Talk:Abstract polytope/Archive 3

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Johnson's approach

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)

Johnson's primary concern here is to provide a bridge between abstract polytopes, which are purely algebraic, and real polytopes which exist as geometrical objects in Euclidean spaces - i.e. coordinate spaces whose coordinates are real numbers. The use of "real" here is therefore consistent with mathematical usage. Johnson focuses more on the distinction between an abstract cube and a real cube, rather than that between say and abstract cube and an abstract hemicube, which is of no help to your distinction between those abstract polytopes which can be realized faithfully in real spaces vs. those which cannot. I do not think this issue has been resolved, so there is probably little to say about it and consequently little need for a simpler term than "realisable fathfully" - and in any case it would be OR and so not allowed on Wikipedia. (We can of course realise a hemicube unfatithfully as say a tiling of the Boy surface). -- Cheers, Steelpillow (Talk) 13:36, 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)

It is possible in a sense for vertices (rank 1 elements) to be coincident in an abstract poset. For example we might consider the regular compound of five cubes abstractly as having several cubes meeting at a single vertex. This poset is not a polytope, because it is not properly connected. If we treat the vertices of each cube as distinct (i.e. not abstractly coincident), then we obtain five proper abstract polyhedra, whose realization has coincident vertices in real space. This is the point which Johnson sought to make. (Note that Johnson's definition of faithfulness does not apply to compound figures, but only to the individual members). Provided any abstract poset - be it a digon or whatever - is monal, dyadic and properly connected, then it is a valid polytope. -- Cheers, Steelpillow (Talk) 13:36, 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)

The key point here is that "classical" polytopes are defined only in Euclidean real space. Therefore, any purely abstract theory has no built-in method of identifying them. To do so remains an outstanding problem as far as I am aware, so for us to beat the big guns at their own game is IMHO not very likely. To disprove your conjecture about shared i-faces, consider the hemidodecahedron. It has 6 (i.e. half of 12) pentagonal faces, no two of which share more than the usual single edge and 2 associated vertices, yet it cannot be realised faithfully in Euclidean space. Nor is the Euler characteristic any greater help. -- Cheers, Steelpillow (Talk) 20:46, 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)

Yes, this is the outstanding problem I mentioned. -- Cheers, Steelpillow (Talk) 09:57, 30 October 2008 (UTC)

One of the comments here got me thinking... Can the classical polytopes be identified purely within the abstract theory? After abt 10 minutes of thinking, I think the answer is 'yes'. As far as I recall, this definition doesn't appear in the literature, but it will do... "a spherical abstract polytope is one which (a) has rank 1, or (b) is a finite universal polytope with spherical facets and vertex figures" and let's go one step further... "A tesselation is an abstract polytope with spherical facets and vertex figures". mike40033 (talk) 00:10, 31 October 2008 (UTC)

Interesting ideas...as this section is getting pretty chaotic, I started a new one below, #Spherical vs Toroidal polytopes.SteveWoolf (talk) 07:15, 31 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)

My one objection to Johnson's polytopes

Or, "Yes, I'm finally taking sides"

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

• is indeed equivalent, and
• is indeed simpler

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

• abstract polytopes
• incidence polytopes
• thin geometries (satisfying certain additional properties)

As well as conflicting notation and terminology from fields such as

• eulerian posets
• order theory
• lattice theory

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 to second this. We absolutely MUST have a standard terminology so that we can focus on ideas and their development. And clearly this now has to be that of the Great Work (satyr intended!), Abstract Regular Polytopes.

Just for the record, I myself am not entirely happy with this status quo, as I have oft expressed. I would have preferred "polytope" to have been defined more strictly and more in conformity with classical pols (though still purely combinatorial). In particular, I think no two faces should be allowed the same vertex sets (and ditto the dual); and two faces should always intersect at a single face, possibly ø - and dually. In my previous work alone, I preferred n-cell to n-face as the general element. I had axis instead of flag, coaxial instead of incident, main cell instead of facet, hierograph instead of Hasse Diagram.

HOWEVER - we must live in the real world. SO - I would like to recommend that we all accept ARP terminology as the standard without further ado, and get on with the ideas. In particular, Guy (SteelPillow), I hope, as perhaps the polytope world's most active contributor, will concur with this practical standard, so that the AP world, at least, doesn't end up in the mess that the classical world seems to have got itself. With a solid consensus, hopefully we can then keep AP articles (and sections in non-AP articles) reasonably coherent across Wikipedia, at least. SteveWoolf (talk) 18:51, 26 November 2008 (UTC)

I agree entirely with mike40033. Well said. It's very much in line with Wikipedia policy, too. I sometimes feel that there are more alternatives to the status quo than there are abstract polytopists - and I doubt if I understand any of them fully. Since I have not read ARP, I feel it wiser not to contribute much detail content myself - though contributing to this discussion does no harm. -- Cheers, Steelpillow (Talk) 23:15, 26 November 2008 (UTC)

"I sometimes feel that there are more alternatives to the status quo than there are abstract polytopists" Depending on definitions, this is literally true. But let's not start inventing definitions of "abstract polytopist" now as a form of "definition invention replacement therapy" mike40033 (talk) 01:17, 27 November 2008 (UTC)

Glad we're in agreement here. So let us try to keep AP articles/sections to the ARP standard. Guy, I think as long as each of us is aware of our limitations and willing to admit mistakes and learn, then all is well. Both of us (I think) have goofed at times, seen it, and now are the wiser, with the end result being a better Wikipedia. Unfortunately those with the most knowledge have the least available time. But I think it's also true that those in the middle sometimes miss the wider picture. In the BBC's Life Story, a truly wonderful film about the discovery of DNA's structure in 1953, neither of the methodical experts - Maurice Wilkins and Rosalind Franklin - got the answer, which was found instead by the two passionate amateurs, Watson and Crick, who skillfully collected the various bits of the puzzle from different (and often most ungracious) experts.SteveWoolf (talk) 06:43, 29 November 2008 (UTC)

Archiving

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)

I think it has served its purpose for now, so I archived it in full. -- Cheers, Steelpillow (Talk) 22:28, 27 December 2008 (UTC)

FINAL DECISION: "Traditional"

The long preceding discussion is in Archive 2.

• ARP uses Traditional the most, though McMullen now prefers Geometric.
• Mike (above) voted Classical > Traditional > Non-abstract > Geometric (best first)
• Guy prefers Geometric, but could accept Traditional
• Tetracube likes "Traditional" (or "Schizophrenic")
• CunningGabe last voted for Traditional
• Steve now votes Traditional

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)

Mailing list

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)

Archimedean Dilemma

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)

Nobody knows Archimedes' precise form of words nor its exact meaning. Our knowledge of his work comes to us through Pappus, and it is not clear to the modern translator whether the term for congruence used by Pappus would have been understood in terms of the overall symmetry or not by Archimedes, and/or by Pappus. We can at least credit Archimedes as being concerned with symmetry, and he evidently did not consider the pseudo thingy because it did not fit the ideas he was trying to express. So describing it as "Archimedean" is quite wrong. Modern mathematicians have elegantly resolved the dilemma by considering the transitivity of flags, which works both geometrically and abstractly. (BTW, there are also two pseudorhombicosidodecahedra, so the correct "incorrect" count is 16). -- Cheers, Steelpillow (Talk) 22:41, 27 December 2008 (UTC)

Abstract Polytope Definition too general?

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)

Spherical polytopes and Topology

There is also a relevant discussion in Talk:Spherical polyhedron#Rigorous Definition regarding Sphericality. SteveWoolf (talk) 14:27, 25 December 2008 (UTC)

Mike's Definition

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)

I assume your definition is recursive and not circular re "spherical"!

By sheer conciseness, your definition is probably better than my 2 below; unfortunately, my maths abilities aren't yet up to understanding "universal". It would however, be gratifying to know if my definitions, possibly amended, were equivalent. SteveWoolf (talk) 07:29, 27 December 2008 (UTC)

"Universal" is a bit like this...

* If you start pasting squares together, edge to edge, four per vertex, you can keep doing this as long as you want. "Eventually", you'd get the tesselation of the plane with squares. This tesselation is the "universal" {4,4}. Or, you could decide after a while, "I quit", fold up your tesselation into an infinitely long tube, or a torus. The latter are not universal.

* If you start pasting pentagons together, three per vertex, you find that you are quickly forced to close the shape up into a dodecahedron. The dodecahedron is the "universal" {5,3}. Or, you could decide to close it up early, twisting it into a tesselation of the projective plane, the hemi-dodecahedron. This latter is not universal, since you forced certain faces together that didn't need to go together.

* If you start pasting dodecahedra together, face-to-face, making each vertex figure a hemi-icosahedron, eventually, it will close up by itself (at about the 50000000th dodecahedron) This is the universal {{5,3},{3,5}/2}. Or, there are a couple of opportunities to close the figure early and get a smaller figure, obtaining non-universal quotients of the universal {{5,3},{3,5}/2}.

* Ok, that example was not very instructional, I know. But the idea is if you build your polytope, face of type K by face, keeping the vertex figures of type L - if you never force any K 's to join others, but only let them join as the geometry dictates, you get the universal {K,L} - if any polytope at all exists with facets K and vertex figures L. On the other hand, if you do force certain faces together before their time, you get a non-universal quotient of the universal {K,L}.

This is not clear from the definition, but it's much more intuitive. mike40033 (talk) 02:00, 29 December 2008 (UTC)

Is the square tiling of the plane truly universal? There is a sponge (or infinite skew) polyhedron {4.6} which has six squares at each vertex - so forcing the vertex closed after just four squares is surely non-universal? Also, in the hyperbolic plane any arbitrarily large values of K and L will yield a polytope or tiling. So how can we define "forcing together before their time"? -- Cheers, Steelpillow (Talk) 18:32, 29 December 2008 (UTC)

It's the universal polytope with squares for facets, and squares for vertex figures. There's no reason a priori reason to force the vertex figures to close with four squares each - but there's also no a priori reason to force the faces themselves to close after four edges. The only good reason is if we happen to be interested in polytopes with square faces, and square vertex figures. Then, it's useful to try to understand the universal one. Again, continuing your line of thought, let me point out that the universal {∞,∞} (with apeirogonal faces, infinitely many per vertex) covers all other polygons. I believe this is called the free polytope of rank 3. mike40033 (talk) 01:14, 30 December 2008 (UTC)

BTW, I didn't intend the above as a definition, just as a guide to intuition. If you get a chance to play with Klikko (see tinyurl.com/9tuos8 for example) or similar toys, try clicking together pentagons, three per vertex, you'll know what I mean by faces being forced together. Actually, I'm sure you already do. mike40033 (talk) 01:14, 30 December 2008 (UTC)

Thanks for that - yes I did have the basic idea from my early polytope days, but pleased to have it matched up with "universal". Is it known when {K,L} exists, or is that question open? HNY SteveWoolf (talk) 03:21, 30 December 2008 (UTC)

This is, in general, a very open question. But a collection of specific cases have been solved. mike40033 (talk) 04:49, 31 December 2008 (UTC)

A Bi-Cube drawn on a sphere.

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)

Steve, you'd better explain to us what you do think "spherical polytope" means. Taken out of context, it is ambiguous. It can mean a polytope inscribed on a spherical surface (the original "metric" meaning), or it can mean a polytope whose surface is a topological sphere and has a planar graph (the abstract meaning). These are not equivalent - for example the tetrahemihexahedron may be inscribed on a sphere (see H.S.M. Coxeter, M.S. Longuet-Higgins, J.C.P. Miller, "Uniform polyhedra", Phil. Trans. 246 A, 1954, pp 401-450) but is not a topological sphere. Neither set of "spherical polytopes" is a subset of the other, although they do overlap. Meanwhile I suspect that the bicube, being structurally the same as a cube, is indeed abstractly universal. -- Cheers, Steelpillow (Talk) 15:49, 30 December 2008 (UTC)

Okay, informally, I thought an n-polytope is "spherical" if it is combinatorially equivalent to a polytope that can be drawn on the surface of an n-sphere. Thus a bicube is spherical. The 3-dimensional Ring of Cubes can't be, and is, "toroidal" - I think.

With the proviso that the image on the sphere is convex, that is correct. It is in effect a geometrical equivalent to the abstract definition in terms of Euler characteristic. Sorry I misunderstood the structure of the "bicube" - with all vertices tangent to a sphere its parts are not two "cubes", but two square frustra. -- Cheers, Steelpillow (Talk) 13:00, 31 December 2008 (UTC)

I don't see how the bicube can be universal going by Mike's informal description of what universal means. In any case, is not a universal polytope necessarily regular? The bicube is not, tho' it has some symmetries. It cannot be "structurally the same as a cube" in any sense useful to us APists. HNY SteveWoolf (talk) 04:21, 31 December 2008 (UTC)

Mm - perhaps a universal polytope needn't be regular if either K and L are not??? But surely this must be true:

A universal polytope {K, L} made up from regular polytopes K joined together with regular vertex figures L is regular.

SteveWoolf (talk) 02:38, 31 December 2008 (UTC)

Yes, you've caught me. My definition only defines regular spherical polytopes. mike40033 (talk) 04:49, 31 December 2008 (UTC)

Okay, I'm sure you agree that we need a fundamental (abstact) definition that includes the irregulars. Can you comment on my Talk:Abstract polytope#2nd Definition of Spherical, based on my Catenation discussion?

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 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:

• Spherical: χ = 2, g = 0,
• Boy surface: χ = 1, k = 1
• Toroidal: χ = 0, g = 1,
• Klein bottle: χ = 0, k = 2
• Higher genus orientable toroids: χ < 0, g > 1,
• For higher genus non-orientable toroids, there is a trade-off between g and k.

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)

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)

Read the section on exchange maps in the article. Proposed definition : a polytope is orientable if no odd sequence of exchange maps returns every flag to itself. mike40033 (talk) 04:51, 31 December 2008 (UTC)

Steve's 1st Definition

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 χ 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 χ 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:

χ = E0 - E1 + E2 - ... +/- E(n-1)

For a polygon (all polygons are 1-spherical), χ = 0

For a spherical polyhedron, χ = 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#-1 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)

Steve's 2nd Definition

See Talk:Abstract polytope#2nd Definition of Spherical. SteveWoolf (talk) 14:48, 25 December 2008 (UTC)

The "Hasse Polytope"

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} $\cup$ {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)

I think (though I'm not completely sure) that this is the polytope $2^{K}$ considered in section 8C of Abstract Regular Polytopes (where K is a polytope). It is in fact part of a more general family described in 8B. McMullen and Schulte's conclusions about $2^{K}$ are for the specific case where K is regular, so without doing some digging, I'm not sure which of the results would carry over to the general case. For the regular case, the vertex figures of $2^{K}$ are isomorphic to K, and the (i+1)-faces are isomorphic to $2^{K_{i}}$, where $K_{i}$ is the i-face of K. In particular, the 2-faces are isomorphic to $2^{K_{1}}$ which is a square as required. Also, if K is an n-simplex, then $2^{K}$ is an (n+1)-cube. -CunningGabe (talk) 18:02, 20 November 2008 (UTC)

Cartesian Products

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)

We ought to find the refs (or you ought to publish) first... Have you proved that the product satisfies all the axioms of an AP? mike40033 (talk) 06:28, 6 November 2008 (UTC)

(a) Have you got time for ref-hunting?

(b) How does one publish? Who/what/where counts as a reference?

(c) Have not proved it, only noticed that it seems to work always, i.e. the result is a polytope. I will see if I can prove it. Usually 3 of the axioms are easy (1,2,4) - it's the the connectedness axiom 3 that's trickier.SteveWoolf (talk) 10:23, 6 November 2008 (UTC)

(a) Sorry, but no... :-(

(b) I'm not sure what counts as publication with respect to the NOR guideline. There must be a guideline on that too... Certainly, an article in a peer-reviewed journal is sufficient, but probably impossible to obtain for most of wikipedia

(c) I think you'll find 3 easy in this case... mike40033 (talk) 06:10, 11 November 2008 (UTC)

Simple and Simplicial polytopes

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.

Agreed, especially in light of the definition of Simplicial polytopes. CunningGabe (talk) 23:46, 20 November 2008 (UTC)

An abstract polytope whose facets and vertex figures are all simplices is a simplex. Look up Hartley M. I., "Polytopes of Finite Type". mike40033 (talk) 01:11, 21 November 2008 (UTC)

The articles Simple polytope and Simplicial polytope should be amended accordingly. SteveWoolf (talk) 06:08, 25 December 2008 (UTC)

Hexagonal Trihedron

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:

6 Vertices: a b c d e f

9 Edges: ab bc cd de ef fa ad be cf

3 Hexagonal 2-Faces: abcdef(a) abefcd(a) adebcf(a)

It's graph is K_3,3 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)

Wow! Congrats! It might be the regular polytope called {6,3}_(1,1). DOes it have symmetries mapping, say {a,ab,abcdef} to any other flag? Can you draw your three hexagons on a torus?mike40033 (talk) 07:22, 7 November 2008 (UTC)

Much appreciated. Don't know the answers, a bit out of my depth, though as time permits I will try to remedy that. I do intend to respond to all raised topics, but hard to keep up with so many! SteveWoolf (talk) 08:14, 7 November 2008 (UTC)

I suspect that it can be found in one of Grünbaum's papers, possibly Polyhedra with hollow faces. I will try and remember to check. -- Cheers, Steelpillow (Talk) 08:56, 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)

Maybe one of you has time to make an article for this, fine. I'm no fan of flat nasties, tho' I will respect their constitutional rights!

Given that Pandora's abstract hemibox has been opened, I wonder if any modern Saccheri has gone further still - I think denying M-S's 4th postulate (the "diamond" property) might result in some truly repugnant beasts! SteveWoolf (talk) 06:55, 17 November 2008 (UTC)

Schulte's PhD dissertation talked about "Regulare Inzidenzkomplexe", or "regular incidence complexes". A regular incidence polytope was a regular incidence complex with the diamond property - now called an abstract polytope. Not much work has been done on these, but there were one or two articles published. mike40033 (talk) 01:08, 21 November 2008 (UTC)

This seems to be part of a nice family of polytopes with 3 facets, 3n edges, and 2n vertices. All 3 facets are 2n-gons. The 1-skeleton of this graph is a 2n-cycle where opposite vertices are joined by an edge. Then the 3 facets are:

1. The "outer" 2n-cycle.
2. The two 2n-cycles formed from all n chords and every other outer edge.

I believe all of these are toroidal maps, but I'm not completely sure. -CunningGabe (talk) 02:40, 21 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)

The Smallest Irregulars

Is this the simplest polytope with no automorphisms?

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)

There is a simpler irregular polytope: the digonal pyramid! This has 3 vertices, 4 edges, and 3 faces. There is an automorphism, however, sending one of the triangular faces to the other.

You are correct that there are polytopes with palindromic face cardinalities that are not self-dual. An example is the digonal prism, which has 2 square faces and 2 digonal faces, 6 edges, and 4 vertices. The vertex figures are all triangles, so the dual has 4 triangular faces (it seems to be a flattened tetrahedron, so to speak). I'm sure there are other examples even if we outlaw digons as facets or vertex figures.

even regular ones. Check out this mike40033 (talk) 08:29, 6 December 2008 (UTC)

I'll have to think more about the amorphic polytope problem - maybe there is a simpler nasty that is also amorphic. -CunningGabe (talk) 13:33, 4 December 2008 (UTC)

Just remembered one that I'm pretty sure is amorphic. Start with two identical pentagons. Add a diagonal to one, so that you divide it into a triangle and a tetragon. Now glue the pentagons together. You get a polyhedron with 3 nonisomorphic faces, 6 edges, and 4 vertices. Such a construction is easily generalized to yield lots of amorphic polyhedra. Again, perhaps the one you've found is the simplest amorphic polyhedron without digonal facets or vertex figures. --CunningGabe (talk) 13:57, 4 December 2008 (UTC)

Surely this has |Aut(P)|=2 ? mike40033 (talk) 01:08, 5 December 2008 (UTC) I think I don't understand your construction. It seems to me that you'd get a polyhedron with 3 faces, 6 edges, and 5 vertices, with |Aut(P)|=2, but you're saying you get something quite different. Can you draw a picture? mike40033 (talk) 01:11, 5 December 2008 (UTC)

Sorry, I meant 5 vertices! And now that I look at it a little more closely, I think I see the nontrivial automorphism. Call the vertices A-E. Then the polyhedron consists of the pentagon ABCDE, the triangle ABC, and the tetragon ACDE. The 5 "outer" edges are each part of the pentagon and part of either the triangle or the tetragon, and the edge AC is part of the triangle and the tetragon. Then there is an automorphism that swaps A with C and D with E. So I guess it's back to the drawing board with me. -CunningGabe (talk) 04:15, 5 December 2008 (UTC)

But you might be able to use this trick anyway, maybe by splitting both faces to break all symmetries (and hopefully not introduce any more). If it can't work with a pentagon, I think it will work with a hexagon... If the hexagons are ABCDEF, what if you split one along AD and the other along AC. There is only one triangle, and one pentagon, sharing edge AC, so the only possible nontrivial automorphism must swap A&C. But this causes trouble for the edge between the two squares... I think... mike40033 (talk) 08:05, 6 December 2008 (UTC)

Atomistic and Coatomistic Lattice Polytopes

I am pleased to report some interim results from my attempts to formalise "nice" and "nasty" polytope concepts:

(1) If a polytope is a lattice, i.e. a poset with meets and joins, then the intersection of any two faces is always a face, and of lower dimension; and ditto the dual.

(2) My "Vertex/Facet uniqueness" property is, in fact, well established in order and lattice theory as "atomistic/coatomistic". Very simply, an atom is a vertex and a coatom is a facet, i.e an (n-1)-face. So in a polytope that is both atomistic and coatomistic, every face has a unique vertex set and a unique facet set.

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.

Can the classical polytopes be identified purely within the abstract theory? After about 10 minutes of thinking, I think the answer is 'yes'. mike40033 (talk) 00:10, 31 October 2008 (UTC) —Preceding unsigned comment added by SteveWoolf (talk • contribs)

I am a bit confused about whether this is equivalent to "Realisable".

The abstract polytope effectively defines the specific equivalence class to which some set of polytopes (of any variety) belong. There is no need for angels. This is a wholly abstract (combinatorial) notion, whereas realizability involves geometrical construction, so they are clearly not the same. Someone has already pointed out that any abstract polytope is realizable in some fashion. I suspect that your quest is to understand what property of a polytope allows it to be realised faithfully. This is equivalent to saying that it is realized as a traditional polytope: what one means by "faithful" is precisely the same as what one means by "traditional". -- Cheers, Steelpillow (Talk) 16:23, 17 December 2008 (UTC)

Convex polytopes, and I think all traditional polytopes, do satisfy the 2 conditions above, i.e. lattice and dually atomistic. One of my questions is whether these 2 are sufficient to characterise which AP's are "traditional".

If so - and it's very much an "if" at the moment - then we could make the useful statement that, amongst the set of all AP's, which includes digons, only those with the "2 properties", which excludes digons, correspond to traditional pols. Furthermore, rather than constantly refer to polytopes having the "2 properties", we would define them, e.g. as "angelic" - OF COURSE, that was only meant tongue-in-cheek, to give you a break from "Nice"! There may well already be such a term - I have written to McMullen again on this whole topic, you can see my letter on my talk page, tho' I'll copy his reply on this page when I receive it.

Your two conditions are not sufficient, for example the hemicube and many other elliptic tilings meet both conditions but are not usually regarded as "traditional" polyhedra. -- Cheers, Steelpillow (Talk) 14:27, 18 December 2008 (UTC)

No, sorry! The hemicube does not satisfy EITHER of the two conditions:

(1) It is not a lattice, because two 2-faces do not meet at a single k-face.

(2) It is not atomistic, since all of the 2-faces and the 3-face have the same vertex set.

What is an elliptic tiling? SteveWoolf (talk) 18:29, 18 December 2008 (UTC)

OK, try the hemicosahedron or hemidodecahedron then. An elliptic tiling is a tiling of the elliptic plane - a projective plane which has positive curvature and finite size. Embeddings of the elliptic plane in Euclidean space include the Boy surface and Steiner's Roman surface, among others. -- Cheers, Steelpillow (Talk) 21:59, 18 December 2008 (UTC)

You are right, the hemi-dodecahedron, and therefore its dual the hemi-icosahedron are both nice. See section below. SteveWoolf (talk) 07:49, 21 December 2008 (UTC)

My ideas about realizability are still fuzzy, tho' I was aware of the concept of faithful realizations. I think I can shed some light here. Any finite polytope can be "Realized" in 2 dimensions, i.e as a graph on a piece of paper, with all its vertices "separated". But this is a "poor" realisation. I think a "good" (faithful) realisation means that an abstract n-polytope is realised in Euclidean n-space, and as a polytope as defined in some "reasonable" manner - which I think would exclude the digon. Okay, not well expressed - but if any AP can be "poorly" realised then the concept is quite useless, so why not just define "realisable" to mean only faithfully.

I don't think you can equate "Faithful" and "Traditional". What I am hoping is that it may turn out that

An abstract polytope is (faithfully) realisable if and only if is a dually atomistic lattice.

That, in my opinion, would be a significant result, helping both to clarify the differences bewteen the Trad and AP worlds and to unify them, as well as offering the trad world the possibility of a standardised definition (which they will undoubtedly argue over till the cows come home, preferably without foot and mouth). Regards SteveWoolf (talk) 19:36, 17 December 2008 (UTC)

More precisely, notice that I equated "faithfully realised" with "traditional". Degenerate figures such as skew polygons or sub-dimensional simplexes are said to be realised unfaithfully. All abstract polytopes are realisable unfaithfully. The existence of figures such as the tetrahemihexahedron (a faithful realisation of a tiling of the elliptic plane) points to the very deep intractability of knowing which APs are capable of faithful realisation, i.e. as traditional polytopes. I am not sure if there is much point in discussing this further here, as we have been round the loop several times and you will find that what I am constantly saying is to be found, if not in ARP, then in other papers by McMullen and Schulte and other authors. -- Cheers, Steelpillow (Talk) 14:27, 18 December 2008 (UTC)

You may well be correct in saying that defining precisely which AP's correspond to traditional pols is "difficult". It may well be an open question. And as Godel taught us, not all math problems can be solved, so this one could be intrinsically unsolvable. Nevertheless I am a little more optimistic. At the very least, I think we can say that, to be realisable an AP must satisfy the 2 conditions, since all authors I have found appear to agree that all traditional polytopes do. Also, the general agreement about the meaning of realisability seems to be that it must be from n- to n- dimensions. Realising into higher dimensions doesn't qualify. Anyway, clearly this topic needs more input from the pros before it becomes worthy of inclusion in the article.SteveWoolf (talk) 18:52, 18 December 2008 (UTC)

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.

Genus is a property of all polytopes, specifically of their abstract combinatorial structure. Moreover, while APs have definable genus, no other combinatorial structures do. It is precisely from the properties of this combinatorial genus that the topological notion of ascribing genus to smooth surfaces arose (by decomposing the surface into a finite tiling, or polytope). You may not like the idea, but it is the foundation of much modern mathematics, including such delights as string theory. It is fundamental to topology that we define classes of polytopes (and other surfaces/spaces) according to genus - the "spherical" class has g=0, the "toroidal" class has g>0, and so on. However, genus has little or nothing to do with "niceness" - unless one's idea of niceness is unusually restrictive - for example Schläfli (I think) rejected some of the regular star polyhedra because they did not obey Euler's formula, in other words he decided they were not nice because their genus was not zero. -- Cheers, Steelpillow (Talk) 16:23, 17 December 2008 (UTC)

As I explained above, Toroids and I are friends now; it took a while. SteveWoolf (talk) 19:45, 17 December 2008 (UTC)

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)

The goal was very much to capture the combinatorial properties of traditional polytopes - sufficiently rigorously (for a change!) to be able to know what among the space travellers might or might not be admissible to the zoo. The deep problem is, which inmates are faithfully realizable and which are aliens? 83.104.46.71 (talk) 18:20, 17 December 2008 (UTC) Sorry, that was me - login timed out. -- Cheers, Steelpillow (Talk) 18:21, 17 December 2008 (UTC)

Just to spice up the discussion... I think the {5,5|3} is faithfully realisable as the vertex set of a stellated dodecahedron. Are stellations "traditional"?

On the other hand, I'd be very surprised if the {3,7}₈ were faithfully realisable in 3D, and equally surprised if didn't satisfy (1) and (2). mike40033 (talk) 00:54, 19 December 2008 (UTC)

{3, 7}

I am quite sure {3, 7} satifies (1) and (2). But why is not realisable? I accept that the realization can't be Euclidean-regular, but who cares? So what if the too-many triangles get all squashed up? This infinite series never reaches 0, only gets arbitrarily close to it:

1, 1/2, 1/4, 1/8, 1/16 ....

So we can keep cramming 'em all in and never run out of room, even on a flat 2-plane! Well, I'm no realization expert, so if I'm wrong, egg away!

I didn't mean the {3,7}, but the {3,7}₈. The latter is a finite quotient of the former. It's definately "realisable" in some manifold, as would be any 3-polytope, but is that what we all wanted to mean by realisable? mike40033 (talk) 00:30, 22 December 2008 (UTC)

P.S. Anyone up for a game of hyperbolic football? SteveWoolf (talk) 10:15, 19 December 2008 (UTC)

Love to! As soon as I figure out how to tie my 4-D bootlaces mike40033 (talk) 00:30, 22 December 2008 (UTC)

I have - temporarily - removed the following related paragraph from the article, until the above topics are finalised.

Geometric polytopes are abstract polytopes that are combinatorially isomorphic to classical polytopes in Euclidean space. They have posets that are also lattices: given any two faces, there is a join (a smallest face containing both faces, which may be the greatest face), and a meet (a largest face both faces contain, which may be the least face).

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)

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)

Conflicting Terminology

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)

The Smallest Toroids

A Regular 3-Toroid created from a circuit of 3 triangular prisms joined on their triangular faces. It has 9 vertices, 18 edges, and 9 square faces.

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?

Mm yes - I guess ONE tetrahedron plus 2 trigonal prisms will work. I don't think TWO tetrahedra plus one trigonal prism does.

Nasty ones?. Well I guess there's some very small "digonal" cases. I think 2 trigonal prisms might work, joined at BOTH triangular faces. But now my ignorance catches up with me, as I don't precisely know what toroidal means - only intuitively. Perhaps one of you can consider whether two digonal prisms works, joined together on BOTH their two ugly digonal faces. This little devil has 4 vertices, 8 edges, and (I think) 4 faces - 2 digonal and 2 square. SteveWoolf (talk) 05:15, 21 December 2008 (UTC)

Isographic vs Isomorphic

This particular toroid can be faithfully realized in Euclidean 4-space as a duoprism. In particular, this is a 3,3-duoprism, the Cartesian product of two triangles.—Tetracube (talk) 22:37, 19 December 2008 (UTC)

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.

The "Ring of 4 Cubes" - a polyhedral toroid. It is isographic, but not isomorphic, to the 4-cube.

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)

Thanks for your feedback. This is indeed quite intriguing. I would note that your "tetra-cube" appears to be half of a tesseract, since two copies of it, suitably deformed into 4-space, can be linked together to enclose a region of 4-space isomorphic to the interior of a tesseract. Or at least, this is what it appears to me when I first thought about it, but then I realized that in order to deform it into the required shape, I would have to introduce square faces where the original joints were (since otherwise it would cease to be a polyhedron---at least, no longer a faithful realization), and in the process the single polyhedral (maximal) face is split into 4. Likewise with the second copy of it, which, taken together, gives the 8 cells of a tesseract.

Now the reason I just went through such great lengths to describe all this, is because I note that if one takes the limiting case of the toroidal polyhedra, one gets a torus, which is topologically equivalent to one of the bounding 3-manifolds of the duocylinder. Two copies of the torus glued together makes the entire duocylinder, just as gluing two toroidal polyhedra, suitably deformed (and folded, in the 4D sense), makes a duoprism. I find this very interesting, because in the limiting case, one can hardly speak of "introducing 2-faces" in order to deform it to the requisite shape matching a duocylinder's bounding manifold, yet in the toroidal case, such a process of deformation, which seems to be a perfectly isomorphic (in the topological sense) operation, creates distinct abstract polyhedra. I suppose one does need to draw a distinction between the surface of a polyhedron and its interior, since the deformation of the interior doesn't create a topologically different object.—Tetracube (talk) 18:01, 20 December 2008 (UTC)

I goofed! 4-cubes have 8 cells as you say, sorry about that!

Clearly you are quite knowledgeable about non-abstract polytopes, which I am not! So I can't comment on a lot of what you said. However, I can tell you about abstract pols. Here, there simply aren't such concepts as "interiors" of any face. A square face, for example, has 4 edges, 4 vertices, and that's it (except the null face); tho' it is also important what it is a part of. Seasons Greetings. SteveWoolf (talk) 21:21, 20 December 2008 (UTC)

Which is what I was driving at: the abstract polytope captures the structure of what in geometric/classical polytopes would be called its "surface" (e.g., how the faces of an icosahedron relate to each other, how their edges connect, etc.), but there is no concept of the "interior" since this is not meaningful for such objects as the 11-cell. The equivalence I noted between toroidal polyhedra and parts of 4-polytopes are based on (topological) equivalences of their interiors, but this isn't captured by the abstract polytope concept. (This reminds me of what Guy/Steelpillow wrote once about hollow/solid polytopes, that often in polytope discussions the interior of a polytope is not consistently handled, leading to seemingly contradictory results, which are really not contradictory once the distinction is properly paid attention to.)

I am not sure if all that is correct. Yes a square face has 1 null, 4 vertices and 4 edges, but it also has a maximal element - it is after all an AP in itself. This maximal element is sometimes called its "body" and mapped onto (realized as) the physical interior. Just what physical region (if any) is treated as "inside" becomes an argument over the details of the mapping, and has nothing to do with the AP itself. However, there remains a linguistic convenience in referring to the maximal element as the "interior". For example an edge element can be talked of as the "interior" between the two vertices of a 1-polytope: this helps distinguish between the whole abstract 1-polytope vs. the edge element alone, both of which are "edges" in one way or another. -- Cheers, Steelpillow (Talk) 14:06, 6 January 2009 (UTC)

This makes one wonder about the precise connection between abstract polytopes and classical polytopes; given some abstract polytope P, such as your "tri-triangular prism", one can "expand" it by introducing the requisite faces and cells to turn it into a 3,3-duoprism (replacing the original maximal face containing the 9 2-faces). This process suggests a reverse operation, where, given some polytope (abstract or otherwise), one can "trim" off a few levels from the top of the Hasse diagram and replace them with a new maximal face. Say, take a 10-polytope, throw away the 8-, 9-, and 10-faces, replacing them with a single maximal face, and you get an 8-polytope with the same faces up to 7 dimensions, at which point the structure terminates. I wonder if this violates any of the conditions of being an abstract polytope? (I was going to generalize this "diminishing" operation to the bottom layers of a polytope as well, until I realized that the result would violate the requirement that every 1-face must be an edge, since one could remove, say, the vertices of an icosahedron and replace it with a minimal face, but one ends up with a structure where an edge contains 3 vertices).

If removing the "top layers" still yields a valid abstract polytope, then we have the interesting situation where a classical n-polytope can have n-1 "diminishings", each of which is identical in structure up to some dimension j<n. For example, if we take only the vertices and faces of a 10-simplex and replace all higher faces with a maximal face, we get a 2-polytope which is a mesh of 165 triangles. If we start from this mesh, and "close" the polytope by recursively forming (j+1)-faces for every closed circuit of j-faces, then we get the 10-simplex back. In such a case, I would argue that it would make more sense to assume such a closure, so that these "diminishings" are really just identical to the original polytope itself. (Although I have the feeling that I'm missing something obvious here. :-))—Tetracube (talk) 22:59, 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)