![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | Archive 2 | Archive 3 | Archive 4 | Archive 5 | Archive 6 |
Unless anything new has been published recently, the theory of abstract polytopes is scattered across many sources, and often expressed in rather convoluted ways. Prof. Norman Johnson has developed a more understandable and succinct approach, which is due for publication in a forthcoming book on uniform polytopes. In the mean time an accessible summary, Polytopes - abstract and real, is available here. It might provide some useful ideas, especially where the present article risks moving into original research ( wp:or). -- Cheers, Steelpillow ( Talk) 10:29, 26 October 2008 (UTC)
To enlarge a little, some key properties which he highlights, and names, are that an abstract polytope is monal, dyadic and properly connected. He also distinguishes facials from faces (facets). -- Cheers, Steelpillow ( Talk) 11:05, 26 October 2008 (UTC)
Conflict of Terminology. If I understand correctly, Johnson considers it most important that we make a clear distinction between very general abstract polytopes (eg Digon, Hemicube) and more familiar ones (eg triangle, cube). I absolutely agree with this (see my #Suggested Improvements to the Definition below, §4). However - "Real" is a term that cannot be used. Coxeter already distinguishes between Real and Complex polytopes, and in any case the term Real as in Real number is used widely throughout mathematics, and especially in the many areas of abstract geometry.
Perhaps Normal would be a better term (and should certainly appeal to Prof Norman Johnson!)
There are clearly some of us who would have preferred the definition of Abstract Polytope to be less general; but as mike40033 aptly pointed out previously, us lesser mortals have to respect, however reluctantly, established terminology in order to have a common language with which to exchange - and thereby advance - ideas. So... let's try to reach a consensus on what to call polytopes that don't include the dastardly digon and the heinous hemicube. SLWoolf ( talk) 05:47, 28 October 2008 (UTC)
Coincident vs Same Vertices/i-Faces. Johnson talks about coincident vertices. In a digon, the two edges have the SAME two vertices. That is what a Digon is. While two separate edges ab and cd in (e.g.) Euclidean space can be coincident, this concept does not carry over into Abstract Polytopes, where the only properties that matter are the combinatorial poset (face lattice) relations. I think the issue here, as far as Abstract Polytopes go, is whether or not two i-faces are allowed to have the SAME vertex sets (and, dually, facet sets) as they do in the digon or the hemicube. To repeat, the concept of coincident vertices or other i-faces has no meaning for an abstract polytope. SLWoolf ( talk) 05:47, 28 October 2008 (UTC)
Thanks for the response. Yes, maybe I misunderstood Johnsons's definition of Real Polytope. Returning to the subject of "Normal" vs other polytopes: I strongly feel that the subject of Abstract Polytopes needs to be self-contained and not merely an adjunct of classical polytopes, in much the same way that topology has become. Euclidean or Real space is merely one of many geometries and has no special place (indeed it is only an approximation to the real universe and fails both in the microworld of subatomic physics and the macroworld of cosmology). Therefore, I think the difference between polytopes such as a cube and those such as a hemicube should be defined abstractly primarily, and not in terms of realisability in Real Space. I think the crucial distinction is whether or not different i-faces can have the same vertices, and dually, (containing) facets. Then, if this condition should turn out to be equivalent to realisability, no-one will be more pleased than myself.
I don't think we should be always intimidated and muzzled (us Woolfs like to howl) by the No Original Research thing - this is only a talk page, and anything of value we "discover" here will probably already have been already covered by the publishing elite - or soon will be, at which point any worthwhile material can be promoted to the article and referenced. SLWoolf ( talk) 06:25, 29 October 2008 (UTC)
Your feedback greatly appreciated. Have scrutinised the hemidodecahedron's incidences and as you say, my "uniqueness" criterion doesn't eliminate it as it does the digon and hemicube. Well, we fall down, then get up the stronger for having tested the terrain. Which leaves me wondering - is there a purely abstract property that is provably equivalent to realisability - is that the outstanding problem you mentioned? I still feel that we need a purely abstract concept of Normal vs Abnormal, but as to what that means, right now my intuition fails me - and intuition is pretty unreliable anyway.
You never know, us amateurs might just see something the pros have missed - I think it has happened now and again. Edwin Hubble, for example. SLWoolf ( talk) 05:04, 30 October 2008 (UTC)
We agreed about archiving and I spent some time doing that. Against Wikipedia policy, you have edited this archive and therefore compromised it as a historical record. Please revert your edit, THEN copy back just the parts of this section that you consider still worthwhile, but please omit the rubbish which only serves to clutter the section and the page. Hope you agree with this. SteveWoolf ( talk) 07:25, 3 November 2008 (UTC)
Johnson presents a definition of an abstraction of the "polytope", which he claims to be equivalent to Schulte & McMullen's abstract polytopes. This is fine, indeed wonderful if his definition
However, he then defines his own terminology and notation. And this is where I put my foot down and say No!!
The literature is already littered with too many sets of terminology for basically the same thing. We have
As well as conflicting notation and terminology from fields such as
The best way to get people to understand more about abstract polytopes is not to invent (putatively) "better" notation, but to unify (ha ha) the existing terminologies, or at least produce an easy way to translate between them.
So sorry, I can't support Johnson's terminology and notation being used in the abstract polytopes article. However, translataions of his ideas into the "standard" terminology would of course be welcome. mike40033 ( talk) 06:27, 26 November 2008 (UTC)
I have archived some of our resolved issues, obsolete waffle, and interminable repetitions and interminable repetitions. Unfortunately, they were very much mixed in with some good ideas which should remain "current". So I decided the best solution was to archive many sections, then copy back the best paragraphs, editing and regrouping as seemed appropriate. What we lose in the historical sequence of our talk (which is still viewable in Archive 2) will be more than compensated by a very much better (usable!) presentation of ideas.
I hope the result is acceptable to all - but inevitably I had to make judgements re what to keep. I tried to be objective, but if anyone feels I have omitted anything of value then go ahead and revive it.
Guy - I left the huge "Johnson's approach" section for you to archive or not as you see fit, but it would be a good idea to condense it to the main points at least. SteveWoolf ( talk) 05:48, 25 December 2008 (UTC)
The long preceding discussion is in Archive 2.
So, let us go with "Traditional", at least until some other Great Work is published. ARP should be our terminology Bible, as we agreed. I hope Guy will not be too disconcerted. It has to be better than our current mish-mash, at least! SteveWoolf ( talk) 13:20, 25 December 2008 (UTC)
If anybody is interested in joining a private mailing list, discussing everything polyhedral from sculpture to abstract polytopes, drop me a line through the Wikipedia mail system. I half-recognise many ideas here, such as products of two polytopes, from previous discussions. -- Cheers, Steelpillow ( Talk) 09:07, 7 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 with the fact that the definition allows such an ugly duckling. mike40033 ( talk) 06:32, 6 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)
There is also a relevant discussion in Talk:Spherical polyhedron#Rigorous Definition regarding Sphericality. SteveWoolf ( talk) 14:27, 25 December 2008 (UTC)
A polytope is spherical if and only if one of the following hold :
mike40033 ( talk) 08:28, 31 October 2008 (UTC)
But now your defined "spherical" concept doesn't at all match what I thought it meant. Consider a "bicube" - two cubes pasted together on one face. This can be "inscribed" on a sphere, or, equivalently, has a planar graph. Does that not make it spherical? Yet it is not universal, as you have "defined" that. Can you clarify this apparent clash of definitions? SteveWoolf ( talk) 03:37, 30 December 2008 (UTC)
SteveWoolf ( talk) 02:38, 31 December 2008 (UTC)
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 χ = 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:
It is trivial to find χ for any abstract polytope. If rank (dimensionality) = 3 and χ = 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)
McMullen/Schulte's ARP p153 says:
(Thanks CunningGabe for that).
However, I feel that we
So here it goes (for simplicity, I'll talk about polyhedra):
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)
See Talk:Abstract polytope#2nd Definition of Spherical. SteveWoolf ( talk) 14:48, 25 December 2008 (UTC)
The article Simplex (Relation to the (n+1)-hypercube), states, correctly, that the Hasse diagram of an n-simplex is the graph (1-skeleton) of an (n+1)cube.
In fact (I think - any dissenters?) not only is any Hasse diagram of an n-polytope P the graph of an (n+1)-polytope, but this latter polytope is easily defined as the set of all sections of P, ordered by inclusion. Note that a section is precisely an (order-theoretic) interval in the original poset.
For example, in the triangle abc, the section ab/a is {a, ab}, while both the sections ab/c and a/ab would be ø, the empty set. This gives multiple ø's in general, which is okay since, in set theory, {a} {a} = {a}, so there's only one null face.
A triangle generates a cube; a square gives a polyhedron with 8 equivalent tetragonal faces, but the edges and vertices and not equivalent. The null polytope gives the 0-polytope, the latter the 1-polytope, then the square.
As a direct consequence of the diamond property, any 2-face of a Hasse Polytope must be a tetragon, and consequently most polytopes are not the Hasse Polytope of another polytope.
Terminology. Alternative names: Meta-polytope, metatope, hierotope... any more suggestions?
Original Research. Is it? (Did I actually discover something?) As I don't have the financial, mental or time resources to easily find out, would any of you be interested in researching this? And what happens to this or any other new ideas, especially worthy ones? Can humbler mortals publish? How? What is a reasonable working definition of "published"?
SteveWoolf ( talk) 11:12, 2 November 2008 (UTC)
As noted in Prism (geometry), the Cartesian Product of two polytopes gives another polytope. For example an n-polygon times an edge (1-polytope) gives an (n+1)-prism. Generalising, an m-polytope P times an n-polytope Q gives an (m+n)-polytope. The faces of the new polytope are precisely the set of pairs FG for each F of P, G of Q, except that each and every Fø and øG pair become the same single new empty face in the new polytope. Otherwise (i.e. when both i, j >=0), each i-face, j-face pair gives an (i+j)-face of the new polytope.
The product of 2 squares is a 4-cube. Multiplying by a point leaves a polytope unchanged (rather like x 1); multiplying by the null polytope gives a null polytope (rather like x 0).
It is easily seen that if P has v vertices and Q has w verices, the product polytope has vw vertices, and therefore no polytope with a prime number of vertices can be a (nontrivial) product polytope. The converse is false - a hexagon is not a product. In fact, if P and Q both have dimension > 0, the product will have at least one tetragonal face, since edge x edge = square.
If there is a consensus to do so, I will be happy to make a section in the article with an elegant, tabulated example showing how all the faces are computed. Though I discovered this independently, I expect there are refs about this somewhere.
Yes/No, because...? SteveWoolf ( talk) 19:07, 5 November 2008 (UTC)
A Simple n-polytope is best defined as an n-polytope all of whose vertex-figures are (n-1)-simplexes. This is equivalent to saying that each vertex is contained in n edges.
The articles Simple polytope and Simplicial polytope should be amended accordingly. SteveWoolf ( talk) 06:08, 25 December 2008 (UTC)
Can hardly keep up with pace of ideas - but that's great, feedback greatly appreciated, and hope that our common interests will always rise above any differences.
Just "discovered" a flat polyhedron as follows:
It's graph is K3,3 of non- Planar graph fame. It satisfies the diamond property, I guess it's properly connected. Anyone encountered this before? SteveWoolf ( talk) 06:10, 7 November 2008 (UTC)
Here's what was said below (moved up here, where it makes more sense to put it):
Grünbaum has kindly sent me a lecture note of his from 2001, in which he describes several (geometrically degenerate) hexagonal trihedra having vertices numbered 1-6 and faces [1,2,3,4,5,6] [1,4,3,6,5,2] and [1,6,3,2,5,4]. The same underlying map (i.e. the associated abstract polytope) is common to all, and is regular. -- Cheers, Steelpillow ( Talk) 21:39, 16 November 2008 (UTC)
There is more about this family and related ones in Grünbaum's note. Email me if you want a copy of the PDF. -- Cheers, Steelpillow ( Talk) 08:28, 21 November 2008 (UTC)
Irregular polytopes are interesting too!
The square pyramid is, I think (any dissenters?), the "simplest" irregular abstract polytope, but it still has many symmetries, i.e. "equivalent" vertices, edges, and faces. I leave "simplest" a bit vague.
What is the "simplest" polytope with no automorphisms - i.e. no two equivalent i-faces? It might be the one shown - can anyone find a simpler one? I'm 99% sure it is "amorphic", i.e. without automorphisms (any dissenters?)
It has 7 vertices, 12 edges and 7 faces and is self-dual. Self-duality clearly implies palindromic face cardinalities, but the converse I'm sure is false (any examples, anyone?)
I'd also be interested in the simplest amorphic 4-polytope - anyone? SteveWoolf ( talk) 07:50, 4 December 2008 (UTC)
I am pleased to report some interim results from my attempts to formalise "nice" and "nasty" polytope concepts:
Since our abstract polytope concept is much more general than the traditional polytopes, my goal here is to formalise, strictly in abstract polytope terminology, which objects in our polytope menagerie correspond to traditional polytopes. Or, to define "angelic" (e.g.!) in such a way that there is precisely one "angelic" abstract polytope for every (equivalence) class of combinatorially equivalent traditional polytopes, and no others.
I am a bit confused about whether this is equivalent to "Realisable".
Of course, the latter class (i.e Traditional polytopes) is ill-defined. And we may therefore need more than one definition. But it is not a hopeless task, because one of the definitions may be more elegant and useful than the others. This could then be modified/qualified at will to include/exclude various classes, such as finite/infinite, spherical/toroidal etc.
So... clearly, traditional polytopes DO satisfy (1) and (2) above. Is that enough? I have also seen it stated that convex polytopes have face lattices that are meet/join distributive. I haven't thought that through yet.
I personally think toroidal polytopes are "nice", though I didn't always. My reason is that genus is a property of a polytope, not a defining characteristic. To say a polytope is an object that [blah, blah, blah...] and has genus 0 smacks of arbitrariness, and, in my view, not the stuff of, or prelude to, great mathematics.
To summarise, I think it is a significant question to frame what the definition of Abstract Polytope might have been, if the goal had been only to capture the combinatorial properties of traditional polytopes, instead of letting extra-terrestial aliens into our polytope zoo. Not to say that the aliens aren't interesting, just not as cuddly as koalas and crocodiles. SteveWoolf ( talk) 07:43, 17 December 2008 (UTC)
I have - temporarily - removed the following related paragraph from the article, until the above topics are finalised.
SteveWoolf ( talk) 06:00, 19 December 2008 (UTC)
Can anyone tell me whether the "Nice" conditions (1) and (2) and Realisability have any "implication" connection, i.e. do any of these three imply each other? SteveWoolf ( talk) 12:05, 11 November 2008 (UTC)
Most unfortunately, the term "Lattice Polytope" often means something quite different - a Euclidean polytope all of whose vertices have integer-value coordinates. I guess the only way to be clear is to say "Lattice-Poset" or "Meet/Join Lattice". In any case the "Integer"-Lattice concept is meaningless for an AP, but we'd better be careful here.
It also seems that "Atomistic" and "Atomic" are confused. We should only use "Atomistic", meaning that every k-face is a unique join of vertices, ie, no 2 faces have the same vertex set. Coatomistic is the dual concept.
Guys, we can't keep referring to properties "1 and 2" ! But many authors are quite clear that (at least most) trad pols do satisfy them. They are clearly important and basic properties of tradional polytopes. So let's choose a term. I hardly think we want to really settle on "Nice" for serious work, nor "Angelic". I suggest "
Proper" as an interim term, until a respected published author decrees otherwise, or we here decide on something else.
So a triangle will be a proper polytope, and a digon will be improper. Some of you may object to some of your revered pols having their image tarnished - but it's better than nasty or "imaginary".
Just to be quite clear: this is just an in-house term, not for article use (at this point)! SteveWoolf ( talk) 22:07, 19 December 2008 (UTC) SteveWoolf ( talk) 03:39, 22 June 2010 (UTC)
I asked myself what were the smallest abstract toroids. I came up with two candidates, both regular polyhedra, tho' I didn't try to find any nasty ones.
The first was the one shown. The second was a "ring of eight tetrahedral cells". It has 8 vertices, 28 edges, and 24 faces. It's graph is K8, the complete graph with 8 vertices, as is also the 7- simplex.
I suspect that every "single-circuit" n-toroid is isographic (has the same graph as) to an (n+1)-spherical - anyone know?
Any smaller ones?
Thanks for the feedback. Actually, for abstract polytopes at least, the 4-polytope (Cartesian product of two triangles) is not the same polytope. The one in my picture, as I very informally defined it, is a polyhedron. What these two do have in common is that they are isographic - both have the same graph, i.e. the same vertices and edges.
I will explain this in detail. If you "glue" two square pyramids together on their square faces, you get an octahedron. The square faces are then no longer faces. This point is crucial. Now, take four cubes, but let them be elongated and with bevilled (45°) ends, so that you can easily arrange them in a "ring of 4 cubes". Now you have a toroidal polyhedron with 4 polyhedral cubic cells - a "TetraCube", perhaps?. It has the same graph as a 4-cube, but the 4-cube has two more 3-cells, which you can think of as the "hole" in your TetraCube (the polytope, that is), and the outside of it. has 4 more cubic cells.
In years past, I puzzled over this at great length. Was this a real difference, or just airy-fairy word-play? The answer is that it is real. Only the vertices and edges are the same. The 2-faces and 3-faces (cells) are different in the two cases, so the poset is different, and not isomorphic.
The confusion arises because when we draw pictures, we normally only draw vertices and edges and leave faces uncoloured.
As a polyhedron, my toroid has NO triangular faces, while the 4-d 3,3-duoprism does.
nI hope you found this interesting, it was a Great Leap foreward for me when I got it! I guess this topic should be in the article. Regards SteveWoolf ( talk) 05:59, 20 December 2008 (UTC)
As I noted earlier, "I suspect that every "single-circuit" n-toroid is isographic (has the same graph as) to an (n+1)-spherical". As you say, you can't just prune the Hasse diagram (poset) at will and always get a valid polytopic poset! Note that Axiom 4 of the AP defn requires that every 1-section is an edge, not only the 1-faces.
Also as I said, any two polyhedra with a "matching" face can be catenated. Clearly this can be continued indefinitely to produce a tree of polyhedral cells. Furthermore, loops are allowed. I suspect spherical structures too - a "ball" of polyhedra, which of course can have loops and "hairs" on it, and so on. I suspect this is true in any dimension ≥ 3. It is not true for n<3; catenating 2 polygons only gives you a bigger polygon. See my section Talk:Abstract polytope#Polytope Catenation. You cannot have a 2-polytope with more than one 2-face - your mesh of 165 triangles isn't a polytope if the higher k-faces are removed. The "1-sections are edges" rule won't work.
Now, any n-polytope, n>3, will contain many 3-cells. There will be many cases of adjacent 3-cells among them, i.e. sharing a 2-face. From these adjacencies you can select many trees or loops etc., which will be multicelled polyhedra. I imagine this holds for any higher dimension than 2.
So, here is one way to create many, many m-pols that can be created from n-pols where m≤n. SteveWoolf ( talk) 04:39, 21 December 2008 (UTC)
![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | Archive 2 | Archive 3 | Archive 4 | Archive 5 | Archive 6 |
Unless anything new has been published recently, the theory of abstract polytopes is scattered across many sources, and often expressed in rather convoluted ways. Prof. Norman Johnson has developed a more understandable and succinct approach, which is due for publication in a forthcoming book on uniform polytopes. In the mean time an accessible summary, Polytopes - abstract and real, is available here. It might provide some useful ideas, especially where the present article risks moving into original research ( wp:or). -- Cheers, Steelpillow ( Talk) 10:29, 26 October 2008 (UTC)
To enlarge a little, some key properties which he highlights, and names, are that an abstract polytope is monal, dyadic and properly connected. He also distinguishes facials from faces (facets). -- Cheers, Steelpillow ( Talk) 11:05, 26 October 2008 (UTC)
Conflict of Terminology. If I understand correctly, Johnson considers it most important that we make a clear distinction between very general abstract polytopes (eg Digon, Hemicube) and more familiar ones (eg triangle, cube). I absolutely agree with this (see my #Suggested Improvements to the Definition below, §4). However - "Real" is a term that cannot be used. Coxeter already distinguishes between Real and Complex polytopes, and in any case the term Real as in Real number is used widely throughout mathematics, and especially in the many areas of abstract geometry.
Perhaps Normal would be a better term (and should certainly appeal to Prof Norman Johnson!)
There are clearly some of us who would have preferred the definition of Abstract Polytope to be less general; but as mike40033 aptly pointed out previously, us lesser mortals have to respect, however reluctantly, established terminology in order to have a common language with which to exchange - and thereby advance - ideas. So... let's try to reach a consensus on what to call polytopes that don't include the dastardly digon and the heinous hemicube. SLWoolf ( talk) 05:47, 28 October 2008 (UTC)
Coincident vs Same Vertices/i-Faces. Johnson talks about coincident vertices. In a digon, the two edges have the SAME two vertices. That is what a Digon is. While two separate edges ab and cd in (e.g.) Euclidean space can be coincident, this concept does not carry over into Abstract Polytopes, where the only properties that matter are the combinatorial poset (face lattice) relations. I think the issue here, as far as Abstract Polytopes go, is whether or not two i-faces are allowed to have the SAME vertex sets (and, dually, facet sets) as they do in the digon or the hemicube. To repeat, the concept of coincident vertices or other i-faces has no meaning for an abstract polytope. SLWoolf ( talk) 05:47, 28 October 2008 (UTC)
Thanks for the response. Yes, maybe I misunderstood Johnsons's definition of Real Polytope. Returning to the subject of "Normal" vs other polytopes: I strongly feel that the subject of Abstract Polytopes needs to be self-contained and not merely an adjunct of classical polytopes, in much the same way that topology has become. Euclidean or Real space is merely one of many geometries and has no special place (indeed it is only an approximation to the real universe and fails both in the microworld of subatomic physics and the macroworld of cosmology). Therefore, I think the difference between polytopes such as a cube and those such as a hemicube should be defined abstractly primarily, and not in terms of realisability in Real Space. I think the crucial distinction is whether or not different i-faces can have the same vertices, and dually, (containing) facets. Then, if this condition should turn out to be equivalent to realisability, no-one will be more pleased than myself.
I don't think we should be always intimidated and muzzled (us Woolfs like to howl) by the No Original Research thing - this is only a talk page, and anything of value we "discover" here will probably already have been already covered by the publishing elite - or soon will be, at which point any worthwhile material can be promoted to the article and referenced. SLWoolf ( talk) 06:25, 29 October 2008 (UTC)
Your feedback greatly appreciated. Have scrutinised the hemidodecahedron's incidences and as you say, my "uniqueness" criterion doesn't eliminate it as it does the digon and hemicube. Well, we fall down, then get up the stronger for having tested the terrain. Which leaves me wondering - is there a purely abstract property that is provably equivalent to realisability - is that the outstanding problem you mentioned? I still feel that we need a purely abstract concept of Normal vs Abnormal, but as to what that means, right now my intuition fails me - and intuition is pretty unreliable anyway.
You never know, us amateurs might just see something the pros have missed - I think it has happened now and again. Edwin Hubble, for example. SLWoolf ( talk) 05:04, 30 October 2008 (UTC)
We agreed about archiving and I spent some time doing that. Against Wikipedia policy, you have edited this archive and therefore compromised it as a historical record. Please revert your edit, THEN copy back just the parts of this section that you consider still worthwhile, but please omit the rubbish which only serves to clutter the section and the page. Hope you agree with this. SteveWoolf ( talk) 07:25, 3 November 2008 (UTC)
Johnson presents a definition of an abstraction of the "polytope", which he claims to be equivalent to Schulte & McMullen's abstract polytopes. This is fine, indeed wonderful if his definition
However, he then defines his own terminology and notation. And this is where I put my foot down and say No!!
The literature is already littered with too many sets of terminology for basically the same thing. We have
As well as conflicting notation and terminology from fields such as
The best way to get people to understand more about abstract polytopes is not to invent (putatively) "better" notation, but to unify (ha ha) the existing terminologies, or at least produce an easy way to translate between them.
So sorry, I can't support Johnson's terminology and notation being used in the abstract polytopes article. However, translataions of his ideas into the "standard" terminology would of course be welcome. mike40033 ( talk) 06:27, 26 November 2008 (UTC)
I have archived some of our resolved issues, obsolete waffle, and interminable repetitions and interminable repetitions. Unfortunately, they were very much mixed in with some good ideas which should remain "current". So I decided the best solution was to archive many sections, then copy back the best paragraphs, editing and regrouping as seemed appropriate. What we lose in the historical sequence of our talk (which is still viewable in Archive 2) will be more than compensated by a very much better (usable!) presentation of ideas.
I hope the result is acceptable to all - but inevitably I had to make judgements re what to keep. I tried to be objective, but if anyone feels I have omitted anything of value then go ahead and revive it.
Guy - I left the huge "Johnson's approach" section for you to archive or not as you see fit, but it would be a good idea to condense it to the main points at least. SteveWoolf ( talk) 05:48, 25 December 2008 (UTC)
The long preceding discussion is in Archive 2.
So, let us go with "Traditional", at least until some other Great Work is published. ARP should be our terminology Bible, as we agreed. I hope Guy will not be too disconcerted. It has to be better than our current mish-mash, at least! SteveWoolf ( talk) 13:20, 25 December 2008 (UTC)
If anybody is interested in joining a private mailing list, discussing everything polyhedral from sculpture to abstract polytopes, drop me a line through the Wikipedia mail system. I half-recognise many ideas here, such as products of two polytopes, from previous discussions. -- Cheers, Steelpillow ( Talk) 09:07, 7 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 with the fact that the definition allows such an ugly duckling. mike40033 ( talk) 06:32, 6 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)
There is also a relevant discussion in Talk:Spherical polyhedron#Rigorous Definition regarding Sphericality. SteveWoolf ( talk) 14:27, 25 December 2008 (UTC)
A polytope is spherical if and only if one of the following hold :
mike40033 ( talk) 08:28, 31 October 2008 (UTC)
But now your defined "spherical" concept doesn't at all match what I thought it meant. Consider a "bicube" - two cubes pasted together on one face. This can be "inscribed" on a sphere, or, equivalently, has a planar graph. Does that not make it spherical? Yet it is not universal, as you have "defined" that. Can you clarify this apparent clash of definitions? SteveWoolf ( talk) 03:37, 30 December 2008 (UTC)
SteveWoolf ( talk) 02:38, 31 December 2008 (UTC)
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 χ = 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:
It is trivial to find χ for any abstract polytope. If rank (dimensionality) = 3 and χ = 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)
McMullen/Schulte's ARP p153 says:
(Thanks CunningGabe for that).
However, I feel that we
So here it goes (for simplicity, I'll talk about polyhedra):
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)
See Talk:Abstract polytope#2nd Definition of Spherical. SteveWoolf ( talk) 14:48, 25 December 2008 (UTC)
The article Simplex (Relation to the (n+1)-hypercube), states, correctly, that the Hasse diagram of an n-simplex is the graph (1-skeleton) of an (n+1)cube.
In fact (I think - any dissenters?) not only is any Hasse diagram of an n-polytope P the graph of an (n+1)-polytope, but this latter polytope is easily defined as the set of all sections of P, ordered by inclusion. Note that a section is precisely an (order-theoretic) interval in the original poset.
For example, in the triangle abc, the section ab/a is {a, ab}, while both the sections ab/c and a/ab would be ø, the empty set. This gives multiple ø's in general, which is okay since, in set theory, {a} {a} = {a}, so there's only one null face.
A triangle generates a cube; a square gives a polyhedron with 8 equivalent tetragonal faces, but the edges and vertices and not equivalent. The null polytope gives the 0-polytope, the latter the 1-polytope, then the square.
As a direct consequence of the diamond property, any 2-face of a Hasse Polytope must be a tetragon, and consequently most polytopes are not the Hasse Polytope of another polytope.
Terminology. Alternative names: Meta-polytope, metatope, hierotope... any more suggestions?
Original Research. Is it? (Did I actually discover something?) As I don't have the financial, mental or time resources to easily find out, would any of you be interested in researching this? And what happens to this or any other new ideas, especially worthy ones? Can humbler mortals publish? How? What is a reasonable working definition of "published"?
SteveWoolf ( talk) 11:12, 2 November 2008 (UTC)
As noted in Prism (geometry), the Cartesian Product of two polytopes gives another polytope. For example an n-polygon times an edge (1-polytope) gives an (n+1)-prism. Generalising, an m-polytope P times an n-polytope Q gives an (m+n)-polytope. The faces of the new polytope are precisely the set of pairs FG for each F of P, G of Q, except that each and every Fø and øG pair become the same single new empty face in the new polytope. Otherwise (i.e. when both i, j >=0), each i-face, j-face pair gives an (i+j)-face of the new polytope.
The product of 2 squares is a 4-cube. Multiplying by a point leaves a polytope unchanged (rather like x 1); multiplying by the null polytope gives a null polytope (rather like x 0).
It is easily seen that if P has v vertices and Q has w verices, the product polytope has vw vertices, and therefore no polytope with a prime number of vertices can be a (nontrivial) product polytope. The converse is false - a hexagon is not a product. In fact, if P and Q both have dimension > 0, the product will have at least one tetragonal face, since edge x edge = square.
If there is a consensus to do so, I will be happy to make a section in the article with an elegant, tabulated example showing how all the faces are computed. Though I discovered this independently, I expect there are refs about this somewhere.
Yes/No, because...? SteveWoolf ( talk) 19:07, 5 November 2008 (UTC)
A Simple n-polytope is best defined as an n-polytope all of whose vertex-figures are (n-1)-simplexes. This is equivalent to saying that each vertex is contained in n edges.
The articles Simple polytope and Simplicial polytope should be amended accordingly. SteveWoolf ( talk) 06:08, 25 December 2008 (UTC)
Can hardly keep up with pace of ideas - but that's great, feedback greatly appreciated, and hope that our common interests will always rise above any differences.
Just "discovered" a flat polyhedron as follows:
It's graph is K3,3 of non- Planar graph fame. It satisfies the diamond property, I guess it's properly connected. Anyone encountered this before? SteveWoolf ( talk) 06:10, 7 November 2008 (UTC)
Here's what was said below (moved up here, where it makes more sense to put it):
Grünbaum has kindly sent me a lecture note of his from 2001, in which he describes several (geometrically degenerate) hexagonal trihedra having vertices numbered 1-6 and faces [1,2,3,4,5,6] [1,4,3,6,5,2] and [1,6,3,2,5,4]. The same underlying map (i.e. the associated abstract polytope) is common to all, and is regular. -- Cheers, Steelpillow ( Talk) 21:39, 16 November 2008 (UTC)
There is more about this family and related ones in Grünbaum's note. Email me if you want a copy of the PDF. -- Cheers, Steelpillow ( Talk) 08:28, 21 November 2008 (UTC)
Irregular polytopes are interesting too!
The square pyramid is, I think (any dissenters?), the "simplest" irregular abstract polytope, but it still has many symmetries, i.e. "equivalent" vertices, edges, and faces. I leave "simplest" a bit vague.
What is the "simplest" polytope with no automorphisms - i.e. no two equivalent i-faces? It might be the one shown - can anyone find a simpler one? I'm 99% sure it is "amorphic", i.e. without automorphisms (any dissenters?)
It has 7 vertices, 12 edges and 7 faces and is self-dual. Self-duality clearly implies palindromic face cardinalities, but the converse I'm sure is false (any examples, anyone?)
I'd also be interested in the simplest amorphic 4-polytope - anyone? SteveWoolf ( talk) 07:50, 4 December 2008 (UTC)
I am pleased to report some interim results from my attempts to formalise "nice" and "nasty" polytope concepts:
Since our abstract polytope concept is much more general than the traditional polytopes, my goal here is to formalise, strictly in abstract polytope terminology, which objects in our polytope menagerie correspond to traditional polytopes. Or, to define "angelic" (e.g.!) in such a way that there is precisely one "angelic" abstract polytope for every (equivalence) class of combinatorially equivalent traditional polytopes, and no others.
I am a bit confused about whether this is equivalent to "Realisable".
Of course, the latter class (i.e Traditional polytopes) is ill-defined. And we may therefore need more than one definition. But it is not a hopeless task, because one of the definitions may be more elegant and useful than the others. This could then be modified/qualified at will to include/exclude various classes, such as finite/infinite, spherical/toroidal etc.
So... clearly, traditional polytopes DO satisfy (1) and (2) above. Is that enough? I have also seen it stated that convex polytopes have face lattices that are meet/join distributive. I haven't thought that through yet.
I personally think toroidal polytopes are "nice", though I didn't always. My reason is that genus is a property of a polytope, not a defining characteristic. To say a polytope is an object that [blah, blah, blah...] and has genus 0 smacks of arbitrariness, and, in my view, not the stuff of, or prelude to, great mathematics.
To summarise, I think it is a significant question to frame what the definition of Abstract Polytope might have been, if the goal had been only to capture the combinatorial properties of traditional polytopes, instead of letting extra-terrestial aliens into our polytope zoo. Not to say that the aliens aren't interesting, just not as cuddly as koalas and crocodiles. SteveWoolf ( talk) 07:43, 17 December 2008 (UTC)
I have - temporarily - removed the following related paragraph from the article, until the above topics are finalised.
SteveWoolf ( talk) 06:00, 19 December 2008 (UTC)
Can anyone tell me whether the "Nice" conditions (1) and (2) and Realisability have any "implication" connection, i.e. do any of these three imply each other? SteveWoolf ( talk) 12:05, 11 November 2008 (UTC)
Most unfortunately, the term "Lattice Polytope" often means something quite different - a Euclidean polytope all of whose vertices have integer-value coordinates. I guess the only way to be clear is to say "Lattice-Poset" or "Meet/Join Lattice". In any case the "Integer"-Lattice concept is meaningless for an AP, but we'd better be careful here.
It also seems that "Atomistic" and "Atomic" are confused. We should only use "Atomistic", meaning that every k-face is a unique join of vertices, ie, no 2 faces have the same vertex set. Coatomistic is the dual concept.
Guys, we can't keep referring to properties "1 and 2" ! But many authors are quite clear that (at least most) trad pols do satisfy them. They are clearly important and basic properties of tradional polytopes. So let's choose a term. I hardly think we want to really settle on "Nice" for serious work, nor "Angelic". I suggest "
Proper" as an interim term, until a respected published author decrees otherwise, or we here decide on something else.
So a triangle will be a proper polytope, and a digon will be improper. Some of you may object to some of your revered pols having their image tarnished - but it's better than nasty or "imaginary".
Just to be quite clear: this is just an in-house term, not for article use (at this point)! SteveWoolf ( talk) 22:07, 19 December 2008 (UTC) SteveWoolf ( talk) 03:39, 22 June 2010 (UTC)
I asked myself what were the smallest abstract toroids. I came up with two candidates, both regular polyhedra, tho' I didn't try to find any nasty ones.
The first was the one shown. The second was a "ring of eight tetrahedral cells". It has 8 vertices, 28 edges, and 24 faces. It's graph is K8, the complete graph with 8 vertices, as is also the 7- simplex.
I suspect that every "single-circuit" n-toroid is isographic (has the same graph as) to an (n+1)-spherical - anyone know?
Any smaller ones?
Thanks for the feedback. Actually, for abstract polytopes at least, the 4-polytope (Cartesian product of two triangles) is not the same polytope. The one in my picture, as I very informally defined it, is a polyhedron. What these two do have in common is that they are isographic - both have the same graph, i.e. the same vertices and edges.
I will explain this in detail. If you "glue" two square pyramids together on their square faces, you get an octahedron. The square faces are then no longer faces. This point is crucial. Now, take four cubes, but let them be elongated and with bevilled (45°) ends, so that you can easily arrange them in a "ring of 4 cubes". Now you have a toroidal polyhedron with 4 polyhedral cubic cells - a "TetraCube", perhaps?. It has the same graph as a 4-cube, but the 4-cube has two more 3-cells, which you can think of as the "hole" in your TetraCube (the polytope, that is), and the outside of it. has 4 more cubic cells.
In years past, I puzzled over this at great length. Was this a real difference, or just airy-fairy word-play? The answer is that it is real. Only the vertices and edges are the same. The 2-faces and 3-faces (cells) are different in the two cases, so the poset is different, and not isomorphic.
The confusion arises because when we draw pictures, we normally only draw vertices and edges and leave faces uncoloured.
As a polyhedron, my toroid has NO triangular faces, while the 4-d 3,3-duoprism does.
nI hope you found this interesting, it was a Great Leap foreward for me when I got it! I guess this topic should be in the article. Regards SteveWoolf ( talk) 05:59, 20 December 2008 (UTC)
As I noted earlier, "I suspect that every "single-circuit" n-toroid is isographic (has the same graph as) to an (n+1)-spherical". As you say, you can't just prune the Hasse diagram (poset) at will and always get a valid polytopic poset! Note that Axiom 4 of the AP defn requires that every 1-section is an edge, not only the 1-faces.
Also as I said, any two polyhedra with a "matching" face can be catenated. Clearly this can be continued indefinitely to produce a tree of polyhedral cells. Furthermore, loops are allowed. I suspect spherical structures too - a "ball" of polyhedra, which of course can have loops and "hairs" on it, and so on. I suspect this is true in any dimension ≥ 3. It is not true for n<3; catenating 2 polygons only gives you a bigger polygon. See my section Talk:Abstract polytope#Polytope Catenation. You cannot have a 2-polytope with more than one 2-face - your mesh of 165 triangles isn't a polytope if the higher k-faces are removed. The "1-sections are edges" rule won't work.
Now, any n-polytope, n>3, will contain many 3-cells. There will be many cases of adjacent 3-cells among them, i.e. sharing a 2-face. From these adjacencies you can select many trees or loops etc., which will be multicelled polyhedra. I imagine this holds for any higher dimension than 2.
So, here is one way to create many, many m-pols that can be created from n-pols where m≤n. SteveWoolf ( talk) 04:39, 21 December 2008 (UTC)