Talk:Abstract polytope/Archive 4

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How nice is the Hemi-dodecahedron?

The hemi-dodecahedron (and also therefore its dual the hemi-icoashedron) is nice.

However, it contains the complete graph K₅. By Kuratowski's theorem it therefore is not planar . Since all traditional polyhedra do have planar graphs, this seems to prove that the hemi-dodecahedron is not realisable in Euclidean 3-space, however.

Well maybe my "proof" is misconceived, but anyone disagree that these two objects are not realisable in Euclidean 3-space?

Are these two TP's ("Traditional Polytopes")? Is realisability in Euclidean 3-space the criterion?

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

Hmm - I think it can be embedded in R^3. This is how I'd do it : First of all, here is a pair of 3x3 matrices that generate the symmetry group of the hemi-dodeca. Then, I'd find out how to express certain standard symmetries of the hemi-dodeca in terms of these matrices. Then, I'd find a symmetry that fixes a vertex of the hemi-dodeca, and a fixed point in R^3 of the corresponding matrix. Then, I'd successively apply the symmetries I had to find the other 9 points, and join them up with edges and pentagons.

mike40033 (talk) 23:56, 6 November 2008 (UTC)

If the these two hemis are not TP's, then we are at this point:

All traditional polytopes are nice, but

Not conversely.

It seems to me that these two hemis are right on the fault line between "nice" and "nasty". My intuition says they're nasty. I'd like to know just how faithfully they are realisable - but can anyone define those last two terms for me? Can't find much via Google.

I suspect the 11-cell and the 57-cell are also nice. SteveWoolf (talk) 13:49, 25 December 2008 (UTC)

What is a Traditional polytope?

One of the comments here got me thinking... 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) SteveWoolf (talk) 04:32, 31 December 2008 (UTC)

Just what is a trad polytope, as generally understood? Nearly all authors I've found seem to limit their definitions to convex cases, which I assume excludes toroids. I for one would definitely include apeirotopes. In what other spaces do TP's exist?

Even tho' there may be no single agreed defn, we MUST have at least a rough one; otherwise we are talking nonsense every time we mention them.

Looks like characterising TP's is a tricky problem - but that doesn't mean unsolvable! SteveWoolf (talk) 05:39, 25 December 2008 (UTC)

As a suggested starting point for tradition, here is Coxeter's definition in "Regular Polytopes". This has the advantage (or otherwise) of reflecting the confused state of current theory. First he defines a convex polytope in Chapter 7:

...we define a 'polytope' as a convex region of n-dimensional space enclosed by a finite number of hyperplanes.

He does not restrict himself to Euclidean space, but also discusses hyperbolic (but, curiously, not elliptic) polytopes. The, in Chapter 14, he discusses star polytopes on the basis that:

...it is natural to extend the definition of a polytope so as to allow non-adjacent cells to intersect, and to admit star-polygons and star-polyhedra as elements.

As a result, he gives no single workable definition for a general polytope (star or otherwise). Nor do his definitions allow apeirotopes, which he calls honeycombs and describes as "degenerate" polytopes. However, since he devotes more than a chapter to the regular and quasiregular varieties, we might still wish to think of them as traditional.

While no two polyhedronists would ever agree on the criteria for "traditional", I think it fair to expect flat faces in Euclidean space (Many people assume convexity because that's where the mainstream math has been done, and a few even quote authorities such as Grünbaum in support of this. But these quotes appear out of context, also it would be hard to deny traditional status to the Kepler-Poinsot stars). I think there is also some value in leaving "traditional" somewhat loosely defined, so that we can say things like, "so-and-so is poorly defined for traditional polyhedra and authorities disagree. However for abstract polytopes ...". -- Cheers, Steelpillow (Talk) 20:36, 21 December 2008 (UTC)

Meanwhile, on this basis the hemidodecahedron and hemicosahedron are not traditional. -- Cheers, Steelpillow (Talk) 20:36, 21 December 2008 (UTC)

Summarising the above, it seems that convex polytopes have been well defined because it was relatively easy. The other classes (stars, infinite, toroids...) seem to have been admitted informally on the basis of having broadly similar properties. Still, surely some of today's more advanced polytopists must have come up with some wider defn of trad pols? But maybe not. I sometimes get the feeling reading some authors' work that they are more interested in getting their PhD by sticking to a narrow topic, or selling books by merely collecting known results - with little genuine passion for knowledge or the "big picture" of the whole field and its wider implications. Still, that just leaves space for any Watson-Crick's to sober them up!

My intuition tells me also that hemi-dod and hemi-ico are not traditional; but since they are nice, I shall be relentlessly searching for another third criterion of niceness to add to the two already. One wonders why M-S and others decided to broaden the polytope concept as much as they did; certainly the 2 "nice" conditions are simple and basic enough.

My feeling is that this "third" criterion will have something to do with topology and/or realizabity. I doubt if the statement by one author that trad pols are meet/join distributive is significant here, but I will check right now to see if our 2 hemi's are distributive. If not, wow! SteveWoolf (talk) 06:58, 22 December 2008 (UTC)

I think it fair to say that modern researchers have been constantly seeking better definitions, and tend to evolve their definitions over time as their thinking progresses. The many metric difficulties have led to increasing focus on the combinatorial properties, to the point that in his 2003 essay, "Are your polyhedra the same as my polyhedra?", Grünbaum defines a real (or traditional) polytope as a faithful realisation of an AP. Compare this with his earlier 1993 definition in "Polyhedra with hollow faces" where he defines "the most general object that we might want to call a polyhedron" in terms of a partially-ordered set of vertex point-pairs (edges) and the distinction between physical and abstract is not yet fully explicit. Or with the definition in his standard reference work "Convex polytopes" as a compact convex set K of real points provided ext K is a finite set.

Cromwell, perhaps the last traditionalist, in his 1997 book "Polyhedra" devotes several pages to the question, "What is a polyhedron?" and eventually arrives at the following:

"Definition. A polyhedron is the union of a finite set of polygons such that

"(i) Any pair of polygons meet only at their sides or corners.

"(ii) Each side of each polygon meets exactly one other polygon along an edge.

"(iii) It is possible to travel from the interior of any polygon to the interior of any other.

"(iv) Let V be any vertex and let F₁, F₂, ..., F_n be the n polygons which meet at V. It is possible to travel over the polygons F_i from one to any other without passing through V.

"In this context, polygon means a planar figure bounded by straight lines that is topologically equivalent to a disc. The restriction excludes the star polygons used by Kepler ..."

Like Coxeter, Cromwell goes on to say that we can generalise polygons and polyhedra to admit star figures, without actually bothering to generalise his definition. I believe this is because it is patently in a form which cannot be so generalised, yet neither Coxeter nor Cromwell shows the slightest concern. How typical of the traditionalists!

Here is another fundamental confusion. Plato treated polyhedra as solids, Kepler as surfaces. If we go for solids (more common with convex definitions), do we include the bounding surface, i.e. is the solid region open or closed? Plato by implication includes it, Grünbaum's Convex polytopes excludes it. There is no modern consensus on either question, and some authorities regard it more as a matter of taste or relevance to the matter in hand than anything. Once again, I recommend the idea that "traditional" should by definition be undefined in mathematical terms - any other attitude will bring you nothing but heartache and contradictions. Meanwhile, good luck with your third criterion - I suspect you will need it! -- Cheers, Steelpillow (Talk) 11:52, 22 December 2008 (UTC)

I would characterise a "traditional" polytope as follows: the main idea being that we begin with a working definition of what a "polygon" is (which I propose to be a planar figure consisting of a set of vertices and a set of edges on them such that every vertex belongs to two edges and every edge contains two vertices), and then defining an (n+1)-polytope to be a figure having n-polytopes as facets, where each (n-2)-face joins precisely two (n-1)-faces. This allows us to begin with star polygons and build up higher polytopes recursively. This definition also allows n-space tesselations and apeirotopes, since the set of facets need not be finite.

As for whether polyhedra are solids or merely the boundary, my personal inclination is to regard them as solid, since otherwise you get hollow facets in the next higher-dimensional polytope (e.g., take a hollow square, fold 6 of them into a cube: you only have a wireframe cube, whereas we usually regard the faces to be filled in. I suppose you could argue it's "richer" to draw the distinction between hollow and non-hollow faces, but then you get a lot of spurious distinctions such as taking 8 cubes, each of which may be in different states of hollowness (both in the 2D sense and in the 3D sense), and folding them into a tesseract, which now has an exponential number of states of hollowness---any of the facets could be hollow, in the 3D sense, and any of the faces can be hollow, in the 2D sense. And while we're at it, we can have a "hollow" edge too---consisting of merely 2 points. It seems geometrically ludicrous at this point, since then a tesseract which is "completely" hollow in all of its faces is but a set of 16 disconnected points.). Of course, when you deal with non-convex polytopes, it gets trickier what exactly is the "solid" bounded by the polytope; I believe this is the underlying motivation behind Coxeter's paying quite some attention to the question of density in Convex Polytopes. Regardless, even with non-convex polygons such as the pentagram, if you consider higher polytopes built from them, it seems quite clear that they need to be considered as (n-1)-dimensionally "solid", since otherwise a pentagram reduces to a set of 5 points and becomes indistinguishable from a pentagon, which is just ridiculous.—Tetracube (talk) 17:55, 22 December 2008 (UTC)

Well... thanks for all your contributions. We sure seem to have opened Pandora's hyperbox, and let the genus (genius?) out of the Klein bottle.

I think Tetracube has quite a good point: we certainly don't think of edges (of trad pols) as ever being without the intervening points, so why do so with polygons and other n-faces. So I agree, a traditional polytope IS a solid, tho' I don't know what implications that may have.

While clearly it is unlikely that there is any definition of trad pol that will please everyone, it is quite likely that there are more criteria that would at least bring us closer, as I believe the two "nice" criteria have already done. I think the main obstacle right now is probably our limited mathematical knowledge and abilities, which I am trying to improve.

Given my particular agenda, namely to try to define in abstract terminology which AP's correspond to TP's and which are "nasty degenerates", we might get away with a somewhat simpler definition than would satisfy a true traditionalist, for a couple of reasons. We don't, I think, have to worry about star-pols, as these are combinatorially equivalent to convex ones (true/false??). And we certainly don't have to worry about skewy (screwy?) objects, or curved faces. As far as I'm aware, the main variations on the convex pol we have to worry about are aperiotopes and toroids, but I guess there's other things out there too that I need educating about.

I have already provisionally defined nice, and in my view, it is useful to state whether or not a given polytope is "nice"; a hemicube is NOT, a hemidodecahedron is. Since this "razor" has not proven sharp enough to separate the hemidodecahedron, which I think we know is a "bit funny", from traditional pols, I think it is a most worthwhile mission to try to make this "funniness" precise. And for that, I think I need to get a grip on just what "realizable" means, if I am correct in thinking that the hemidodecahedron is not realizable (or even realiSable).SteveWoolf (talk) 23:18, 22 December 2008 (UTC)

An important family to consider are polytopes having non-orientable surfaces. These include the tetrahemihexahedron, which is a polyhedron corresponding to a tiling of the projective plane, oddities such as the cuploids, and continue on up among the higher toroids. When you come to consider the density of interior regions, you will get some nasty surprises. Meanwhile, two of the regular stars are isomorphic to their convex counterparts, the other two are (orientable) toroids. -- Cheers, Steelpillow (Talk) 23:24, 23 December 2008 (UTC)

Digon Derivatives

Peace and Good Will to these??

A few nasty creatures for you to digest over Xmas. They are quite tasty with either cranberry sauce or Bisto.

The left figure is both

• The Hasse Diagram of the Digon, and
• (The graph of) an "nasty" polyhedron with V=6, E=6, F=4 (squares). Yes, Virginia, it is an AP. Note that this beast has two 2-valent vertices, proving that once we stray from the path of righteousness, there is no evil from which we are safe!

The figure on the right is (please check...) the Hasse of a digonal dihedron, and presumably the graph of an "nasty" 4-tope. It clearly suggest an infinite series of di-'s in any dimension, all with only two k-faces for each 0 ≤ k ≤ n-1. SteveWoolf (talk) 13:14, 22 December 2008 (UTC)

What software do you use to make all these pretty pictures? mike40033 (talk) 03:35, 23 December 2008 (UTC)

Thanks - The software is none other than the humble MS Paint! They are all done manually. I do however take care to make my lines go to the centre of the vertices, and use a standard size vertex circle, with radius chosen to give to best round look, which you can copy. I use line size 3. Tom Ruen helpfully advised me to save in .png (lossless) format. I usually spend time trying to get the best layout.SteveWoolf (talk) 09:23, 23 December 2008 (UTC)

I can see a polyhedron in the right hand figure, I suppose you could make a 4-polytope out of it with the same graph easily enough, eg, its dichoron. mike40033 (talk) 03:37, 23 December 2008 (UTC)

As noted in my section Talk:Abstract polytope#The "Hasse Polytope", I have conjectured that the Hasse diagram of an n-tope is the graph of an (n+1)-tope (all of whose sections are easy to derive). Therefore, the Hasse of the digonal dihedron should be the graph of a 4D object.SteveWoolf (talk) 09:23, 23 December 2008 (UTC)

I agree that both figures are valid Hasse diagrams of abstract polytopes. In particular, for any n, there is an n-polytope where there are exactly 2 i-faces for each i (other than -1 and n). Also, I believe that for the left figure, you mean it is the graph of a 3-polytope with 6 vertices, 8 edges, and 4 faces.

Digonal Trihedron. It has V=2, E=3, F=3; the faces are digons.

Also, for any m, there is a dihedral m-gon with m vertices, m edges, and 2 faces. We also have its dual, with 2 vertices, m edges, and m faces. It's also possible to show that whenever V=E, then F=2 (and by duality, whenever F=E, then V=2). --CunningGabe (talk) 16:52, 23 December 2008 (UTC)

I tried to create a hemitetrahedron, but it's impossible - you would need edges with 3 vertices, which conflicts with the diamond property (Axiom 4). However, I tweaked it and got the Digonal Trihedron. SteveWoolf (talk) 11:24, 23 December 2008 (UTC)

Polytope Catenation

Simple Catenation

Take two identical square pyramids. "Glue" them together on their square faces. Now you've got an octahedron.

More formally, and generalising to n dimensions, take any two n-polytopes P, Q with isomorphic (n-1)-faces F in P, G in Q. Then there is a new n-polytope R defined by

• Every i-face of P, excluding F and the n-face, is in R
• Every i-face of Q, excluding G and the n-face, is in R
• R has a new n-face which contains all the other faces.

So, 4 faces go and 1 new face is created (other than for n < 1).

This definition works for ALL dimensions, but is trivial for small n:

n = -1: Null polytopes don't have (n-1) faces, so no cases exist.

n = 0 : Two 0-pols make the null-polytope (precisely following the rules).

n = 1 : Two 1-pols ab + bc combine to form the 1-pol ac.

n = 2 : Polygons: a p-gon and a q-gon combine to make a (p+q-2)-gon.

This concept leads naturally to yet another classification of polytopes - those that are "primitive" and those that are the concatenations of other polytopes.

Well, I'm quite sure the above is true for the "nicer" polytopes at least - I'm not so sure about all those funny flat ones and other nasties.

This is such a simple concept that there must be existing terminology and material somewhere - can anyone help? Despite repeated attempts, I haven't been able to find any material on this subject.

Interesting - to be precise, it seems that we want not only to remove the faces F and G, but also identify their sections (i.e. subfaces) under the isomorphism. That is, if we have a triangle ABC and a triangle 123, if we glue together side AB to side 12, our new triangle should have all 6 edges except for those two, but also vertex A and vertex 1 should be joined to a single vertex, as should vertex B and vertex 2.

The two matching facets F, G are "removed" from the result. Their two sets of subfaces, which are isomorphic, become a single set isomorphic to both. In your example, vertex A and 1 become the same vertex; ditto 2 and B.SteveWoolf (talk) 09:03, 24 December 2008 (UTC)

I'll take a look and see if I can find any prior work - I'm actually working with Schulte, so I'm very happy that there's currently a lively discussion going on! -CunningGabe (talk) 19:31, 10 November 2008 (UTC)

Yes, please look it up... You may find that there's not much in the abstract polytopes literature, but the construction dates waaaaaay back in the non-abstract theory. You'll normally get a non-regular polytope. I'm sure there's some worthwhile research problems around the idea, too...mike40033 (talk) 00:31, 11 November 2008 (UTC)

Complex Catenation

Generalising the idea, we can simultaneously join on more than one pair of faces, creating toroids. It seems to me that joining on p pairs of faces would give a polytope of genus p-1 - provided that the original polytopes were of genus 0.

We can therefore construct all kinds layouts - chains, trees, loops etc. of n-polytopes. I conjecture that any reasonably well-behaved graph can be the layout of a polytope catenation - including a polytopic graph.(Reasonably well-behaved might include only allowing "discrete" graphs, i.e. those with the property that there is a finite path between any 2 connected points.)

For example, take a cube. For each of its vertices, assign an m-polytope, m a constant ≥ 3, in such a way that each facet of each m-pol can be matched with one on each adjacent vertex of the cube. Now "glue" the matching faces together, dispensing with the original cube. This should, I think, create a "super-polytope"... but always? Under what conditions?

Regular polytopic layout graphs with regular vertex polytopes might often give regular super-polytopes. SteveWoolf (talk) 03:16, 25 December 2008 (UTC)

2nd Definition of Spherical

I still maintain that it would be better mathematics to have a definition of spherical (as well as other topological types) in pure AP terminology, and then prove its equivalence to "cross-discipline" definitions. And the polytope catenation concept seems to suggest one.

• Define a polytope to be primitive if it is not the catenation (simple or complex) of two other polytopes.

• Define a Catenation Graph or C-graph of a polytope P to be a graph that has a vertex for each primitive polytope in the catenation, and an edge wherever the primitives are glued together. Thus the C-graph of a primitive pol would have just one vertex, an octahedron would have just an edge (line segment). I suspect a given polytope may sometimes have more than one C-graph.

• Then, a polytope is spherical if it is finite and it has an acyclic C-graph, i.e. containing no loops.

Chains and trees, such as an octahedron or five cubes stuck together to make an X shape, are therefore spherical.

Well, topology is not my forte - but this seems like an interesting idea. SteveWoolf (talk) 05:32, 31 December 2008 (UTC)

Presumably this is your "second" definition that you asked about above? I have been trying to find a handy reference for the following, but failed to. Let's hope I get this right. Consider the Euler characteristic χ = V - E + F - .... This is unsatisfactory because an abstract polytope also has a null member and a maximal member. Let O be the count of null members and P the count of maximal members (i.e. O = P = 1), which we may incorporate into the Euler formula. We now have a "complete" Euler characteristic χ' (which I do not know the name or symbol for so I have made them up) = - O + V - E + F ... +/- P where the sign of P alternates with the number of dimensions. A geometrical (traditional) polytope is spherical if it can be realized as a convex figure inscribed on a sphere. A necessary and sufficient condition for this is that it has χ' = 0. Thus, we DEFINE an abstract polytope as spherical if, and only if, it has χ' = 0. -- Cheers, Steelpillow (Talk) 13:34, 31 December 2008 (UTC)

We've been here before! I don't feel that this is a definition of spherical - it is an equivalent property. I will try to illustrate. We do not define a right angle as that angle such that opp² + adj² = hyp². If we did, you would then prove that 4 right angles make a circle. But that is the wrong way round. First, you define a circle as 360° (and a right angle as a quarter of that), then you prove that right-angle triangles satify Pythagoras' Theorem.

Why does it matter? For one thing, if you prove outside AP theory that these two are equivalent, how can you be confident that such a definition works inside it? At some point you have to justify any definition of spherical by showing that it matches one's intuitions of what sphericality is, and this must be done within AP theory, in my view.

Okay, I'll concede that if you can prove - strictly within AP theory - that pols with the right Euler Ch, and no others, have all the properties we associate with sphericality, that would lend weight to the Euler Ch defn. But those "spherical" properties would themselves need formal (AP) definition, so you'll still be in the same place.

My 2nd definition is, I think, quite intuitive, once you are comfortable with the (easy) catenation concept. It may not stand up, I can't say. But if it turns out to be equivalent (or provably not) to the Euler characteristic defn then I think we have made progress.

To keep things down-to-earth and try to avoid interminable waffle, what about the dreaded digon and dihedron? They certainly can be inscribed on a circle/sphere respectively. What is their Euler characteristic? I guess they are spherical by my defn 2 - I'll look into that. HNY SteveWoolf (talk) 17:53, 31 December 2008 (UTC)

We have indeed been here before. I hoped that my fuller exposition of the standard definition of a spherical polytope might help you. Evidently I have still not made it clear enough. The proof that for a closed surface, a value of 0 fo χ' equates to sphericality is one of the fundamental proofs of topology - it is almost the starting point (The historical starting point was Euler's original proof that for spherical polyhedra V - E + F = 2). The desired abstraction from geometry is precisely that which leads us to abstract this Euler model and use it to define abstract sphericality. This abstraction appears to be just what you are seeking to achieve and it has already been done. I really do not know how to make this any clearer. For the digon and any typical dihedron you can work out χ' for yourself, it is not hard. If you have trouble, draw the Hasse diagram and work from that. You should find that they are spherical. Even the simple 1-sphere (closed line segment) and 0-sphere (point) are abstractly spherical. Look, as a starter I'll do the digon for you: it has O=1, V=2, E=2, P=1, which yields χ' = - O + V - E + P = -1 + 2 - 2 + 1 = 0. -- Cheers, Steelpillow (Talk) 00:02, 1 January 2009 (UTC)

Some questions/thoughts that pop into my mind...

* is the catenation graph uniquely defined, for a particular polytope? Eg, what if there is a polytope that can be built in more than one way, giving different catenation graphs?

* no loops translates to simply connected in higher dimensions.

* Can you explain to me again (sorry!) how this excludes projective polytopes like the hemi-icosahedron?

mike40033 (talk) 02:48, 2 January 2009 (UTC)

Yes, I suspect that there are polytopes have more than one Cat-graph, tho' I don't have an example yet. Then again, the Cat-graph may always be unique - that would be an impressive result.

I don't recollect saying that the catenation concept "excludes projective polytopes like the hemi-icosahedron?" anywhere. I don't doubt though that it gets messy for "non-traditional" pols - or more interesting, depending on your taste! HNY SteveWoolf (talk) 04:47, 2 January 2009 (UTC)

No, but if spherical is defined as being finite with an acyclic catenation graph, then the hemicube is going to trip up the definer.... mike40033 (talk) 04:24, 4 January 2009 (UTC)

Tesselations of Tesselations

A regular polyhedral tesselation of apeirogons, with 3-valent vertices; each number is a 2-face. The dual is a tesselation of triangles, with infinitely many meeting at each vertex.

"A tesselation is an abstract polytope with spherical facets and vertex figures". mike40033 (talk) 00:10, 31 October 2008 (UTC)

This seems to imply that tesselations must have finite facets, but I don't think that's true.

The infinite tree graph shown (incomplete - there was not enought space in the margin) is the graph of a regular Apeirogonal Apeirohedron. For each p ≥ 3, there is an Apeirogonal Apeirohedron with all vertices p-valent. There are no endpoints or closed (finite) polygons. The cases p = 0, 1 and 2 give a point, edge, and apeirogon repectively.

The original paper on Uniform polyhedra (H.S.M. Coxeter (et al), Uniform Polyhedra, Phil. Trans. 246 A (1954) pp. 401-450.), includes tabulated data on the uniform tilings (tessellations) of the plane. Some of these involve apeirogons and appear to be non-convex. -- Cheers, Steelpillow (Talk) 12:23, 28 December 2008 (UTC)

Listed at Uniform_tiling#Expanded_lists_of_uniform_tilings (including infinite faces) Tom Ruen (talk) 03:44, 30 December 2008 (UTC)

Duals have Infinitely-Valent points

Ah... a problem? Not really. We already have infinitely-valent vertices in the Hasse diagrams of tesselations. There is nothing in our AP definition to exclude this possibility. So, I say they're legal! I admit it's a bit time consuming to draw infinite pictures though. SteveWoolf (talk) 07:17, 27 December 2008 (UTC)

Future Directions

Regrettably, I may not have so much time now - I have to concentrate on music. I will hopefully find some time occasionally, and will try to answer the question "Is the hemidodecahedron poset lattice meet/join distributive?". I suspect yes, but if not, I think that'd be an important result - another refinement to our TP/AP sieve. I have decided to write a computer program to do this - but that needs time.

In my view pinpointing precisely how the hemidodecahedron differs from (abstract versions of) traditional pols is a most crucial issue, which I suspect is closely related - or perhaps the same as - faithful realisability, which I have yet to see defined.

You should have seen just such a definition when you read Johnson's Polyhedra - abstract and real - try reading it again. For another definition, see Grünbaum's Are your polyhedra the same as my polyhedra? (pdf), which I also referenced earlier. Note that such definitions of faithfulness are based upon physical geometrical properties and say nothing about the AP to be realised, and so are only loosely (if at all) related to the abstract problem of faithful realisability. -- Cheers, Steelpillow (Talk) 12:43, 28 December 2008 (UTC)

Ta - will reread them with that in mind. HNY SteveWoolf (talk) 15:24, 28 December 2008 (UTC)

I have just recorded a 60's pop style song of mine which you can here hear. Those of you who only like Classical music will not be amused! Happy New Year. SteveWoolf (talk) 08:34, 27 December 2008 (UTC)

Is the hemidodecahedron meet/join distributive?

I am pleased to report that I have written my Polytope Analyser program and it seems to show that the Hemi-dodecahedron is not meet/join distributive. Using the notation in the Hemi-dodecahedron article's picture:

a ^ (b v d) = a, while

(a ^ b) v (a ^ d) = ø

where ^ = meet, and v = join.

So if I haven't made a mistake, this property at least separates the hemi-dodecahedron from the traditional polytopes, which I believe are always meet/join distributive.

Sorry - looks like I blew it! (Glad I realised it first, at least!) The cube seems to fail also, i.e. is not meet/join distributive. Consider a cube abcdefgh with the "usual" vertex labelling (i.e. ae, bf, cg, dh are edges). Then, as before, we also have

a ^ (b v d) = a, while

(a ^ b) v (a ^ d) = ø

It seems then that meet/join distributivity is not so common after all, which leads one to ask: just which pols are MJD? I'll look into it....

Meanwhile, I guess my desire to kick the hemidodecahedron out of nice society didn't come off this time - but I'LL BE BACK!!!SteveWoolf (talk) 03:53, 19 February 2009 (UTC)

I am quite confused by all of this. The hemi-dodecahedron picture shows a, b and d as vertices of pentagon abdce(a). In projective geometry at least, the meet of two lines is their point of intersection and the join of two points is the line through them. The join of b and d is a line across the face of the pentagon. a does not lie on this line, so its meet with a is null - the empty set ø if you like - and not a as stated above. Is this different from what you mean? -- Cheers, Steelpillow (Talk) 22:28, 18 February 2009 (UTC)

This is a good example of where Abstract Theory clashes with Traditional Geometry. You will find both Meet and Join discussed at length in the excellent article Lattice (order). The meet of two faces is the unique largest face contained in both; dually, the join is the unique smallest face that contains both. The definition of a lattice implies that meets and joins always exist. As far as I am aware, all traditional pols are lattices, unless you stretch the defn of traditional too far!!! SteveWoolf (talk) 03:32, 19 February 2009 (UTC)

Treating the AP as a complete space in its own right, the two approaches are in fact consistent with each other. I was, perhaps naïvely, drawing a diagonal line across one face. -- Cheers, Steelpillow (Talk) 09:43, 19 February 2009 (UTC)

Spherical of rank 4...

I asked a relevant question here. Since it will get archived soon, I've cut-and-paste the question and answers below... mike40033 (talk) 04:13, 2 January 2009 (UTC)

NB - it seems the discussion is still going on. mike40033 (talk) 04:14, 2 January 2009 (UTC)

S³ vs P³

How does a topologist distinguish these manifolds? mike40033 (talk) 04:55, 31 December 2008 (UTC)

I assume P³ is the real projective 3-space. Then S³ is simply connected, while P³ has the cyclic group of order 2 as fundamental group. —Preceding unsigned comment added by Aenar (talk • contribs) 13:11, 31 December 2008 (UTC)

You probably allude to the fact that singular homology doesn't see the difference: but the cohomology algebra does, for The cup-length is 1 for the 3-sphere and 3 for the real P³; an elementary invariant is also the Lyusternik–Schnirelmann category: it's 2 for S ³, 4 for P³. Related with this, any $C^{1}$ function on P³ has at least 4 critical points, whereas a function on S³ may have only 2, a maximum and a minimum point.--PMajer (talk) 14:06, 31 December 2008 (UTC)

But integral singular homology does see the difference, unless by P³ you mean something other than real projective 3-space. Algebraist 17:36, 31 December 2008 (UTC)

Yes you are perfectly right, sorry. The real P³, of course. Singular homology does not distinguish between P³ and S²xS¹, this is the big match: the Poincaré polynomial of both is 1+t+t²+t³; but then cohomology algebra works, since the cup-length of S²xS¹ is 2. There is in fact a function on S²xS¹ with only 3 critical points (degenerate). Thank you for correcting me. --PMajer (talk) 20:39, 31 December 2008 (UTC)

Thanks! This should be helpful... mike40033 (talk) 04:11, 2 January 2009 (UTC)

Topological or geometrical constraint?

Is there a finite regular abstract polyhedron (3-polytope) with hexagonal faces? I've tried to construct such a polyhedron, but have not been successful. Obviously, such a polyhedron is impossible in Euclidean space, but is it at least possible in the abstract realm? (Note that the polyhedron is required to be non-degenerate, i.e., must have more than 1 hexagonal face, and finite, i.e., not the trivial plane tiling.) If this is impossible, why? I'm trying to understand whether the non-existence of this polyhedron is caused by a topological reason or merely a geometrical one.—Tetracube (talk) 22:54, 14 January 2009 (UTC)

Yes, there is at least one. Branko Grünbaum sent me some lecture notes, in which he describes a regular hexagonal trihedron and remarks that it is listed as W#9.2 in the Wilson catalog (whatever that might be). If you number the vertices from 1 to 6, the faces A, B and C may be identified as:

A = [1,2,3,4,5,6]
B = [1,4,3,6,5,2]
C = [1,6,3,2,5,4]

It has 9 edges, hence Euler characteristic 6-9+3=0, hence genus 1. It is orientable, so is topologically a torus.-- Cheers, Steelpillow (Talk) 11:47, 15 January 2009 (UTC)

See my section Talk:Abstract polytope#Hexagonal Trihedron. Don't have time too see if this is the same...SteveWoolf (talk) 02:35, 9 February 2009 (UTC)

Very interesting! Thanks. Are there any others (esp. with more than 6 vertices)?—Tetracube (talk) 19:30, 15 January 2009 (UTC)

I do not recall seeing any others described. However, it seems to me that the translation unit of any periodic tiling of the Euclidean plane may be adapted to tile a torus n times, since these both have the same genus. It also seems to me that for such polyhedra, abstract regularity would be preserved. I don't know the permissible values of n. but I suspect that any non-prime would be a good start (for n = a.b, arrange the units in an a x b array in the plane, and fold/join into a torus). Since there is a regular hexagonal tiling, I am guessing that we may create an infinite number of regular toroidal polyhedra.Things might get weirder in the hyperbolic plan, with four or more hexagons at a vertex - any solutions here would correspond to toroids of higher genus. Of course I may be quite wrong. -- Cheers, Steelpillow (Talk) 10:14, 16 January 2009 (UTC)

There are, indeed, infinitely many regular toroids {6,3}. There's also some fascinating polytopes related distantly to the 57-cell, which have type {6,5}. You can see a small collection of others here (see the section "Facet of...") mike40033 (talk) 04:41, 2 February 2009 (UTC)

Reply from Egon Schulte

I have received a reply to my pre-Xmas letter to E. Schulte, which appears below in full. I have not yet had time to reflect on it, or integrate it into the article. However, I hope to do this within the next few weeks.

Meanwhile I am working/playing on creating software to answer questions about polytopes - eg is a given pol a lattice? Atomistic? Meet/join distributive? Or even a polytope? I can't say yet whether this will be useful, but if it is I'll make it available.

Dear Steve:

Thank you for your email of December 17 regarding Wikipedia's Abstract Polytope article. I am sorry about the late response. Let me say upfront that I am happy to see you and others getting involved in the very worthwhile project of making the concept of abstract polytopes more accessible to a wider audience via Wikipedia. This is quite some effort! I went through the current official page on Abstract Polytopes (but took only a very brief look at the Talk page). I printed it out on January 16. Overall it reads well but there are a few points that would benefit from further clarification. I understand that the exposition shouldn't be too technical. Please find attached my comments and view them as suggestions for changes.

With best wishes, Egon

---

COMMENTS

Paragraph 2.1.: Polytopes as posets

It wasn't clear to me if this paragraph talks about traditional polytopes or abstract polytopes. It more accurately describes traditional polytopes. If the paragraph is meant to refer to abstract polytopes, then two issue need to be addressed briefly. They are addressed later in the text but it would be good to already mention them here.

First, the partial order < (or <=) may not be given by inclusion of vertex-sets (this is addressed in Paragraph 5.1). Second, "dimension" should be replaced by "rank". The convention is to reserve the term "dimension" for geometric realizations and to use "rank" for the combinatorial structure. For example, a skew hexagon in space is realization of dimension 3 of the standard hexagon, which is a polytope of rank 2. The situation is the same for any non-planar polygon. A polytope can have very many realizations and their dimensions may vary. The dimension of a realization is, by definition, the dimension of the ambient space (or more exactly, the smallest subspace that contains the realization).

Paragraph 2.2: Graphical representation

It may be better to replace "higher dimensional faces" by "faces of higher rank", for the reasons just mentioned. Similarly, throughout the entire text, replace "dimension" by "rank" whenever the discussion is about abstract polytopes rather than geometric polytopes.

Paragraph 2.3: Dimension or rank

Replace the sentence "The terms rank and dimension are equivalent" by "For convex polytopes, the terms rank and dimension are equivalent". Or if you want more generality, "For geometric polytopes, the terms rank and dimension are often (but not always) equivalent". The above example of the skew hexagon shows that they are not always equivalent.

Paragraph 3: Formal definition

The third axiom needs to be changed as follows:

It is strongly connected. That is, any flag can be changed into any other flag by changing just one face at a time, and the same is true for any two flags of every section of the polytope. Note that this is stronger than the current version. The additional condition on the sections says that the polytope is also locally connected, not only globally.

Paragraph 5.2: Incidence matrices

Omit the insert "or incidence geometry" in the second paragraph??

Paragraph 6.1: Duality

It is better to say, "Hence, the Hasse diagram of a self-dual polytope must be symmetrical about the horizontal axis half-way between the top and bottom. (The current term (n-1)/2 makes only sense when n is odd.)

Paragraph 6.3: Regularity

Replace the sentence beginning with "Equivalently" in the second paragraph, by "In particular, any two k-faces F,G of an n-polytope are "the same", i.e. .....". (Flag-transitivity is much stronger than the property described in this sentence.)

I would also continue the first sentence of the third paragraph as follows. "This is a weaker condition than regularity for traditional polytopes, in that it refers to the (combinatorial) automorphism group, not the (geometric) symmetry group. For example, any abstract polygon is .......

Paragraph 7: Realizations.

I would rewrite the first sentence as "Any traditional polytopes is an example of a realization [not bold-faced here] of its underlying abstract polytope". Then change the first sentence of the second paragraph as follows.

More generally, a realization [now bold-faced] of a regular abstract polytope is a collection of points in real space (corresponding to the vertices of the polytope), together with the face structure induced on it by the polytope, which is at least as symmetrical as the original abstract polytope; that is, all combinatorial automorphisms of the abstract polytopes have been realized by geometric symmetries. For example, the set ..........

Small point: In the second sentence of the first paragraph, replace "piecewise manifold" by "piecewise linear manifold".

Paragraph 8: The amalgamation problem

I would rewrite the first two sentences as follows: The basic theory of the combinatorial structures which are now known as "abstract polytopes" (but were originally called "incidence polytopes"), was first described in Egon Schulte's doctoral dissertation, although earlier work by Branko Gr"unbaum, Ludwig Danzer, H.S.M.Coxeter and Jacques Tis laid the groundwork. Since then, research in the theory of abstract polytopes has focused mostly on regular polytopes, that is, ...... (Note that I included Ludwig Danzer here. In the second sentence, I changed "exclusively" to "mostly".)

Further down, where one talks about {4,3}: Replace "joined three per edge" by "joined three per vertex".

One thing: The link "Egon Schulte" connects to a page where my name appears several times as "Egone Schulte".

Paragraph 8.1: Local topology

You could end the whole paragraph by saying, "However, much progress has been made on the complete classification of the locally toroidal regular polytopes", and then refer to my book with McMullen for reference.

Paragraph 11:

Typo in my name. It should be "Schulte".

THAT'S IT!

Egon Schulte,

http://www.math.neu.edu/~schulte

SteveWoolf (talk) 02:21, 9 February 2009 (UTC)

OK I have pretty much done all that, including redrawing the Hasse diagram. -- Cheers, Steelpillow (Talk) 21:21, 16 February 2009 (UTC)

Thanks for that, tho' haven't had time for detailed review. I am not happy, however, with the change to the Hasse diagram. Certainly many polytopes are not atomistic, i.e. cannot be represented only by vertex sets. But many are, such as the payramid. So why not use vertex sets in those cases? Later we can give an example of a non-atomistic pol, such as the digon or hemicube. I think the effect of the change is to make a gentle introduction more obscure. Let people understand the shift from the traditional to the abstract definition before you hit them with the more advanced concepts. SteveWoolf (talk) 03:09, 18 February 2009 (UTC)

As Schulte says above, "the partial order < (or <=) may not be given by inclusion of vertex-sets (this is addressed in Paragraph 5.1)". An AP is an abstract poset comprising unique elements of given rank. An element of any given rank does not per se comprise a combination of elements of lower rank. Such a vertex-combinatorial structure is just one of many applications for the AP and, as pointed out in Section 5.1, is not necessarily consistent with the geometric application. Also, inverting the diagram to obtain the dual polytope leaves us with a face-combinatorial structure which is less intuitive. Several of us discussed all this a while ago with Johnson and Grünbaum, and now Schulte's corroboration leaves the point pretty unarguable. If you would like to discuss the vertex-combinatorial approach in a separate section, the original diagram is still available. -- Cheers, Steelpillow (Talk) 12:30, 18 February 2009 (UTC)

Unanswered questions

My equivalent of Hilbert's 23 problems or the Millennium Prize Problems!

A number of questions have so far eluded my best efforts to solve them:

• Is there an example of a poset that satisfies all the polytope axioms except axiom 2? If not, then axiom 2 should be provable from the other 3.

• For a polytope P (rather than any poset) are the 2 "nice" properties independent, or does either imply the other?

These two properties are

(1) P is a lattice, i.e. any 2 elements have a meet and a join. This means that any two faces meet at a face (of lower rank); ditto the dual.

(2) P is atomistic, i.e. every face has a unique vertex set; and coatomistic, i.e. ditto the dual.

• The hemidodecahedron is "nice" as defined above, and yet it is not the face-set of a "traditional" polytope, so it's still "a bit funny". Is there another criterion, definable in pure AP terminology (no talk of realizability here please) that can exclude the hemidodecahedron (and others)?

• Is there an AP-definable property, (which may be a set of properties), eg called "Serious" , so that it can be shown to equivalent to "Realizable".

• Is there an AP-definable property, (which may be a set of properties), eg called "Sensible" , so that the set of Sensible polytopes matches the set of abstract versions of Traditional polytopes, for some "reasonable" defn of Traditional?

• Are the last two questions essentially the same?

• How can "Spherical" be defined in purely AP terms?

hmm... how about some "split and merge" approach - eg, define two operations on an abstract polytope whose facets are spherical (assuming this is well-defined in the lower rank) - one operation splits a facet (or several), another that merges two neighboring facets. Then, define the regular spherical polytopes as I did before - the universal locally spherical regular polytopes. Then, define any other locally spherical polytope to be spherical if there is a sequence of split and merge operations that transform it into a regular locally spherical polytope, or vice-versa. eg, the triangular prism is regular, because you can split the three square faces into triangles in just the right way to get an octahedron. The square pyramid is more complicated, and shows this idea has a whole lot of technical details to be filled in. mike40033 (talk) 08:28, 8 July 2010 (UTC)

Most interesting! For example, we can modify a cube by adding a diagonal to a face - and vice versa. Is that your idea? I have seen this method used to prove Euler's V+F=E+2 rule, so it looks promising. Has a nice parallel with topology's "continuous" concepts.

I think we can easily solve the square pyramid problem: allow vertex truncation! Truncating the "apex" (i.e. the vertex with the square vertex figure) immediately gives a nice spherical cube! Will these 4 operations suffice? (Including de-truncating - is that the same as stellation?) I'll play with it.

We will definitely need, I think, some rule that either states explicitly that we can only ever tweak to another polytope of equal rank, or that implies this.

This might generalise quite easily to all topological types - toroids, projectives... i.e. two pols are of the same type if there is a (finite!) sequence of "allowed" operations that transform one to the other.

If this works, I think someone will need to formally prove that our definition is consistent with topology, i.e. that any polytope "reachable" from another by our operations is also toplogically spherical (or, more generally, of the same topological type). SteveWoolf (talk) 19:46, 9 July 2010 (UTC)

if it works for spheres, it may work fine for other topological types... It may be a matter of coming up with the right set of operations. I'm sure this has been done before in a different context, eg, asking a topologist for advice might be a good idea - or an expert at simplicial complexes... mike40033 (talk) 01:04, 14 July 2010 (UTC)

• More generally, is there a way, using purely AP terms, to divide polytopes into different topological types? Or, equivalently, how can we define 2 AP's to be of the same topological type?

In the 2008 paper "Problems on Polytopes, Their Groups, and Realizations" by Egon Schulte∗ and Asia Ivi´c Weiss ist stated

In contrast to the traditional theory where a convex polytope is locally and globally spherical, it is a very subtle problem to define the topological type of an abstract polytope (see [42, Chapter 6]). In fact, this cannot be done unambiguously, except in certain cases

- a somewhat gloomy result. Still, we can always try, unless of course the impossibility is proven. And it may still be possible to achieve this aim for "nice" polytopes.

If none of us lesser mortals can solve these, I'll send them to Egon Schulte in due course - and to anyone else that any of you recommend.

SteveWoolf (talk) 21:24, 3 July 2010 (UTC)

If anyone else wishes to add further questions, go ahead! SteveWoolf (talk) 07:50, 6 July 2010 (UTC)

Standards and Definitions

• Any consensus reached about these, for Abstract Polytope topics, should be recorded here.
• Discussions about these should be carried out in another section.

General

• The terminology used in "Abstract Regular Polytopes" should be used wherever possible.
• American spellings (e.g. Realization) should be used in the article for technical terms.

Standard Terms

(Terms defined in this section are suitable for use in the article).

• The term "Traditional" should be used for non-abstract polytopes, i.e. what are often called Classical, Geometric, Ordinary etc. See Talk:Abstract polytope#FINAL DECISION: "Traditional". This decision may require review occasionally.

• An abstract polytope is flat if its vertex set is the same as the vertex set of a facet. SteveWoolf (talk) 04:55, 25 December 2008 (UTC)

Unofficial terms "Nice" and "Nasty"

(Talk page only).

An abstract polytope is nice if

(1) It is a lattice, i.e. any 2 elements have a meet and a join. This means that any two faces meet at a face (of lower rank); ditto the dual.

(2) It is atomistic, i.e. every face has a unique vertex set; and coatomistic, i.e. ditto the dual.

Otherwise it is nasty. SteveWoolf (talk) 07:04, 10 July 2010 (UTC)

Abbreviations

(Talk page only).

AP Abstract Polytope

TP Traditional Polytope (non-abstract)

ARP "Abstract Regular Polytopes" book (McMullen & Schulte, Cambridge U Press, can't recall the year)

M-S McMullen/Shulte

Pol Polytope

n-D n-dimensional, e.g. 4-D

SteveWoolf (talk) 05:40, 21 December 2008 (UTC)

Flat vs "Nasty" Polytopes

By definition, all flat polytopes are "nasty". But is the converse true? My guess, not. Anyone know?

The digonal prism is a nasty polytope which is not flat. In general, any polytope with a flat face is nasty, but not all such polytopes are themselves flat. mike40033 (talk) 00:18, 30 June 2010 (UTC)

Thanks for that, Mike. SteveWoolf (talk) 14:47, 3 July 2010 (UTC)

Is "flat face" synonymous with "degenerate element"? —Tamfang (talk) 23:42, 24 July 2010 (UTC)

no, degenerate means something even stronger than flat. A degenerate regular polytope has a digon as a section somehere, ie, its Schlafli symbol contains a 2, but there are lots of flat polytopes that are not degenerate. mike40033 (talk) 07:33, 26 July 2010 (UTC)

Archive of 1 Aug 2010

Have archived (hopefully) concluded discussions. By all means revive anything you consider worthwhile. Will remove this section after a while. SteveWoolf (talk) 00:28, 1 August 2010 (UTC)

Mike's Trihedron

A trihedron (left) which is atomistic, and it's dual which is not.

In recent discussions, we have learned that Hasse diagrams show only "consecutive" incidences - i.e. those with consecutive rank. Drawing all incidences produces a confusing mess - let us call this the "Spaghetti" (diagram), until someone proposes a better name.

I incorrectly remarked that the spaghetti of a 1-polytope (see picture, left) was not the graph of a polytope; Mike has pointed out that it is in fact (the graph of) a "nasty" trihedron with 3 faces: abd, bcd, abcd. It's dual has 2 digonal faces (pink, green) and 2 triangular faces (azure, yellow).

If Mike or anyone else prefers a different name for this polytope, nominate away!

These polytopes are good examples of small irregular, non self-dual, nasty polytopes.

It seems possible that ALL spaghettis of polytopes are themselves graphs of polytopes, and probably all "nasty". If so, this a rich vein of new polytopes for those that like the nasties.

What software do you use to draw these cool diagrams? mike40033 (talk) 07:27, 26 July 2010 (UTC)

Thanks and same as before - the humble MS paint! Which is a good example of how the simplest technology (or even the lack of it) sometimes does the best job. Besides being a musician, I dabble in art occasionally. I like M C Escher especially - google his work "Belvedere" for example and take a close look at it.SteveWoolf (talk) 14:48, 26 July 2010 (UTC)

"Nice" and "Nasty" terminology

Since we use "nice" and "nasty" a lot, do we need a more respectable term? Any suggestions? Meet/join lattice and dually atomistic doesn't exactly roll easily off the tongue, not to mention the grammatical problems. Personally I would prefer some term that connotes the English words traditional, normal, well-behaved, sensible. It should ideally be a word with an English opposite - "nice" eg doesn't have one. Those that like "nasty" polytopes but don't like the idea of a derogatory term (eg "nasty") might reflect on the bad names given to many numbers: negative, irrational, imaginary - and not forgetting transcendental (eg pi).

Some time with a Thesaurus produced

• Conforming/Non-conforming (too long)
• Civil/Uncivil
• Polite/Impolite
• Realistic/Unrealistic (too long, but good connotations)
• Pure/Impure (nice and short)
• Tidy/Untidy
• Safe/Unsafe
• Clear/Unclear

None of the above really inspires me, though!

I am still unsure about the relationship of the two nice properties (lattice, dually atomistic); if they turned out to be equivalent that would greatly simply our problem! But my guess is that they're NOT always, even if often.

I believe 'polite' is being used for a different concept in an upcoming article by someone. mike40033 (talk) 07:28, 26 July 2010 (UTC)

I like tidy. —Tamfang (talk) 22:54, 26 July 2010 (UTC)

An Aside.... A Puzzle....

An Apeirogonal Apeirohedron. Converting this to a digraph gives a solution - with infinitely many teams!

Here's a nice puzzle, not directly related to polytopes. I saw on the sports page of the newspaper a headline "Team X loses four out of 7 games", and I wondered, isn't it obvious that in a sporting contest, at least one team will lose 4 out of 7 games? (Assuming no draws, of course).

It turns out it's not so, so here's the puzzle :

Show, by giving an example, that it is possible for there to be a sports contest, where each team plays three games, there are no lossesdraws, and each team wins two out of their 3 games.

Have fun! mike40033 (talk) 07:54, 26 July 2010 (UTC)

By "no losses" do you mean "no ties"? —Tamfang (talk) 22:56, 26 July 2010 (UTC)

I'm confused too: if A plays B and A wins, doesn't B lose? If you meant no draws, I can only find cases with infinite numbers of teams. Make my Apeirogonal Apeirohedron drawing a Directed graph by adding an "arrow" to each edge in the upper half of the picture going away from the "centre", and going towards the centre if the edge is in the lower half. The "middle" edge (between 2-faces 1 and 2) goes UP. Then the directed edge A-->--B means A beats B. SteveWoolf (talk) 02:29, 28 July 2010 (UTC)

Yes, I mean no draws/ties... And you got it :-)

of course, I realised after I posed the puzzle, that there's a simpler solution, as illustrated by the diagram shown here : mike40033 (talk) 06:54, 28 July 2010 (UTC)

Another solution to the puzzle

Ooh you are a norty boy - I think. There is no diagram there. Which is the same as a diagram with zero teams and zero games - a solution, of course. For the benefit of normal people (i.e. all those except those not well-versed in modern math), it is in fact true (in logic and mathematics) that:

Everyone in an empty room is more intelligent than Einstein.

It is also true that

If 2 plus 2 equalled 5, then the moon would be made of cheese.

though the latter does assume that 2 + 2 = 4. If, conversely (perversely?) you don't accept the latter, then we are no longer in a position to learn anything new to science about the moon from 2 + 2 = 5 , or why the sea is boiling hot, and whether pigs have wings.

If 2 + 2 equals 5 and also 4, then mathematics is completely without contradictions, and in fact 2 + 2 = 6 and no other value

is another great truth.

Ah - I've just noticed "Image:Blank.svg|thumb|240px|Another solution to the puzzle" so you really are a norty boy! SteveWoolf (talk) 17:02, 28 July 2010 (UTC)

tres tres norty. For the same reasons you discuss, to find evidence that all crows are black, it is as logical to examine crows and observe their blackness as it is to examine red things and observe their non-crowness. mike40033 (talk) 04:14, 29 July 2010 (UTC)

"Duoprism" vs "Product polytope"

Please see my comments on this topic at Talk:Duoprism#"Duoprism" term and its definition. I am hoping to bring this topic to our AP article, so we need to tidy up things, I think. Some of you may wish to weigh in there. SteveWoolf (talk) 01:00, 29 July 2010 (UTC)

The Fanohedron

Two isomorphic graphs of the Fano plane. The graphs are polyhedral.

Browsing around, I came across the Fano plane, usually drawn as in the picture (left). This graph is isomorphic to the right hand picture. (To see that, the 3 outer points in the left picture are on the inner "circle" in the right picture). Not knowing anything about projective geometry, isn't the 2nd picture "better" because no lines cross? (Tho' the first is, I think. more attractive).

The graphs are indeed polyhedral. The fanohedron (my term - is there an official one?) is a "nice" irregular spherical deltahedron (with 10 triangular faces - in the right picture, the "outside" triangle is a face). I guess it would have quite a few interesting properties, including having 168 symmetries (one for each our of the weak - hour week joke for today). SteveWoolf (talk) 01:01, 1 August 2010 (UTC)

(moved from Mobihedra section)

The Fano plane is a projective configuration of 7 lines and 7 points. In the usual representation, we don't bother to connect the straight lines into loops, that is understood: only the one line is drawn as a circle, for convenience. In some formulations, the Fano plane is nothing less than the simplest possible projective geometry. Your fanohedron, while fun, has a different number of lines/edges and a different topology.

Is the Fano plane article incorrect then? Because the two graphs in the picture do have the same number of edges (15) and are isomorphic (as graphs), tho' of course my fanohedron has 2-faces also. Clearly I must spend time reading up on projective geometry. SteveWoolf (talk) 00:17, 1 August 2010 (UTC)

Well I guess projective geometry is about straight lines (among other things) so the above two diagrams are not the same in projective geometry, only isomorphic as graphs and AP's. If anyone can recommend an easy to read intro to projective geometry that is available online, I'd be grateful. SteveWoolf (talk) 11:29, 5 August 2010 (UTC)

Hemiprims

The hemiprism derived from a trigonal prism. It has 6 vertices, 9 edges, and 4 polygonal faces - 1 hexagonal (blue) and 3 square (red, green, grey).

Take an n-gonal 3-prism, eg a hexagonal prism (n=6), in the form of a circle of paper tape, with 6 edges drawn on it across the width of the tape. Now snip it along a drawn edge, twist one edge, and rejoin on the same edge to make a mobius strip. You still have 6 square 2-faces, but now only one dodecagonal 2-face where before you had two hexagonal ones. Is this a polytope?

I think yes always - but "nasty".

For n = 2, the digonal prism, already nasty, becomes the hemicube.

For n = 3, the trigonal prism gives a "mobihedron" hemiprism (see picture). It is "nasty" polyhedron - being atomistic, but not coatomistic. It is not a meet/join lattice. It has 6 equivalent vertices, 9 edges and 4 faces (3 square, 1 hexagonal). Its dual is not atomistic.

It's graph is K 3,3, famous in Kuratowski's theorem since no graph "containing" it can be planar - i.e. drawable without lines crossing. All "nice" polyhedra have planar graphs.

I don't know whether or not other polytopes have the same graph, but if so, they are almost certainly nasty.

The famous Utilities problem reduces to this graph also - 3 houses can't all be joined to Water, Electricity and Gas without pipes or wires crossing. (Isomorphically, 3 husbands cannot all construct secret paths to the houses of the others' wives without risk of being discovered en route to their sinful trysts).

In general, the graph of the "mobified" prism will be an n-gon with all pairs of opposite vertexes joined by a diagonal. SteveWoolf (talk) 07:21, 31 July 2010 (UTC)

The "mobihedra" hemiprisms are all tilings, or decompositions, of the projective plane - such figures are coming to be known as projective polytopes. "The mobihedra can be thought of as hemi-prisms. Each 2n-gonal "mobihedron" hemiprism may be derived either via the n-gonal mobius band construction or by removing half the faces of a 2n-gonal prism and identifying opposite exposed edges.

I have investigated your second construction idea, and you are indeed correct - mobifying an n-gonal prism as above gives the same polytope as hemi-ing a 2n-gonal prism. Don't think it works for n=1 - I'm pretty sure the digonal prism can't be hemi-ed, and indeed there are no 1-gons to mobify.

Most interesting, and thanks for that. So indeed, hemiprism is the correct name. If you have any refs, this could/should be a separate article - if you want to create it, I'd be happy to create pictures or otherwise assist. SteveWoolf (talk) 02:21, 1 August 2010 (UTC)

The hemi-ed digon prism is not a valid AP. My use of "hemi" here is more a description than an accepted name, though it follows the logic of the well-known "hemi" polyhedra. In crystallography a hemiprism is something different. BTW, I've just come across this reference: McMullen, P.; "Locally projective regular polytopes", Journal of Combinatorial Theory, Series A, Volume 65, Issue 1, January 1994, Pages 1-10. The abstract reads, "A regular polytope P is called locally projective if its minimal sections which are not spherical are projective spaces, and if these sections, together with its Schläfli type, determine the polytope. In this paper is described, for each n ≥ 4, a family of n or n − 1 (as n is odd or even) locally projective regular polytopes. They are shown to be universal of their type." Might interest somebody. — Cheers, Steelpillow (Talk) 17:07, 1 August 2010 (UTC)

Well I think the use of "hemi" as a prefix is well established, so I would have thought it was okay. Up to the rest of you. In any case we can use it in discussion, or with quotes in the AP article.

Projective geometry is a fascinating discipline. A hundred years ago Klein's Erlangen programme put it at the heart of symmetry, and hence of all geometry, where it has remained. It has the simplest axiomatic expression of all geometries. However it can be somewhat alien to Euclidean or Cartesian modes of analysis, so tends to be avoided by modern fashion. Unwisely in my opinion. — Cheers, Steelpillow (Talk) 21:05, 31 July 2010 (UTC)

Nailing the Nasties

Since the hemidodecahedron does have my two "nice" properties, yet presumably can be proved not to be (faithfully) realisable, it must presumably have (or lack) some property that prevents its being realisable. Yes I've raised this often - but I can't rest without being able to characterise trad pols in AP terms. I am considering the possibility that there is no reason other than cultural brainwashing to feel that the nasties aren't really kosher - but I do still feel strongly that they're a "bit funny".

$20 (Icosi) to the first person who comes up with a reasonable third "nice" property that excludes the hemidod (and hemi-icosahedron); the property must be AP-theoretic - no use of maps into Euclidean space &c. And of course, it must admit all trad pols including apeirotopes and toroids. By reasonable I mean it must be useful and can't be arbitrary, eg "doesn't have 10 vertices, 15 edges, and 6 faces" doesn't qualify!

But why (arbitrarily) include the toroids and exclude the projective planes?

Well toroids are realisable are they not? If not, can you tell me the simplest non-realisable one?

do we mean, here, realisable in the AP sense, or just that you can make a model of one in Euclidean space? mike40033 (talk) 05:17, 5 August 2010 (UTC)

Well realisability and trad pol theory in general are not my forte. Is there a meaning of realisable in an "AP sense"? I thought that was the holy grail we were seeking. Re Toroids, I was assuming that toroids created from trad pols (eg sphericals) would be "well-behaved", tho' those with non-traditional "aspects" would not be. Anyway, I'm fumbling and stumbling (and mumbling and bumbling) in the dark here, yet I still feel most strongly there is something important missing in the shift from trad to AP theory that should be addressed, and that the successful achievement of this will greatly improve our knowledge of the subject.SteveWoolf (talk) 14:56, 5 August 2010 (UTC)

The word 'realisable' has a definition. I don't think it's very good, but it is the standard definition. The correct definition may be found in ARP, the foggy one in my memory goes something like "There is a map from the vertex set of the AP to a set of points in R^n (n will not in general be the rank of the AP) such that all automorphisms of the polytope correspond to symmetries of the set of points", Eg, any polytope with a finite number of vertices can be "realised" at worst as a set of points {(1,0,...,0), ...., (0,0,0,...,1,....,0), ...., (0,...,0,1)} mike40033 (talk) 00:37, 6 August 2010 (UTC)

A polytope of n vertices can be "realised" as the vertices of an (n-1) simplex. Presumably, we then define the other faces (edges etc) as whatever the original had. Anyway, I don't think that counts as "realisable" - what use is a property that is always true? About as useful as saying that if you code Space=00, A=01,..., Z=26 then π (3.14159...) contains, somewhere, the complete works of William Shakespeare, or a recipe for apple π. (The problem is, we don't know where, though of course somewhere in π there will be a statement telling you...).

My personal view is that there are no infinities in the real universe - that space is finite and quantised - there are a finite number of points between two given points, like the Tube map. If I only had a brain of infinite size, tho', my viewpoint might be different. SteveWoolf (talk) 06:38, 6 August 2010 (UTC)

Ok, how about : "Any sequence of flags such that (i) consecutive flags in the sequence differ by exactly one element, and (ii) the sequence begins and ends with the same flag, must have an odd number of terms" mike40033 (talk) 08:18, 2 August 2010 (UTC)

Remarkable! I will investigate asap.

A bit pressed for time at the moment - deciding whether your razor works (admits most trad pols while barring the hemidod) looks quite tricky, for me at least. But I shall get to it asap, and will honour my prize committment! SteveWoolf (talk) 03:16, 5 August 2010 (UTC)

my razor cuts away all the non-orientable polytopes... :-) If you decide that this isn't cheating, you may honor the prize commitment by making a donation of $X to a tax-deductible charity which aims to alleviate poverty in developing countries, where X=20/(1-M), and M is your marginal tax rate... :-) mike40033 (talk) 00:38, 6 August 2010 (UTC)

Ok, nice idea. Which organisation, in your opinion, is the best at using dollars in the most productive way? Better to teach a man to fish than to give him a fish; better a fence at the cliff top than an ambulance at the bottom. The economic crisis, and Chernobyl amnesia has spawned a huge resurgence of nuclear power, which Greenpeace is fighting - tho' alleviating poverty is not their primary goal. (I would like time to try to understand your razor, tho', tho' your donation calculation wasn't too taxing). SteveWoolf (talk) 05:55, 6 August 2010 (UTC)

I'll leave the choice of charity to you, of course, though if I had to choose, I would pick World Vision. Let me know if I can do anything to help you understand the razor - another way of saying it is "any sequence of exchange maps that takes a flag to itself has even length".. this has more jargon in it, but is more illuminating once you are comfortable with the jargon... Are you familiar with the concept of the 'barycentric subdivision' ? You split each face (or facet) into triangles (or simplices) in such a way that the triangles (or simplices, and so on throughout this paragraph) and flags are in one-to-one correspondence. Then, flags that differ by exactly one element are adjacent triangles (triangles joined by an edge), or, exchange maps map triangles to adjacent triangles. Eg, the barycentric subdivision of the cube has each face split into 8 triangles, all of which have, as vertices, (i) the centre of the face, (ii) the midpoint of an edge, and (iii) a vertex. If a poly escapes my razor, the triangles can be painted checkerboard-style in alternating black and white. The hemicube does not escape. mike40033 (talk) 07:51, 9 August 2010 (UTC)

Thanks Mike. I have had to take a break and catch up with some work. However, I will get to your razor, and will honor the pledge - unless I can disprove it. But I may need some time to get to it, hopefully within a month or so. SteveWoolf (talk) 00:45, 12 August 2010 (UTC)

An observation from surface topology land. If an AP is a topological surface (all its faces are simply connected, i.e. topological balls) then whether it can be subdivided black-and-white in this way depends on its orientability. If an AP has non-simple faces (i.e. is locally projective or locally hyperbolic/toroidal) then I have no idea what happens - I would suspect that locally projective APs would fail, but locally hyperbolics might be able to fall either way. — Cheers, Steelpillow (Talk) 20:14, 9 August 2010 (UTC)

Yes, that's exactly what happens. This 'razor' cuts out the hemi-icosahedron and its dual (how sad, they were such nice polytopes :-) and passes the toroids.... but there are many other polytopes with (and without) much stranger topologies that it leaves intact. It fulfils the letter of the challenge, but not the spirit. I hope, for the sake of (at least one or two of) the world's poor, that Steve doesn't read this admission until too late, or thinks it's ok anyway.... (Hi Steve!) mike40033 (talk) 05:19, 11 August 2010 (UTC)

It also cuts out the non-orientable but faithfully realisable tetrahemihexahedron and all those others described by Apéry and Brehm (see below). Unless I misunderstand, that doesn't fulfil the letter of the challenge. — Cheers, Steelpillow (Talk) 16:26, 11 August 2010 (UTC)

*sigh* back to the drawing board.... mike40033 (talk) 03:10, 12 August 2010 (UTC)

Investigating Mike's axe - a discovery (for me)! Given a polytope P, define a graph whose vertices are the flags of P, and xy is an edge if flags x and y are adjacent as flags. Is this always the graph of a polytope? An n-polygon generates a 2n-polygon, including the digon, which gives a square, and the apeirogon which gives another apeirogon. The tetrahedron gives, I think (confidence: fair), a Truncated octahedron with 24 equivalent vertices with triangular vertex figures, an Archimedean solid. Can anyone enlighten on this general "flagotope" topic? SteveWoolf (talk) 13:17, 7 August 2010 (UTC)

Hmmm.... I'm trying to think how the higher-rank faces might be defined to make this work... hmmm.... blank.... blank.... need more caffeine.... mike40033 (talk) 07:59, 9 August 2010 (UTC)

Ok, this works - but I had to lean on some technical results about flag actions to prove it - Let P be an abstract polytope, and let F be the set of its flags. For any flag Phi, and for any subset S of {0,..,d-1}, let F(Phi,S) be the set of flags that can be obtained from Phi by successively changing elements with ranks in S. Then, the set of all F(Phi,S) (and the empty set) with partial order 'is a subset of' is a polytope, with the vertices and edges you describe. mike40033 (talk) 08:13, 9 August 2010 (UTC)

Ah - a question: are projective polytopes and (faithfully) unrealisable polytopes one and the same class? Or are there projectives which are realisable, or unrealisables that are not projective? SteveWoolf (talk) 19:07, 1 August 2010 (UTC)

Suppose this conjecture were true. Then, since projective polytopes are unrealisable, then locally projective polytopes are also unrealisable. However, locally projective polytopes are not projective. Contradiction! mike40033 (talk) 00:18, 3 August 2010 (UTC)

Okay, then is realisabilty equivalent to "neither projective nor locally projective" for example? Or "not having any projective sections"? SteveWoolf (talk) 05:10, 3 August 2010 (UTC)

Faithfully realisable examples of the projective plane include the uniform Tetrahemihexahedron. Apéry included several polyhedral examples in his book on "Models of the projective plane", and Brehm published an article on "Minimal polyhedral models of the projective plane" in which he described some novel 9-hedra. Some non-orientable toroids can also be faithfully realised as uniform polyhedra, while some of the simpler orientable toroids (such as {4, 4} where the count of squares is just 4), cannot.

Many (all?) locally projective polytopes do not form a closed surface (something I only found out recently). One might argue that therefore they cannot bound a definite "interior" and hence cannot be realised faithfully.

The process of realisation is poorly understood. My view is that until it can be properly formalised to provide a consistent approach to both AP theory and Euclidean geometry, then there can be little chance of establishing deeper formal correspondences across the divide. Not a popular view perhaps, but if anybody does manage to build that sound mathematical bridge, no-one will be more delighted than I.

— Cheers, Steelpillow (Talk) 09:19, 4 August 2010 (UTC)

The Tetrahemihexahedron looks interesting - clearly fits nicely into 3-space - I will draw a Hasse and convince myself it really is a polytope - and see how "nice" it is.

The gulf of unsolved problems reminds me of Godel, who proved that there are mathematical truths that cannot be proved, dealing the death blow to Hilbert's 1900 dream of wrapping up once and forever all the remaining loose ends of mathematics. On the other hand, the recent proof of Fermat's Last Theorem after 350+ years encourages us not to give up too easily.

SteveWoolf (talk) 03:22, 5 August 2010 (UTC)

The Tetrahemihexahedron

I have researched the Tetrahemihexahedron and can state (confidence: high) that

• It is an abstract polytope
• It is both atomistic and coatomistic - no two faces have the same vertices, ditto the dual
• It is neither a meet nor a join semi-lattice.

Yes! We have finally answered one of the Unanswered Questions above. There are atomistic pols which are not meet/join lattices - the 2 "nice" properties are not equivalent.

Actually, it is only a partial answer. Are there meet semi-lattice polytopes that are not atomistic or not coatomistic? Actually, there are of course four nice properties (so far) - atomistic, coatomistic, meet semi-lattice, join semi-lattice. My guess is that they're all independent, but it's only a guess. There are probably numerous theorems of the form "A polytope which has property N1 and which is X is also N2" where N1 and N2 are one of the four nice properties and where X is some other property.

To see that the TH3 is not a "meet-pol" or "join-pol":

Let the 3-face be abmnxy, where a,b are opposite, ditto m,n and x,y. The 3 square faces are then ambn, axby, mxny. Any two of these intersect at two separate points, NOT along an edge.

Now take a pair of opposite points say a,b. No edge has them both. But they are both in the faces ambn and axby. So there is no unique smallest superface, i.e. no join.

So thanks, Guy, for introducing me to the Tetrahemihexahedron, my life will be forever the better.

But is it a Traditional polytope?

Given the failure of the faces to intersect "nicely" - is this creature actually a polytope (in the non-AP sense) ??? If so, why is the hemicube not one? I had thought ALL trad pols were (meet/join) lattices - is this not so? Having widened "polytope" in AP theory, are we now rewriting history and calling all AP's also trad pols? Or am I Xumbling in the dark again? Let there be light! SteveWoolf (talk) 07:20, 6 August 2010 (UTC)

The geometric tetrahemihexahedron has long been accepted as one of the traditional uniform polyhedra. It is fully three-dimensional, has flat faces and straight edges and, although elements may intersect, none overlap or coincide. So by most definitions it is a faithful realisation.

The hemicube cannot be realised in this way, and therefore is not a traditional polyhedron. My favourite realisation maps the faces onto the Petrie polygons of the regular tetrahedron - these faces are skew, so the realisation is not faithful. "Filling in" the face interiors with their minimal surface (a parabolic hyperboloid) looks cool.

I assume you mean this object - yes it's very nice.

One way to understand the tetrahemihexahedron is as a "hemi cuboctahedron". Its dual is therefore a "hemi rhombic dodecahedron" - a hexahedron which, as it happens, cannot be realised faithfully in Euclidean space and so is not traditional. Does this dual yield any insights into nice/nasty properties?

— Cheers, Steelpillow (Talk) 21:45, 6 August 2010 (UTC)

I haven't checked whether the tetrahemihexahedron ("TH3") is the hemi of the cuboctahedron, but as I have carefully researched and said (above), the TH3 is both atomistic and coatomistic, and neither a meet nor a join semi-lattice. Since atomistic and coatomistic are dual properties, as are "meet-pol" and "join-pol", it follows that the dual of TH3 is also both atomistic and coatomistic, and neither a meet nor a join semi-lattice.

We now have the "interesting" situation that, from what you say, a trad pol has a dual which is not a trad pol. This seems quite bizarre to me at least - and seems to lend weight to my suggestion that TH3 is not a trad pol. Certainly it's "long acceptance" as one doesn't carry much mathematical force, though of course traditional means precisely that. Are there many examples of trad pols without trad duals? SteveWoolf (talk) 20:08, 7 August 2010 (UTC)

Ouch! Caught out by my own forgetfulness. I owe you an apology. The "hemi rhombic dodecahedron" can be realised in distorted form as a trad pol, but its symmetries cannot be realised geometrically at the same time (the reasons for this get mildly complicated). The M&S definition of faithfulness requires symmetries to be realised too hence, strictly, the hemi rhombic dodecahedron trad pol is not faithful. Given the latter, I forgot that the former is not necessarily implied. — Cheers, Steelpillow (Talk) 21:15, 7 August 2010 (UTC)

If Steelpillow's original comment were correct, then the rest of the hemipolyhedra (not just the uniform ones!) and Miller's monster would all be trad pols without trad duals. Double sharp (talk) 09:02, 24 July 2012 (UTC)

True but perhaps uninteresting. Whether a given figure, such as a hemi dual, has a realisable form is so far as I know quite arbitrary. The point of greater interest is that none of the hemi duals can be faithfully realised. — Cheers, Steelpillow (Talk) 19:54, 24 July 2012 (UTC)

Example abstract quads

I think that the lead illustration of abstract quads could include some more general examples. The present diagram suggests that abstract polytopes only represent real polytopes, which is incorrect. Some suggested additions/substitutions:

• A cyclically ordered set of point pairs, not sure how it should be represented but something like: {{a,b}- {b,c} - {c,d} - {d,a} -}. I think this is important, to illustrate that these figures are essentially abstract.

• A simple arrangement of four infinite lines and four crossing points, something like this:

    |     |
  --o-----o----
    |     |
    |     |
  --o-----o----
    |     |

Even in elementary geometry, such figures are often described as polygons.

• A Petrie polygon of a regular tetrahedron. This is a skew (non-planar) example.

• A vertex figure of an octahedron, comprising the vertex and the connected edges and sides - I would also include the body of the figure to illustrate the maximal element, but I suspect that might be controversial. This would help to prevent any initial assumption on the part of the reader that abstract rank and real dimension are identified.

What do folks think? — Cheers, Steelpillow (Talk) 15:09, 1 January 2011 (UTC)

not sure.... I suspect I'm too familiar with abstract notations to be sure what a non-technical reader can swallow in the intro... mike40033 (talk) 03:25, 13 January 2011 (UTC)

Clarity shouldn't be sacrificed for efficiency – but it shouldn't be sacrificed for inefficiency, either.

This article tries so hard to give the reader a gentle introduction to abstract polytopes that it completely falls on its face with irrelevancies.

The London Tube Map, for instance, is a huge distraction. Worst of all, the reader has virtually no idea of what an abstract polytope is until after having to read far too many words.

It would be so much better to start with an actual example of an abstract polytope by taking a common polytope – like the cube – and reducing it to its 0-, 1-, 2-, and 3-faces, each expressed as a subset of the 8 vertices, forming a partially ordered set, and arranged in layers according to their dimension (possibly leaving the minus-one dimensional layer of the null set for a later formalization).

Then it can be observed that each 0-face lying in a 2-face (or each 1-face lying in the 3-face) has exactly two faces of the in-between dimension as the in-between elements of the partially ordered set. Etc.

As it currently stands, the article takes so long to get to the point that its attempt at a gentle introduction is far more confusing than helpful.Daqu (talk) 20:25, 15 June 2014 (UTC)

I broadly agree with you. I would only caution that the treatment of a j-face as a set of (j−1)-faces is not fundamental but is just one of several interpretations or applications: this idea should not be introduced until later. — Cheers, Steelpillow (Talk) 21:08, 15 June 2014 (UTC)

In no way did I intend to suggest thinking of "a j-face as a set of (j−1)-faces".Daqu (talk) 13:59, 18 June 2014 (UTC)

I should have written "...of a j-face as a set of k-faces where k < j...". The treatment of a j-face as a set of 0-faces (e.g. the j-faces of the cube as subsets of the 8 vertices) is not foundational and can cause confusion if introduced too early. — Cheers, Steelpillow (Talk) 14:46, 18 June 2014 (UTC)

I definitely agree that it's not foundational. But it could easily be made clear that the partial order in the case of that example is valid for that example and not necessarily others. Then another example of an abstract polytope that isn't an ordinary polytope could be given next, for comparison.Daqu (talk) 07:46, 26 July 2014 (UTC)

[Updated] On re-reading your original post, I realise I just suggested much the same thing, whether with cubes, squares, or whatever, so I have rewritten this comment. The only danger is that we might end up distracting ourselves from the core topic too soon, by explaining why we are not using a core example. Something like a real Euclidean polygon or polyhedron is obvious enough not to need explanation, but anything else may not be familiar to all visitors. Having said that, there might be a useful progression of ideas here. The familiar geometrical ABC notation is typographically vertex-combinatorial but, in Euclidean space at least, typically represents the object between the vertices. Taking this understanding as our starting point: 1) a real polytope, say a square ABCD; 2) its vertices A, B, C, D, sides AB, BC, CD, DA and interior ABCD, all tabulated as ranked by dimension and typographically still using the vertex-combinatorial description; 3) the fully abstracted expression with elements ø,a,b,c,d,e,f,g,h,z including the Hasse diagram and an explanation of how the ABCD incidences are now denoted by the links and not the names. In this way, the explanation builds itself as we progress. — Cheers, Steelpillow (Talk) 09:51, 26 July 2014 (UTC)

Something along these lines:

A geometrical polytope such as the square ABCD has vertices A, B, C, D, sides AB, BC, CD, DA and a region or body ABCD. These individual elements may be ranked according to dimension:

Image for effect only. Needs aligning with the text

Elements of a square

Type	Dimensions	Elements
Region	2	ABCD
Side	1	AB, BC, CD, DA
Vertex	0	A, B, C, D

The incidence relations between these elements may be represented as the partial ordering of a generalised or abstracted set P, in which the partial ordering is most clearly expressed as a Hasse diagram. Note that the diagram includes the empty set ø, which is a member of every set. The incidence relations are expressed by the links, so there is no need to label the sides e, f, g, h and region z with the connected vertices.

The dimension of any element is now described by the ranking, from -1 (for the empty set) to n (for the maximal element of an n-polytope).

Such a partially ordered set, or poset, retains no vestige of real geometry and expresses the structure of the polytope in a wholly abstract way.

Any good? — Cheers, Steelpillow (Talk) 10:14, 26 July 2014 (UTC)

Monogon

The section on The simplest polytopes does not include the monogon, a simple closed loop with a single vertex. What part of the definition of an abstract polytope does this figure fail to meet, or is it in fact a valid polytope? I think it would be useful to explain all this, if only to illustrate the rather obscure definition for those of us who struggle with the jargon. — Cheers, Steelpillow (Talk) 10:21, 30 August 2013 (UTC)

Below is a crude Hasse diagram of it. I would guess that the "double-bond" between the edge and vertex is the problem, but how does that relate to the definition?

 Max
  |
 Edge
  ||
Vertex
  |
 Min

— Cheers, Steelpillow (Talk) 10:25, 30 August 2013 (UTC)

Between Max and Vertex, there only exists a single element, Edge. The definition of a regular polytope requires that there exist exactly two elements between incident elements whose rank differs by 2. It's nothing to do with "how many times" Edge and Vertex are connected - remember, abstract polytopes are just partially ordered sets. In your diagram, Edge > Vertex, and Max > Edge. If you ask "how many times is Edge > Vertex", your diagram suggests the answer is "2", but actually it's not a meaningful question. mike40033 (talk) 07:16, 16 May 2014 (UTC)

By the way: What should we call a two-sided polygon? I would suggest we just let bigons be bigons. :-D mike40033 (talk) 07:19, 16 May 2014 (UTC)

If by "a two-sided polygon" you mean what I guess you mean, then it's called a Dihedron. The editor who uses the pseudonym "JamesBWatson" (talk) 14:38, 29 May 2015 (UTC)

A two-sided polygon is a digon. A dihedron is a two-sided polyhedron. — Cheers, Steelpillow (Talk) 18:42, 29 May 2015 (UTC)

Yes, of course. For some reason I took "two sided" as meaning "considered as having two faces, facing in opposite directions, but coincident", whereas obviously it meant "with two edges". The editor who uses the pseudonym "JamesBWatson" (talk) 19:23, 29 May 2015 (UTC)

My point is that the formal definition given in the article does not express this in an immediately obvious way. Here it is:

An abstract polytope is a partially ordered set, whose elements we call faces, satisfying the 4 axioms:

1. It has a least face and a greatest face.
2. All flags contain the same number of faces.
3. It is strongly connected.
4. Every 1-section is a line segment.

An n-polytope is a polytope of rank n.

Would I be right in saying that the connection is that the monogon has two 1-sections which are not line segments? Also, I believe that the line segment itself does meet the definition and is therefore also an abstract polytope - in fact, axiom 4 mandates that it is the simplest one possible. Am I correct? Either way, I think it would be helpful to explain these examples in the article. — Cheers, Steelpillow (Talk) 08:26, 16 May 2014 (UTC)

The reason we talk of digons is all Greek to me. :-p — Cheers, Steelpillow (Talk) 08:26, 16 May 2014 (UTC)

The 4th axiom is unnecessarily hard to understand. Better to speak of what lies between a k-face and a comparable (k+2)-face.

Also: Yes, axiom 4 means that for any k-face and comparable (k+2)-face, there must be exactly two (k+1)-faces between them in the partial order. So a monogon fails to be an abstract polytope. (Though Steelpillow's creative idea of a "double-bond" sounds like a useful generalization.) And Yes, displaying the monogon as one almost-abstract polytope that fails to be one would be an excellent idea.

And as regards ". . . those of us who struggle with the jargon":

What number does the prefix jar- indicate?Daqu (talk) 20:38, 15 June 2014 (UTC)

LOL. I don't know, I'll see if Jar-Jar Binks knows. — Cheers, Steelpillow (Talk) 20:58, 15 June 2014 (UTC)

Clarity is needlessly sacrificed for efficiency

The formal definition of an abstract polytope is given as follows:

"An abstract polytope is a partially ordered set, whose elements we call faces, satisfying the 4 axioms:

1. It has a least face and a greatest face.

2. All flags contain the same number of faces.

3. It is strongly connected.

4. Every 1-section is a line segment."

Axiom 4. is a very obscure way to say, equally rigorously, that if the ranks of two faces a > b differ by 2, then there are exactly 2 faces that lie strictly between a and b.

An example would help as well: Axiom 4. is true for every actual polytope, such as the 3-dimensional 4-sided pyramid P. The 3-cell of P, and any of its edges, have exactly two 2-faces containing that edge. Likewise, for any 2-face of P and any vertex of that 2-face, there are exactly two edges of the 2-face that contain the vertex.

Also: The definition of "rank of a poset" as given on the portion of the article under Rank is not adequate to making clear what the rank of a section means. In fact it confuses the issue by assigning a fixed rank to each face. In fact the rank of a face depends on which poset it is being considered as a face of (such as a section of a much larger poset).Daqu (talk) 06:49, 5 July 2012 (UTC)

Agreed. Worse, a line segment is defined earlier as a particular poset. That is quite wrong - abstract theory has no need of line segments - and, doubly wrong, that definition is inconsistent with Axiom 4 anyway. As given, a line segment has four elements of ranks -1, 0, 0 and +1 while in truth a 1-section has four elements of rank (say) n-1, n, n and n+1. A neat example of the confusion you also note. — Cheers, Steelpillow (Talk) 20:09, 5 July 2012 (UTC)

Agreed. I was trying to understand why the definition of a polytope would not apply to a Rank 1 polytope with three elements at Rank 0. It was not possible to understand from this. I had to go to this page: to understand. That document describes Axiom 4 as the "diamond condition" which is much clearer. Also, there is no definition of a 1-Section. I had to guess at the meaning. 58.34.50.139 (talk) 08:01, 14 January 2016 (UTC)

a polytope can only be fully described using vertex notation if every face has a unique set of vertices.

Surely every polytope satisfies the quoted property, including the digon. I think what is meant is "a polytope can only be fully described using vertex notation if every face Is uniquely determined by a set of vertices." -lethe ^{talk +} 18:49, 6 November 2016 (UTC) -lethe ^{talk +} 18:49, 6 November 2016 (UTC)

The two sides of the digon do share the same set of vertices, neither has a set unique to it. But strictly, the face is a distinct element of rank 2 while each vertex is a distinct element of rank 0 and neither can "have" or "be determined by" the other rank. What they do have is an incidence relation, defined within the partial ordering. The so-called vertex notation blurs this key fact and in my opinion whoever made such a meal of it here did the article a disservice. I have edited the section to correct and clarify a little, but much needs to be tidied elsewhere to give an accurate and clear explanation. — Cheers, Steelpillow (Talk) 21:03, 6 November 2016 (UTC)

"Our theory" ?????

From the Introductory concepts section:

"We use the term face to refer to ..."

"We shall define a polytope, then ..."

"When F < G, we say that ..."

An encyclopedia article is not intended as a personal memoir.

Thus the use of "we", etc., is inappropriate and distances the reader.

"If [abstract polytopes' having a "least" face] seems strange at first, the feeling is quickly dispelled on seeing the elegant symmetry which this concept brings to our theory."

Here again, "our theory" is inappropriate. Etc.Daqu (talk) 06:13, 20 December 2016 (UTC)

Agreed. There is a lot of non-encyclopedic style to be ironed out, a lot of the article reads more like a text book than an encyclopedia - WP:NOTTEXTBOOK. — Cheers, Steelpillow (Talk) 10:30, 20 December 2016 (UTC)

Anyway, I made a start. — Cheers, Steelpillow (Talk) 11:22, 20 December 2016 (UTC)

Role of the empty set

The article currently states that "Just as the number zero is necessary in mathematics, so also set theory requires an empty set which, technically, every set contains. In an abstract polytope this is known as the least or null face and is a subface of all the others. Since the least face is one level below the vertices or 0-faces, its rank is −1 and may be denoted as F−1. It is not usually realized." The first sentence is not correct: the empty set {} or ∅ is certainly a subset of every set but it is not normally an element of the set. Therefore a set does not normally "contain" ∅ though it does "include" it. The empty set can be added as an element but this must be done explicitly, as {∅, ...}. For example {∅} is a set of cardinality 1 containing the empty set as an element. It is sometimes said that the rank −1 element of an abstract polytope is, or gets gets realized as, the "null polytope", but is that a property of the realization or of the abstract set itself? There appears no a priori reason why that rank −1 element should be the empty set. Can anybody find a reference which clarifies the correct treatment here? — Cheers, Steelpillow (Talk) 16:11, 3 October 2019 (UTC)

[Update] Several papers by McMullen and/or Schulte describe the rank −1 element as "improper" but stop short of identifying it with any set-theoretic construct. Schulte remarks that the dual polytope is obtained simply by reversing the ranking. I do not have access to their definitive book on Abstract Regular Polytopes. However Johnson (Geometries and Transformations) is explicit that the rank −1 element is the empty set ∅ and that the null polytope is {∅}. But as far as I can see a simple reversal of ranking will not then produce the dual polytope, as the empty set then gains maximal rank, and Johnson does not appear to make that claim. So on the face of it we have two incompatible definitions to deal with, though we need sight of McMullen & Schulte's book or other suitable reference to be sure that this conclusion is not my original research. Does anybody here have access to a copy? — Cheers, Steelpillow (Talk) 09:53, 4 October 2019 (UTC)

Somebody off-wiki has cited me the relevant passage from McMullen and Schulte:

"A flag of an n-polytope P is a maximal subset of pairwise incident faces of P; thus, it is of the form {F₋₁, F₀, . . . , F_n−1, F_n}, with F₋₁ ⊂ F_{0 ⊂ ··· ⊂ Fn−1 ⊂ Fn.}

"Here we introduce the conventions F₋₁ := ∅ and F_n := P for an n-polytope P; the inclusions are strict, so that dim F_j = j for each j = 0, . . . , n − 1. The improper faces ∅ and P are often omitted from the specification of a flag, since they belong to all of them. The family of flags of P is denoted F(P)."

So it is a "convention" that F₋₁ := ∅. Thus, one can say something like, "while every set has the empty set ∅ as a subset, by convention an abstract polytope also contains ∅ as an element through the identification F₋₁ := ∅." I will make that change. The inconsistency which I see in that convention is confirmed to be WP:OR so we must pass it by. — Cheers, Steelpillow (Talk) 09:32, 9 October 2019 (UTC)

Face lattice

The relationship between abstract polytope and face lattice should be clarified. Currently that term links to Convex polytope § The face lattice, but surely non-convex polytopes have face lattices too. Should it redirect here instead? Watchduck (quack) 22:40, 8 October 2019 (UTC)

The set-theoretic descriptions do appear to be identical. However I have only ever seen the term "face lattice" in discussion of convex polytopes. I have added cross-links to both articles, but I am reluctant to change the redirect unless someone can find/cite its usage for the non-convex case. — Cheers, Steelpillow (Talk) 09:06, 9 October 2019 (UTC)

Oops, apparently not quite identical. I have come across this from Schulte:

Face lattice of a polytope: The set F(P) of all (proper and improper) faces of P, ordered by inclusion. As a partially ordered set, this is a ranked lattice. Also, F(P) \ {P} is called the boundary complex of P.

I have no idea how or why this differs from P itself, so will have to do some more reading and head-scratching. — Cheers, Steelpillow (Talk) 12:37, 10 October 2019 (UTC)

Simplex and hypercube

Face lattices of triangle and tetrahedron (universe in center, empty faces not shown)

It would be good to add the relationship between n-simplex and (n+1)-hypercube to the examples. See Simplex § Relation to the (n + 1)-hypercube, Hypercube § Relation to (n−1)-simplices. Watchduck (quack) 22:40, 8 October 2019 (UTC)

I think that is too detailed and off-topic an observation for this discussion. It applies as much to concrete geometric figures and is more relevant to the duality of polytopes. We really do not need to explain here that the equilateral triangle is both a vertex figure of the cube and a face of the octahedron. — Cheers, Steelpillow (Talk) 09:14, 9 October 2019 (UTC)

Maybe we are talking past each other here. I was talking about the cube as the face lattice of the triangle, not about the triangle as vertex figure of the cube. (If that is somehow the same, then I don't get it right now.)

I am not sure if bringing up the face lattices of hypercubes will add to or reduce the (supposed) confusion, but according to 0xDE the face lattice of the square is the tetragonal trapezohedron: Bit tricks for wildcard strings and hypercube face lattices, Face incidence polytopes

Assuming that this article gets a section about face lattices, mentioning that would seem reasonable to me. Watchduck (quack) 23:04, 10 October 2019 (UTC)

I'm not convinced that this is sufficiently important to mention here, but maybe we should have a separate article on face lattices where it can be mentioned? My blog posts don't count as reliably published sources, but the comments on the second one have pointers to some sources that might be used for this. —David Eppstein (talk) 23:30, 10 October 2019 (UTC)

OK, so since this is meant to be a subtopic of the face lattice conversation, I am editing the heading accordingly, I hope that is OK.

Definition and distinction

One thing still confuses me: if the Hasse diagram can be read as capturing both a polytope and a face lattice, and both are defined as partially-ordered sets, then what is the distinction between the two? Here are a couple of Hasse diagrams for the square pyramid:

Abstract polytope from this article.

Face lattice from the convex polytope article.

They are in essence the same diagram, just with different naming conventions. I seem to recall that at one time the second diagram was even used for the abstract polytope article.

According to Schulte, if P is some polytope then its face lattice is F(P) and (while we are at it) it also has a boundary complex F(P)\{P}. But I find Schulte's treatment confusing because he is not above adopting such conventions as F := F/F₋₁, i.e. defining a face as a certain associated section. Johnson (2018) calls F/F₋₁ its span <F> and elsewhere has explicitly cautioned against McMullen & Schulte's convention. His reason is of course that the fine distinctions between such entities get lost and the theory loses consistency. On the other hand the right hand "combinatorial" labelling was the original conception of abstract polytopes (and still used by Johnson), with the left hand generic "discrete-object" labelling developed later and adopted by McMullen & Schulte. Historically there has also been a divide between the combinatorialists who understand abc as essentially the point set {a,b,c} and the geometers who understand it as the triangular region bounded by points a, b and c.

In dealing with such inconsistencies we must be careful to remain precise but also to avoid WP:OR. Returning to the face lattice, how then does it differ from the abstract polytope itself? — Cheers, Steelpillow (Talk) 09:51, 11 October 2019 (UTC)