Polytopes"SteveWoolf (talk) 06:48, 14 November 2008 (UTC)

The Hemicube back in the dock
Poor harassed Hemicube - no sooner acquitted of one crime than surrounded by yet one more juicy scandal. And pray what nasty is it accused of now, you ask?

This: In a "nice" polyhedron, two (2-)faces always intersect precisely at some i-face - either at an edge, a vertex, or the (-1)-face (i.e. not at all). More generally, any two j-faces of an n-polytope intersect at an i-face for some i, -1<=i<=j, and ditto for the dual.

(NB: Intersect has the same meaning as meet of a poset, and also coincides with both classic-geometric and set-theoretic meanings.)

The hemicube fails this criterion, also the digon. All the hemis, I suspect.(I'm wrong - see below)

So - is there a standard existing definition for this property? How does this property relate to other "niceties" - flatness, realisabilty, monality, VFUness? SteveWoolf (talk) 06:45, 6 November 2008 (UTC)


 * I think the hemi-icosa and hemi-dodeca escape this one. Lucky, because they (and the 11-cell and the 57-cell) certainly seem nice to me... mike40033 (talk) 06:42, 6 November 2008 (UTC)


 * I can (half!) confirm that the hemi-dodeca is "nice" in the above sense: in fact, every pair of distinct faces intersects at an edge. Oops, forgot to check the dual - I'LL BE BACK....SteveWoolf (talk) 16:18, 6 November 2008 (UTC)


 * Back soon, I hope.... :-) mike40033 (talk) 23:56, 6 November 2008 (UTC)

Can anyone tell me if this so-far-nice hemi-dodeca is 100% so? Is it realisable in normal 3-space? Any other skeletons lurking in its closet? My intuition tells me it's nasty somehow, but look where intuition leads people!SteveWoolf (talk) 16:32, 6 November 2008 (UTC)


 * It's extremely nice. It tesselates the projective plane with six pentagons, and so proves that maps on the projective plane can need up to 6 colours to colour them. As for embeddings in euclidean space, I've never found that the most important thing, so realisation theory is not my forte... Since it has 10 vertices, it can be faithfully embedded in 9-dimensional space (with skew-polygonal faces), but I'd have to look it up to discover how to embed it in R^3..... mike40033 (talk) 23:56, 6 November 2008 (UTC)


 * Hmm - I think it can be embedded in R^3. This is how I'd do it : First of all, here is a pair of 3x3 matrices that generate the symmetry group of the hemi-dodeca. Then, I'd find out how to express certain standard symmetries of the hemi-dodeca in terms of these matrices. Then, I'd find a symmetry that fixes a vertex of the hemi-dodeca, and a fixed point in R^3 of the corresponding matrix. Then, I'd successively apply the symmetries I had to find the other 9 points, and join them up with edges and pentagons. —Preceding unsigned comment added by Mike40033 (talk • contribs) 00:29, 7 November 2008 (UTC)

What is a Simple polytope?
Guys[, gals] - take a look and the short Simple polytope article, and see my lone comment in the Discussion. Can you make sense out of it, or is it wrong? I thought simple meant spherical vs toroidal....SteveWoolf (talk) 13:11, 6 November 2008 (UTC)


 * The definition wouldn't work n the field of abstract polytopes, but is otherwise fine. I'd have expressed it differently.
 * Shortest possible definition : a simple polytope is dual to a simplicial polytope.
 * More clear, perhaps : a simple polytope is one whose vertex figures are simplices (triangles, tetrahedra, etc)
 * So a simple polyhedron is one where each vertex figure (the shape you get when you chop off a vertex) is a triangle - hence, prisms, the dodecahedron, etc. The list in the article is not complete. A simple 4-polytope is one where each vertex figure is a tetrahedron.
 * In the article, they say "a d-polytope is simple if each vertex has d edges". This is equivalent to saying : "a d-polytope is simple if each vertex figure has d vertices". For spherical polytopes, you can prove : "a (d-1)-polytope with d vertices is a simplex", but that doesn't hold in general. Our good friend the hemicube is another example of a 3-polytope with 4 vertices (and there are others) so any 4-polytope with hemicubes for vertex figures would satisfy this definition, but I don't think we'd want to call it simple, or its dual simplicial. —Preceding unsigned comment added by Mike40033 (talk • contribs) 00:06, 7 November 2008 (UTC)
 * I think I understand your confusion, Steve, and I've clarified the definition. A d-dimensional simple polytope is a d-dimensional polytope such that each vertex has d edges. Thus the icosahedron is not simple, because it is 3-dimensional, but each vertex has 5 edges. -CunningGabe (talk) 18:15, 20 November 2008 (UTC)


 * Well I think I'm clear now - but wouldn't the best defn of a simple n-pol be that all vertex-figures are (n-1)-simplexes? To give an analogy, we wouldn't define a cube as a platonic solid with 8 vertices. Looks to me like things are a bit upside down here, especially given the dual defn of Simplicial polytope. SteveWoolf (talk) 22:34, 20 November 2008 (UTC)


 * Agreed, especially in light of the definition of Simplicial polytopes. On another note, I wonder if the claim made in the article (that polytopes that are both simple and simplicial are either simplices or polygons) is true of abstract polytopes as well? I'll think about it when I have a moment and see if I think it is true or if I can come up with a counter-example. -CunningGabe (talk) 23:46, 20 November 2008 (UTC)


 * Yes. An abstract polytope whose facets and vertex figures are all simplices is a simplex. I proved this once, and put it in an article, but the anonymous reviewer said that it was an obvious well-known fact, and told me to take out the proof. The theorem is still there in the article, but no proof. Look up Hartley M. I., "Polytopes of Finite Type". mike40033 (talk) 01:11, 21 November 2008 (UTC)

"Niceness" and Realizabilty
(Separated from Polytope Catenation section SteveWoolf (talk) 09:03, 24 December 2008 (UTC))


 * Just so you know where I'm at, I'm not a great lover of flat polytopes and other nasties, though I have no problem with irregular ones. I'm currently wrestling with several different abstract definitions of a "nicer" subset, but I'm a dyed-in-the-wool purist about abstract definitions; my unshakeable conviction is that abstract polytope theory must stand on it's own foundations and not be some mere offshoot of Euclidean/classical theory. More specifically, I feel that a "nice" polytope would satisfy
 * (a) No two faces have the same vertex set or the same facet set
 * (b) The poset is a lattice, i.e. any two distinct j-faces intersect at precisely one i-face, i<j, and ditto the dual.
 * Can you or Shulte tell me whether (a) and (b) and Realisability have any "implication" connection, i.e. do any of these three imply each other?SteveWoolf (talk) 12:05, 11 November 2008 (UTC)
 * I've played with this a bit now, and I'm still not sure what implies what (if any), though here is one example: there is an abstract polyhedron with 3 facets, 5 edges, and 4 vertices, that is essentially two triangles glued onto a square. This satisfies that no two faces have the same vertex set (though there are pairs of faces with the same facet set), but there are pairs of faces without a meet or join. It might be possible to build a 4-polytope based on this that satisfies both halves of (a) without satisfying (b).
 * As far as realizability is concerned - I am principally a combinatorialist and algebraist, so I don't know much (nor care to learn much!) about these geometric notions :) -CunningGabe (talk) 15:17, 11 November 2008 (UTC)

Classifying Polytopes
Or, Free the Degenerate 13!

Well hopefully we all agree that whether or not the polytope net has been cast too wide, it dont matter no more. The real point is: Those of us who like playing around with degenerates can do so, and a very noble pastime it is too. And those of us who care to specialise in some more selective subclass of pols can do that - be it Regulars, Sphericals, Monals, VFU's, 3-polytopes, or Schmolytopes. And who knows, as we speak, some modern Saccheri heretic might actually be denying our Great Definitions's 4th axiom, just to see how "repugnant" are the implications. Great, long live all heresy, other than questioning "Long live all heresy", of course.

I am fairly sure that polytopes with only two facets would not pass my two niceness concepts - Vertex/Facet uniqueness and Meet/Join (lattice) property.

Do you have a definition of "degenerate"? I haven't seen one. I suspect it will relate very closely to my two nice properties, or to Johnson's monal. I'm fairly sure by monal he means no two different faces F,F' have the same face subsets, i.e. that the sections F/&oslash;, F'/&oslash; are always different. My VFU, however, demands different vertex sets, which, a bit paradoxically, is a stronger condition.

I share CunningGabe's distaste for classical theory, with its messy, vague definitions. However, I do think we owe it to ourselves to precisely characterise which abstract polytopes are also classicals. But what is "are". And how can we characterise a classical structure if they don't have a standard definition? Well maybe they do. But I do feel very strongly that abstract theory is on the verge of a Great Leap Foreward - but it really does need to divorce itself completely from it's aged ex. SteveWoolf (talk) 07:53, 14 November 2008 (UTC)

Examples good and bad
Several figures have caused discussion as to their correctness. I'll report my findings/thoughts more or less together below. -- Cheers, Steelpillow (Talk) 21:39, 16 November 2008 (UTC)

Hexagonal trihedron
Moved this bit up to the appropriate section -- Cheers, Steelpillow (Talk) 08:16, 21 November 2008 (UTC)

Dihedron
CunningGabe sent me a definitive quotation from McMullen and Schulte:

"It follows easily from the defining properties (P1),...,(P4) that the Hasse diagram of a 2-polytope P is of one of the two kinds illustrated in Figure 2A2. According as P is finite or infinite, it is identical to the diagram of a convex p-gon {p}, 2 <= p < infinity, or the apeirogon {infinity}..."

So clearly the dihedron {2} is valid and I was wrong. I have not had time to revisit the issue thoroughly, but it is far more likely that I misunderstood Johnson's "monal" property than that so eminent a mathematician as Prof. Johnson would be wrong: Johnson is clear that his definition is fully equivalent to McMullen and Schulte's. -- Cheers, Steelpillow (Talk)


 * umm - wouldn't {2} be a digon? Wouldn't a dihedron be a {p,2} ? mike40033 (talk) 06:13, 26 November 2008 (UTC)
 * Glad someone's awake. I don't think my slip affects the conclusion. -- Cheers, Steelpillow (Talk) 22:49, 26 November 2008 (UTC)


 * If you can get Johnson's original definition of monal, i.e. his own words, and quote it here, we may be able to resolve the discrepancy. Clearly, if Johnson's and M-S's defns are equivalent, then monal cannot be as I attempted to define it above. SteveWoolf (talk) 05:34, 17 November 2008 (UTC)

"The monal property is the abstract property that relates each j-face of an n-polytope to a unique j-facial (its "span"), as well as to a unique j-cofacial (its "cospan") that includes the j face and all the k-faces incident with it for k > j. In the [Hasse] diagram this means that no node can be the top (or bottom) of two different subdiagrams each representing a valid j-facial (or j-cofacial). In other words, one node cannot be superimposed on another.".

You might find this more intelligible if you follow the link which I have given you several times already, to here - I am not going to quote yards of the stuff. -- Cheers, Steelpillow (Talk) 19:36, 17 November 2008 (UTC)


 * Haven't forgotten about this - just a question of finding time! SteveWoolf (talk) 06:45, 9 December 2008 (UTC)

Square pyramid
This figure was discussed on the Regular polyhedron discussion page, which is less concerned with the abstract properties.

The regularity of polyhedra, whether geometric or abstract, is nowadays usually defined in terms of their symmetries. The most recent, and possibly the most elegant, is that a polyhedron, indeed any polytope, is regular iff (if and only if) its symmetries are transitive on its flags. Like all the more useful definitions, this applies equally to geometric and to abstract polytopes.

The square pyramid is not transitive on its flags, and so is neither abstractly nor geometrically regular. -- Cheers, Steelpillow (Talk) 21:39, 16 November 2008 (UTC)


 * There is no question about the square pyramid's irregularity - it cannot be regular even abstractly. The issue is whether Regular polytope's alternative defn works for abstract pols. McMullen-Shulte state in ARP p11, last sentence, that all abstract polygons are regular. Since all the facets and vertex figures of an abstract square pyramid are polygons, though not identical, they too are all abstractly regular - so the criterion clearly is insufficient in the abstract case, so the article should qualify its assertion. You were, as far as I can tell, right and I wrong in the classical case.


 * The point is, of course, is that classical and abstract polytopes are not the same, and nor does regularity have the same meaning.


 * (For the benefit of the RP's talk readers, I'll copy this there also.)


 * Assuming you are clear now, would you care to amend the article? Regards SteveWoolf (talk) 05:59, 17 November 2008 (UTC)

OK I have put some basics in the Abstract polytopes section. There is clearly more to be said, but I do not know enough abstract theory to say more than I have. -- Cheers, Steelpillow (Talk) 20:18, 17 November 2008 (UTC)

Tidied it up, see my comments there. Regards SteveWoolf (talk) 03:13, 18 November 2008 (UTC)

Clearer Definition
Okay, I finally put the Axiom 4 into English by first defining edge and section, which involved a little restructuring. Having done this with care, please be equally careful to keep concepts and links coherent when editing. Both edge and section are linked to their precise definitions. Hope you all agree that the definition is now more accessible and elegant, and that it is better to introduce basic concepts first, rather than be dazed by intimidating formulae in the middle of a definition. SteveWoolf (talk) 07:33, 8 December 2008 (UTC)

Sections
Another term that I think needs better context is "section". An abstract section of a polyhedron is not at all the same thing that a draughtsman understands by the term. For example we can cut a square section through a regular tetrahedron, but if I understand correctly no such abstract section exists. -- Cheers, Steelpillow (Talk) 18:28, 8 December 2008 (UTC)


 * In abstract terms, there is the concept of a "cut", which is at the same time more general, and less general than a 'geometric' section. Less general because the definition has only been given for regular APs. More general, because it can do a lot more than just slice a polyhedron and look at the cross-section. mike40033 (talk) 00:35, 9 December 2008 (UTC)


 * Not quite sure what you are seeking for "better context". Recursively, for an n-polytope P:


 * The n-face of P is a section
 * Any k-face of a section or its dual is a section


 * But that is more complicated than the "closed interval" concept, which is well established both in order theory and both classical and abstract polytope texts, and has only scant connection with English usage. SteveWoolf (talk) 05:02, 9 December 2008 (UTC)

By "better context" I mean that we should explain that in abstract theory,"section" (to which we may add words like "cut", "vertex figure", "regular" and possibly others, mean different things from the usual geometric meanings. -- Cheers, Steelpillow (Talk) 14:00, 9 December 2008 (UTC)


 * Done for "section"; don't forget, the introduction clearly states that abstract concepts are not always the same, so we don't necessarily have to labour (in the non-physical work sense) this point (in the non-geometric sense) every time (in the non-temporal sense). But I agree in this case, since the concept is very different. Regards SteveWoolf (talk) 17:20, 9 December 2008 (UTC)


 * Done for regularity, with some other minor improvements. SteveWoolf (talk) 07:32, 10 December 2008 (UTC)

Improvements to Article
"Minimal" and "Maximal" are not the best terms, as they do NOT imply uniqueness. (An order can have many minimal/maximal elements). ARP uses "Least" and "Greatest" which admittedly are a little less aesthetic, but are consistent with standard (Partial) Order Theory.

I have made the "Polytopes as Posets" section both much clearer and more rigorous, and moved the incidence definition to the Incidence Matrix section. SteveWoolf (talk) 22:25, 12 December 2008 (UTC)

I have made the introduction more readable to non-experts by deferring (i.e. moving) the more advanced concepts further down the article. Nothing has been deleted.

I strongly feel that my 5 quads picture should be at the top, because it immediately and instantaneously gets the concept across before the more difficult task of understanding the text, at least to anyone capable of understanding the text. It is much less difficult task to scroll up from the next section that refers to it. It also, I think, makes the article more appealing at first glance, which I think matters. SteveWoolf (talk) 12:30, 15 December 2008 (UTC)

There are inaccurate statements in the "Realizations" section. Four vertices do not make a square, this also requires edges, at least. The intended meaning is clear anyway, of course. I will probably fix this at some stage, unless one of you cares to first. The problem for me is the lack of simple non-abstract defn of polytope. SteveWoolf (talk) 13:38, 15 December 2008 (UTC)


 * The intro paragraph is badly worded. It first says that an abstract polytope is a structure that captures the combinatorial structure of a classical polytope, and then contradicts itself by saying that an abstract polytope encompasses more than just classical polytopes. It really should state at the outset that an abstract polytope is a generalization of a classical polytope, and is more inclusive than a classical polytope.


 * Also, please try to avoid textbook-style wording and tone (such as using "we"); as much as possible, try to word it as an encyclopedic description of the subject rather than a pedagogical introduction to the subject. See WP:style.


 * (I hope I'm not coming across as being nitpicky; I'm just trying to improve the article as you are.)&mdash;Tetracube (talk) 18:06, 15 December 2008 (UTC)


 * P.S., see also WP:NOT.&mdash;Tetracube (talk) 18:38, 15 December 2008 (UTC)


 * Ta for feedback, points taken, and your diplomatic remarks much appreciated.


 * I was aware of the "contradiction". But keep in mind that the introduction is - an introduction. Here, simplicity and clarity should be the paramount objectives. It is not a formal definition, and clearly says so. The second statement doesn't so much contradict as qualify the preceding one.


 * I have added "informally speaking" and "in fact" which I hope will meet your objection. I can't see how to improve it further without losing reading clarity. Hope this is okay for you now, and do always feel free to comment, bearing in mind that one sometimes ends up with egg on the face, as I have a few times! We shall try to avoid "we". Season's Greetings SteveWoolf (talk) 20:30, 15 December 2008 (UTC)


 * Regarding realizations : the article now says that the pyramid we all met in primary school is a realization of the abstract polytope with thegive Hasse diagram. This "realization" has infinitely many points. But if I remember rightly, the realization of a regular AP as defined in ARP is a mapping from the set of vertices of the AP onto some Euclidean space - so it can have only as many points as the AP has vertices. Again, if I remember correctly, ARP says something like "this might not be a satisfactory definition to everyone, but it's the one we are working with here". I may have put my foot into my mouth here, but this means that if I remember correctly, the article is currently incorrect regarding realizations. mike40033 (talk) 07:40, 17 December 2008 (UTC)


 * I am currently musing over precisely what "Realization" means. However, I am not sure your objection is correct. It seems that a realization of an AP is a traditional pol such that the face posets of both are isomorphic, ie a bijective order-preserving map exists. It is not required that all the internal points of a trad pol be mapped. I think we only have to worry about the vertices of the trad pol. So if you had foot and mouth, so please don't visit any farms.


 * In any case, the Realizations section probably needs a rewrite, but I shall not do this myself until I really think I know what I'm talking about, as foot and mouth is not my favourite disease, and takes a while from which to recover. SteveWoolf (talk) 12:27, 17 December 2008 (UTC)


 * A realization of an abstract polytope is just some figure in some real space (usually Euclidean), whose structure parallels that of the poset. Mike40033 is correct in saying that only the vertex points count as structurally significant points - those along a real edge for example will only count as a single unit, i.e. the structural line segment which they form. -- Cheers, Steelpillow (Talk) 15:46, 17 December 2008 (UTC)

Redundant Axiom?
Seems to me that Axiom 2 of the definition is unnecessary. 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. The new definition would become extremely concise and elegant, as follows:

An abstract polytope P is a partially ordered set satisfying:


 * 1) P has a unique minimal face and a unique maximal face.


 * 2) If F &le; H and Rank(H) &minus; Rank (F) = 2, there are precisely 2 faces G such that F < G < H.


 * 3) P is strongly connected, that is, any flag can be changed into any other by changing just one face at a time.

An abstract polytope P is an (abstract) n-polytope if its maximal face has rank n.

Note that I defined Rank earlier as starting at -1 for the minimal face.

I moved 4 -> 2 since it seems more fundamental than the connectedness axiom.


 * I'm reflecting on another idea - just an idea at this point. If "section" and "edge" [i.e. line segment] were defined previously, in the Poset intro bit, then the diamond axiom could be stated extremely lucidly (and with its original intention) as "Every 1-section is an edge". Then the definition would be much shorter again, and be all English words! Pretty cool, huh? (Of course, it's a bit like electric vehicles in the city - you've just shifted the pollution somewhere else (probably a non-marginal constituency (non-swing state(sorry for the nested parens(how many brackets must... oh, brackets, schmackets.

Just to be really clear here: both of these proposed more concise and elegant definitions (of an abstract polytope) depend upon my previous definitions of rank, face, flag. And for the "Every 1-section is an edge" option, edge (more correctly, line segment) would be defined as a poset (isomorphic to) {&oslash;, a, b, ab} with the "usual" ordering.

Good? Bad? Heresy? SteveWoolf (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 a Question...
As expressed above, it seems to me that our standard AP defn is a bit more general than simply "the combinatorial properties of a classical polytope". So my question is this: what additional "nice" properties would an abstract polytope have to have so that classical and abstract polytopes do correspond?

I suspect these two properties may suffice:


 * (1) It is atomic - each face has a unique vertex set, ditto the dual
 * (2) It is a lattice, i.e. any two faces have a meet and a join which are faces. Thus two 2-faces cannot intersect at two disjoint vertices, for example.

I am assuming here that classical polytopes include apeirotopes (infinite polytopes).

Can anyone enlighten?


 * This still allows the 11-cell and the 57-cell, I believe, and many more inscrutable abstract polytopes. mike40033 (talk) 01:19, 27 November 2008 (UTC)


 * Okay, accepting that that's true - what properties do these two have, or lack, that might make us think they're not quite as "nice" as "classical" polytopes? I presume it's that they "don't fit well into Euclidean space", which is presumbably another way of saying they're not realisable. Is that right?SteveWoolf (talk) 06:15, 29 November 2008 (UTC)


 * Personally, I think they're much nicer than the classical polytopes... :-) mike40033 (talk) 13:17, 29 November 2008 (UTC)


 * You suggest that my atomic + lattice conditions together are not so narrow because they "still" allow the 11-cell and 57-cell, which you describe as "inscrutable". But then you say they are "nice(r than classical pols)", which seems to contradict the previous sentiment. So may I ask again, what is about 11/57 that we might want them excluded from a more select group of nice pols, and why do you feel they are "inscrutable"?


 * I can dodge this on syntactic grounds... I could claim that I said your definition allows "other polytopes that are more inscrutable than the 11/57", which doesn't imply that the 11/57 themselves have positive inscrutability. Or, I could claim I only meant "more polytopes, which happen to be inscrutable". Or, I could claim that inscrutability doesn't contradict the fact that I, personally, feel they are nice. Nonetheless, the 11- and 57-cells have facets that would need some explaining to someoone only familiar with classical geometry. If you don't want to exclude them on this basis, then why try to exclude the digon? mike40033 (talk) 08:03, 12 December 2008 (UTC)


 * Which leads to another question which is troubling me (and perhaps others too): has our AP definition really captured the combinatorial aspects of classical polytopes - at least the consensus of what is a classical pol - or have we widened it in the process?  If the latter, then should we not define more precisely this discepancy? Apologies if my tenacious hammering away at some definition(s) of "nice" is getting repetitive, but I feel strongly that they are needed.SteveWoolf (talk) 05:45, 8 December 2008 (UTC)


 * If the definition of an AP did not widen the definition of a polytope, it would be a dead-end topic, research-wise. That's human nature, I'm afraid. mike40033 (talk) 08:03, 12 December 2008 (UTC)
 * Don't think that follows. Numbers have been redefined in "modern" mathematics - Peano axioms, Cauchy sequences &c. But Real numbers aren't a different set than before. And their generalisations have different names, e.g. complex, quaternion, cardinal numbers (alephs etc.).SteveWoolf (talk) 05:06, 15 December 2008 (UTC)

My personal preoccupation is to decide precisely what objects I wish to study. What is "interesting"? Positive integers? All integers? Rationals? Reals? Complex Numbers? Well, they all have interest of course. But generalisation is a double-edge sword. Too little and you miss out on all kinds of exotic objects. Too much, and you miss out on the very special properties of narrower classes. I believe there was a long debate before the definition of a topological space was finally agreed upon, and clearly the results have been productive.

There seems little reason to think that the universe is Euclidean or even continuous. I think a discrete, "Tube Map" (though in more dimensions) type model is much more likely, and my money's on AP's being very relevant in the micro- and macro- worlds. SteveWoolf (talk) 19:24, 26 November 2008 (UTC)


 * The nub of this is to define a classical polytope sufficiently rigorously in the first place. What kinds of space do you wish to consider? Is a "spherical polygon" marked on a 2-sphere acceptable? What untidy features do you allow or disallow? Overlapping faces? 180 deg dihedrals? Slits? Whiskers? What about Petrie polygons, those skew zigzags around the middles of uniform polytopes? Digons, dihedra and hosohedra? Hemi polyhedra which can be realised faithfully, surely yes - but what about those that can't? Is duality a prerequisite? Digging deeper, do you fundamentally define a polyhedron as a solid, a surface, a skeleton or a point set? When one comes to ideas such as area, volume, and inside vs. outside, all these approaches can yield startlingly different results. Euclid, Kepler, Coxeter, Gr&uuml;nbaum and Johnson all give different - and usually incompatible - treatments. There is no single "right" answer, no consensus, in sight (though I continue to work on one). Only when you have defined your special kind of "classical" polytope, can you start to compare its structures with abstract ones. -- Cheers, Steelpillow (Talk) 23:42, 26 November 2008 (UTC)


 * Yes, you have hit the nail on the head; how can we formalise a concept that isn't well defined? (How do you catch a wave and pin it down?) I was aware of this while posing my question, but vainly hoped I could ignore it! It does seem at first that there can never be one definitive answer. The church endeavours to define goodness, but many people today have different ideas, and saying only God can define it is a bit circular. Nevertheless, the defn of topological space was agreed on, and is now a most successful concept. And for better or for worse, we also have a standard AP defn. But besides the general topological space, there are many variations, for example, Hausdorff spaces. And within AP, there will be narrower concepts (eg Regular Pols) - and there are wider ones (eg all posets). And each of us will decide what is interesting and worthwhile.


 * However - there is an objective criterion also - what is useful, and how much. For example, noone would say that complex numbers are not useful - they allow ALL algebraic equations to be solved, and are most useful in electronics. Yet who would dream of saying that their usefulness even begins to compare with that of the real numbers? Now clearly a vast number, perhaps most, AP theorists consider Regular Pols to be a crucially important specialisation - the irregulars seem to have the status of untouchables!  I suspect they will become more important. (If the universe is a large finite spherical polytope, it can't be regular. But it might be "mostly regular", like a mirror ball in a disco, which has mostly square faces but has a problem near the "poles". And the irregularities might correspond to matter.... Just musing!)


 * So I think there are important classes of AP's (other than Regulars) waiting to be defined, and whose study might produce a rush of new ideas. As a way of moving this discussion forewards, would any of you care to define one or more polytope properties that you consider interesting? For example, we can call a polytope primitive if it cannot be formed by joining two others as I defined earlier, eg the cube, but not the octahedron.SteveWoolf (talk) 06:15, 29 November 2008 (UTC)


 * You might find this useful: Gr&uuml;nbaum, B.; Are your polyhedra the same as my polyhedra?, Discrete and Computational Geometry: The Goodman-Pollack Festschrift. B. Aronov, S. Basu, J. Pach, and Sharir, M., eds. Springer, New York 2003, pp. 461 – 488. -- Cheers, Steelpillow (Talk) 22:08, 29 November 2008 (UTC)
 * Thanks for that, looks interesting and actually readable. Will get to it this week hopefully. Regards SteveWoolf (talk) 02:34, 2 December 2008 (UTC)

Spherical polytopes
A polytope is spherical if and only if one of the following hold :


 * it has rank 1
 * it is finite, and has rank 2, OR
 * it is finite and universal, with spherical facets and vertex figures.

mike40033 (talk) 08:28, 31 October 2008 (UTC)


 * Can you define "universal" in this context? -- Cheers, Steelpillow (Talk) 11:27, 2 November 2008 (UTC)


 * I've put the definition into the article now. mike40033 (talk) 05:21, 3 November 2008 (UTC)

Until such time as I grasp this stuff, if any of you care to comment on the discussion in Talk:Spherical polyhedron regarding Sphericality, it could be helpful. SteveWoolf (talk) 12:39, 15 November 2008 (UTC)

Topological definitions
The usual topological definitions are based on the genus of a polytope, both orientable g and non-orientable k. These are related to the Euler characteristic &chi; = V - E + F - C ..., where V, E, F, C etc. are the number of vertices, edges, faces, cells, and so on.

I am not sure how it works for higher polytopes, but for polyhedra we get:


 * Spherical: &chi; = 2, g = 0,
 * Toroidal: &chi; = 0, g = 1,
 * Klein bottle: &chi; = 0, k = 2
 * Boy surface: &chi; = 1, k = 1
 * For higher genus non-orientable toroids, there is a trade-off between g and k.

It is trivial to find &chi; for any abstract polytope. If rank (dimensionality) = 3 and &chi; = 2 then we have a spherical polyhedron. I do not know if the equivalent works in higher dimensions.

Establishing the orientability of an abstract polytope is a trickier problem, and I am not sure if it has been solved.

-- Cheers, Steelpillow (Talk) 13:38, 1 November 2008 (UTC)


 * Orientablility of an AP is solved. I have a wonderful definition of it, but this margin of time does not contain enough moments for me to write it out. mike40033 (talk) 05:14, 3 November 2008 (UTC)


 * Guess we'll have to await its rediscovery 357 years from now, which is quite a Wiles. SteveWoolf (talk) 12:18, 15 November 2008 (UTC)

Have Abstract Defn of Spherical
Well, I hope...

McMullen/Schulte's ARP p153 says:


 * "An abstract n-polytope P is called spherical if it is isomorphic to the face-lattice of a [classical] convex n-polytope Q"

(Thanks CunningGabe for that).

However, I feel that we
 * (a) Need to define this is a purely abstract manner, without recourse to classical ancestors, and then,
 * (b) Prove that the abstract version is equivalent to the "cross-cultural" (i.e defining abstract terms with classical concepts) one.

So here it goes (for simplicity, I'll talk about polyhedra):


 * Stealing directly from topology, we shall define a polyhedron P to be spherical if and only if any "closed loop can be shrunk to a point" - i.e. the object has no "holes". We must, of course, make this quite clear and precise.


 * A loop, then, is any polygon, not in general a 2-face, all of whose edges are in P. (Note that loops may not have repeated vertices - no figures of 8!)


 * In general, such a loop "passes through" many 2-faces. So a loop is a concatenation of arcs, so that all the edges of a given arc belong to the same face of P


 * Given a loop, we can make a new loop by replacing an arc with precisely the edges from the same face that were NOT included before. By this method, loops can be contracted or expanded. Call such an operation a continuous step.


 * Then, a polyhedron is spherical if any loop can be shrunk to a single face by a series of continuous steps. (Rather elegantly, this is equivalent to shrinking to a point - a point is a null loop, obtained by "flipping" the last face).

So, make my day, shoot it down!

Does it work - even for polyhedra? I suspect that toroids have a hole - you can't shrink to a point! Does it generalise to n dimensions? If it does work, I doubt if it's original. And if it's full of holes - well it was fun anyway. SteveWoolf (talk) 04:20, 18 November 2008 (UTC)


 * The Euclidean plane shares the same property that any closed loop may be shrunk to a point. In the (flat) projective plane, this is also true of finite closed loops. A more useful definition is that the Euler characteristic &chi; equals that of a sphere, which is equivalent to a topological genus g = 0. -- Cheers, Steelpillow (Talk) 13:10, 18 November 2008 (UTC)


 * Ta for that. Maybe that is the way to go - but then I have two questions. First, what is the definition of Euler characteristic for an abstract polytope? Second, if it's an expression involving numbers of i-faces for each dimension i, how do you justify saying that Euler Char = 0 means they are spherical? To illustrate my point, you wouldn't define a right-angle triangle as one satisfying Pythagoras's Theorem x2+y2=z2. You define it as having a right angle, then prove the theorem. SteveWoolf (talk) 05:07, 19 November 2008 (UTC)


 * The Euler characteristic &chi; is found in exactly the same way for any polytope, abstract or geometric. For an n-polytope having elements Ek of dimensionality k from k=0 (vertices) to k=n-1:
 * &chi; = E0 - E1 + E2 - ... +/- E(n-1)
 * For a polygon (all polygons are 1-spherical), &chi; = 0
 * For a spherical polyhedron, &chi; = 2
 * Euler proved it for spherical polyhedra, but I do not know who generalised it to non-spherical topologies or higher dimensions. With regard to definition, in topology shapes can get so twisted up that it is impractical to tell from visual inspection when two shapes are topologically alike. The ideas of genus and orientability were developed to provide a rigorous approach (genus is closely related to the Euler characteristic), and we now indeed define two surfaces as having the same topology (e.g. that of a sphere) if they have the same genus and orientability. You should find all this and more if you follow the links I put in my previous remarks. Only if you follow these links and read up on the subject will you understand why it is so - this discussion is no place for these proofs. -- Cheers, Steelpillow (Talk) 22:01, 19 November 2008 (UTC)
 * The generalisatation of V+F=E+2 to n dimensions is known as the Dehn–Sommerville equations. If you include ALL dimensions including -1 and n, the alternating sum of cardinalities is always zero -you avoid that messy flipping between 0 and 2. (This to me is yet another powerful argument for regarding the null and global faces as legitimate - there are still skeptics out there - see Talk:Simplex). Okay, I will follow your links and see if I can understand everything. (Is "my" definition of spherical equivalent to the Euler characteristic one? Is every spherical abstract n-pol topologically "the same" as a Euclidean n-sphere? Must be yes, hopefully!) SteveWoolf (talk) 03:59, 20 November 2008 (UTC)


 * I'm not sure. If it applies in the topology of smooth surfaces, i.e. if a spherical surface is one on which any loop may be shrunk to a point, then yes it applies for polytopes too (in fact I have recently come to think of polytopes as being piecewise surfaces). But what about the Euclidean plane? That doesn't strike me as a topological sphere, yet shrinking loops always works. -- Cheers, Steelpillow (Talk) 21:59, 20 November 2008 (UTC)


 * Indeed you are right. The Euclidean Plane is not a sphere. The similarity is that spheres are "locally 2D" or planar. The same can be said for infinite 3-D tesselations, such as the hexagonal honeycomb, which is locally 2D. A limited part of it is similar to a spherical polyhedron. So - I need to read up on topology. Anyway, if we sometimes throw out half-baked ideas, I guess it's okay - hopefully someone corrects us and we learn. SteveWoolf (talk) 22:21, 20 November 2008 (UTC)


 * What about the hemi-icosahedron? (Ans : eliminated via the euler characteristic) or infinite polytopes? (Ans: define spherical polytopes to be finite) Or the hemi-24-cell? (Ans: I have no answer. How would a topologist distinguish S^3 from P^3???) mike40033 (talk) 06:09, 26 November 2008 (UTC)


 * I still prefer the definition I gave earlier (can't find it now)... A polytope is spherical if and only if
 * * it is finite and has rank 1 or 2, or
 * * it is finite, universal, and has spherical facets and vertex figures.
 * Objections? mike40033 (talk) 06:09, 26 November 2008 (UTC)


 * What does "universal" mean? -- Cheers, Steelpillow (Talk) 22:50, 26 November 2008 (UTC)


 * Is this still not in the article? I could swear I've put it in twice now.. mike40033 (talk) 07:54, 12 December 2008 (UTC)


 * Summary: the definition in terms of topological genus is the standard one, and is both simpler and safer. Your first condition is a part of that theory - for 0 and 1 dimensions, there is no topological distinction between spheres and polytopes. Your second must surely arise as a theorem, and has probably been done to death decades ago. -- Cheers, Steelpillow (Talk) 13:32, 8 December 2008 (UTC)


 * Simpler? How so? Maybe for polyhedra (though 'simpler' is a subjective term) but what do you do in higher ranks? And Safer? As in less likely to throw up obtuse counterexamples to our intuition? The discussion in this section makes me a skeptic.... :-) mike40033 (talk) 07:57, 12 December 2008 (UTC)


 * It is simpler to express, because it is a one-liner: "A spherical abstract polyhedron is one having genus=0". That is it. done. Simple. It is safer, partly because it works for all kinds of fancy complicated polyhedra whose "sphericality" or realizability is not at all obvious from visual inspection, and partly because unexpected issues such as accumulation points (see below) do not upset it. For higher dimensions, the Dehn-Somerville equation (discussed above) generalises this definition. If you still find this standard approach unsatisfactory, then you have a great deal of homework to do - starting from Leonhard Euler and working your way forward to the present day. I shall not discuss this topic further. -- Cheers, Steelpillow (Talk) 14:24, 12 December 2008 (UTC)


 * Come to think of it, the "finite" bit is wrong. Provided there is an accumulation point on the sphere, the abstract polytope may be infinite (an apeirotope), for example Spidron polyhedra. -- Cheers, Steelpillow (Talk) 13:37, 8 December 2008 (UTC)