Abstract polytopes and realization

[steelpillow]

Abstract polytope theory makes brief appearances on this forum but does not seem to have a proper focus. It is a set-theoretic description of polytopes structures, shorn of all geometry. To create a geometrical figure, the abstract set must be "realized" - mapped or injected into some containing space. This thread is intended to cover both abstract theory and the various approaches to realization.

To kick off, a more formal definition: An abstract polytope is a partially-ordered set (poset) of members or elements. The elements are ranked according to their dimension, with an incidence relation existing between pairs of elements in adjacent ranks. The partial-ordering must meet three conditions, which Norman Johnson has described as monal, dyadic and properly-connected. This basically means just two elements of any given dimension meeting on (incident with) the same element of one dimension lower, and no cheating with duplicate elements, structures, etc.

Such an abstract structure may be described in a Hasse diagram. In particular, the dyadic requirement leads to a characteristic embedding of diamond shapes in the diagram, leading to its common name as the diamond condition. Here is the Hasse diagram of the cube:



Note the need for a maximal (body) and a minimal (nullity) element for the set theory to work properly.

Dualising an abstract polytope is stunningly simple; you turn the Hasse diagram upside down (more correctly, you reverse the order of ranking). That is it, it is now the abstract octahedron!

Early versions of the theory treated everything as sets of vertex points; an edge was the set of its two ends, for example, while the nullity was the empty set. However dualising is a pain because the dual has different vertices from the original and the sets of vertices all have to be relabelled. In fact, abstract algebra becomes most tractable and elegant when you treat each element as an atomic object in its own right.

It is sometimes argued that the nullity must be the empty set, because every set contains the empty set. But this commits the elementary blunder of confusing membership with containment. Any set contains all possible combinations of its elements, but not all those combinations need be members of it; the empty set is always contained, but it is not necessarily a member. And the Hasse diagram shows only the members. So modern approaches treat the nullity as just like any other member - I think of it as the null polytope or nullon.

Things get really interesting in four dimensions and above, where abstract theory becomes more general than traditional topology. For example the 11-cell and 57-cell regular polychora are not properly constructed topological manifolds, but they are valid abstract polytopes.

I expand on the basics at https://www.steelpillow.com/polyhedra/abstract/abstract.html

[mr_e_man]

My approach to realization is what you call configurations (not necessarily regular). Each rank k element is realized as a whole k-dimensional subspace. Compared to other approaches, this simplifies the description of an element: it's just a system of linear equations. It also clarifies the distinction between an element and its span (your term; not linear algebra). A rank 2 element, a plane, is obviously not a polygon. Rather, a rank 2 element, together with the rank 1 and 0 and -1 elements incident with it, is a polygon. The Polytope Wiki doesn't maintain this distinction.

I'd like to know what you think of my notion of insiding, described here: viewtopic.php?f=25&t=2437 . Basically, to each incident pair of elements with ranks k and k+1, we assign one of the two unit vectors contained in the k+1 subspace and perpendicular to the k subspace. And any three of the four insiding vectors in a dyad determine the fourth vector.

Did you change your "Filling Polytopes" page? I think I remember a picture of a 1-polytope as part of an oval (or stadium), connecting the two endpoints by going through infinity. But now I see nothing about that. Was it somewhere else?

[wendy]

The trouble with abstract polytopes, is that they deal with a limited sample that works in their theory. By weaving the theory in and out, it is possible to make all sorts of wild associations, such as the nulloid is an empty set, or that 'top closure' is required.

Things that one might think are polytopes in an abstract sense, like the desarge configuration, complex polytopes, and the pano diagrams, or some of the things in Coxeter-Moser, do not fall in the study of 'abstract polytopes'. Instead, we are presented with a structured set containing the complete Hasse antitegum of a simple polytope. Anything that does not pass this test, and many besides, are simply not polytopes.

The nulloid or 'namon' is a common down incidence to all other other elements, but is specifically no down incidence on things that are not part of the polytopes. Polytopes are an instance of multitope. When i draw a multitope, i select that certian points represent vertices, edges, hedra, and so on. These are down-incident on the name i select. Things that have not been selected are dot down-incident on the name. This behaviour means that the namon is not the same as an empty set. Anything is down-incident on the empty set.

An other failing of set theory vers the hasse diagram, is that the up incidence does not correspond to union or complement. This is why we have 'posets'. For example, the intersection of two surtopes is the common or shared parts. The complement of a surtope does not correspond to any element of the figure, and certainly not to those elements that are up-incident of the element. The intersection of incidence and set theory is purely on the down-incidence.

It is possible to construct a stable Hasse diagram on two line segments connected at a vertex. A hasse diagram simple shows the incidance of a multitope.

Code: Select all

      o      o
     / \    / \
    /   \  /   \
   o      o     o
    \     |    /
      \   |  /
        \ | /         o----o----o
          o


The figure on the left is the hasse diagram of the one on the right. There is no top closure. But its a faithful representation of the incidences. Note that the figure on the left is a multitope. It is a connection of surtopes that has a name. It is also dyadic. But the portions of set theory that require top closure will fail, because multitopes don't demand top closure.

The polytope rule demanding closure of portions of a plane, tells us that surtopes do not have to obey this rule either. We can apply the surtope constructions to things like cylinders and spheres. Noth here the down-incidence of the faces of a cylinder fall on two circles-arcs. However the circles-arcs are complete latrixes, that is, they is no need to bound them. The chained incidences thus ends at +1 rather than -1. The edges are directly incident on the namon, by reason that i divided these elements to form a cylinder.

Note that the cylinder correctly follows the prism rule, that is (e+2v)(h+e)= c+3h+2e. or 1,2 × 1,1,0 = 1,3,2,0. That is, it's a solid (c) bound by 3 faces (3h), seperated by 2 edges. Since the edges do not intersect, that is where direct incidence stops.

On the other hand, we note a circle is h+e+n (the n is suppressed in prism product), and multiplying it by a point (v+n), we get c+2h+e+v+n, that is 1,1,0,1 × 1,1 = 1,2,1,1,1. It has two hedra, an edge, and a vertex. But the vertex is not incident on the edge, and the edge functions as a complete space to close direct down-incidence there.

Abstract polytopes to my mind imposes far to many limitations on the modelled polytopes to be of any general utility.

[mr_e_man]

From your linked page:

Introduction

[...]

The simplest valid abstract polytope is the 1-polytope or ditelon. Below that, the 0-polytope or monon corresponds to a point, while the −1-polytope or nullon is a null polytope analogous to the empty set ∅ or the number 0. These two sets are too simple to be dyadic and so are not, strictly, abstract polytopes. However it can sometimes be convenient to treat them as such.

Rather, they're too simple to not be dyadic. They are perfectly valid.
For any two elements x < z with ranks differing by 2, there exist exactly 2 elements y such that x < y < z.
In the 0-polytope or (-1)-polytope, there are no ranks differing by 2, so this is a vacuous truth.

Non-simple (anaploid) pieces

Schulte notes that; "In some sense [the dyadic] condition says that P is topologically real. Note that the condition is violated for nonreal complex polytopes." In other words the diamond or dyadic property of an abstract polytope implies a structure in real space, as the property is violated by complex polytopes.

His remark arises because the dyadic property is also a necessary consequence of the proper decomposition of a manifold, be it given a real or Riemannian (complex) metric. It illustrates the profound distinction between a manifold and a configuration, reaching down to the topological properties of the object and quite independent of the metric subsequently applied to its form. It thus lends support to the notion of a real polytope as a piecewise manifold, as distinct from a real configuration.

[...]

However the abstract definition allows such non-simple pieces. I shal call such topologically non-simple objects anaploid from the classical Greek απλους (aplois) meaning simple. For such polytopes, it is often not clear what the topology of the polytope surface might be. One approach is just to abandon any idea of realising the body, but then it is no longer possible to treat the polytope as a manifold and the justification for insisting that it be dyadic is lost.

My justification for dyadicity, as I use configurations and not manifolds, is that it's essential for insiding.


I don't understand the monal property. Can you give some examples of posets that aren't monal? Your description Johnson's description may need to be re-worded:

Abstract polytopes

[...]

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

[steelpillow]

My apologies for the delay in replying. I had to come out of retirement and start a new job shortly after posting, so I have been rather busy, not to mention exhausted, for some time.
Replies to all your posts follow. I would caution that I update arbitrary web pages periodically - which is one reason I publish them on a web site and not somewhere more permanent. For example the term "anaploid" for a non-simple polytope is one I coined recently, as "non-simple" is just so ugly.

On insiding, it seems to me that:
The general idea appears valid. However if we treat a line as extending across infinity then we are in a space, such as a projective one, where the space itself also does so, and a line does not divide the space in two.
Projective space also offers a duality between a segment on a line and an angle in a point. We can use this to construct the polar reciprocal of some polytope (e.g. a conic in the plane or a quadric in 3-space), which we recognise as an example of the dual polytope. But inside vs. outside is a fickle thing, which only gains rigour for convex figures where the centre of reciprocation lies inside the polytope. What we usually do for non-convex and/or eccentric arrangements, in both Euclidean and projective spaces, is to take that region which does not cross infinity, and treat that as the "inside". In this procedure, insiding vectors may arbitrarily reverese direction.
Non-orientable polytopes are always non-convex, so insiding vectors are always unreliable. The best we can then do is to take the directed insiding function as a parity flip. In this, reversing it has no effect.
On the "stadium" oval drawing of a projective edge or line, I do appear to have removed it during some update. I must have decided it was a distraction from the main thread of the essay.

On configurations and complex polytopes:
The fundamental distinction between a real configuration and a real polytope is that a polytope is dyadic - abstractly, the "diamond condition" - with just two vertices on an edge, etc. A configuration has three or more vertices on an edge, with polytopes being regarded as trivially degenerate. Whether the polytope is a topological surface or, like a configuration, an incidence complex of sub-spaces has often been considered irrelevant. For example I have a 19th century school textbook which takes this view. In abstract theory, the difference is just your choice of realization. Complex polytopes cannot be realized as smooth, closed manifolds, and are in fact configurations (Coxeter was forced to acknowledge this a few chapters in to his "Regular complex polytopes"). Another difference is that by definition configurations are regular, and so the issue of confusion ony applies to regular polytopes. Irregular equivalents (real or complex) instead may be treated as more general arrangements of sub-spaces, such that they form incidence complexes.

On the monal property:
This is Johnson's term for the condition which means, in effect, that a given element does not appear twice in the Hasse diagram.

[mr_e_man]

However if we treat a line as extending across infinity then we are in a space, such as a projective one, where the space itself also does so, and a line does not divide the space in two.

Not necessarily.

We could take the double-cover of the projective plane (equivalent to the sphere), so that a line (a great circle) does divide it in two. However, this implies the existence of a second Euclidean plane antipodal to the first one, and connected to it by a line/circle at infinity.

Or we could forget about infinity, and let the inside-out line segment be disconnected.

Or we could consider the inside-out line segment as having density -1 in the finite part, and density 0 in the infinite part. Note that insiding only cares about differences in density, so this is equivalent to having density 0 in the finite part and density +1 in the infinite part.

Projective space

Projective space is defined by adding 1 to the dimension, and representing a point as a line through the origin, i.e. an equivalence class of vectors related by non-zero scalars.

I generally prefer to represent a point as a ray from the origin, i.e. an equivalence class of vectors related by positive scalars. This is equivalent to working in spheric space (but not using its metric).

Euclidean space Eⁿ can be embedded in the vector space Rⁿ⁺¹ as {(x₁, ... , x_n, x_n+1) : x_n+1 = 1}. Transformations of Eⁿ (translations, rotations, reflections) are represented by linear transformations of Rⁿ⁺¹. Each point in Eⁿ corresponds to a ray (x₁, ... , x_n, x_n+1) with x_n+1 > 0.

Spheric space Sⁿ can be embedded in Rⁿ⁺¹ as {(x₁, ... , x_n, x_n+1) : x₁² + ... + x_n² + x_n+1² = 1}. Isometries of Sⁿ are also represented by linear transformations of Rⁿ⁺¹. Each point in Sⁿ corresponds to a ray (x₁, ... , x_n, x_n+1) with no restrictions (other than the vector being non-zero).

Hyperbolic space Hⁿ can be embedded in Rⁿ⁺¹ as {(x₁, ... , x_n, x_n+1) : x₁² + ... + x_n² - x_n+1² = -1, x_n+1 > 0}. Isometries of Hⁿ are also represented by linear transformations of Rⁿ⁺¹. Each point in Hⁿ corresponds to a ray (x₁, ... , x_n, x_n+1) with x_n+1 > √(x₁² + ... + x_n²). Of course the hyperbolic metric is not the usual metric on Rⁿ⁺¹; it has a minus sign.

And in all three cases, geodesic subspaces of the n-dimensional space (e.g. great circles in Sⁿ) correspond to linear subspaces of Rⁿ⁺¹.

Insiding makes sense in this general setting, using linear algebra, but not using the vector dot product. (In other words, the concepts of unit vectors and orthogonality are not necessary.) I suppose the details belong on a different page.

duality

I have some more thoughts on this also, but for now I'll just say that a polytope in Euclidean space should not have its dual in Euclidean space (unless it has a centre of symmetry).

Non-orientable polytopes are always non-convex, so insiding vectors are always unreliable.

It's true that any non-orientable polytope cannot be given an insiding (that obeys the rules I've specified).

But it's also true that any orientable polytope can be given an insiding. It doesn't matter whether it's convex, or what its topology is otherwise.

On the monal property:
This is Johnson's term for the condition which means, in effect, that a given element does not appear twice in the Hasse diagram.

It still doesn't make sense. By construction, the Hasse diagram has exactly one node representing each element. I don't see any way for something to not be monal.

[steelpillow]

Some general replies.

Wendy studies many constructs which are not widely regarded as polytopes. Configurations are perhaps the classic example, as they are not dyadic - they do not have a definable surface. Coxeter wrote on "Regular complex polytopes" but had to admit in Chapter 6 that they were not polytopes at all, but Configurations. He never did explain why he left the self-contradiction in place and did not title his work "Complex configurations." One supposes that he and Shephard had already invested themselves too deeply in trying to extend polytopes theory into the complex domain, as in his first 5 chapters. Wendy's description of a "multitope" is also not what a conventional "polytope" is. All are examples of what is generally known as an incidence complex.

Abstract theory is in fact broader than conventioinal definitions, as I noted at the end of my original post. But of course, if your studies extend beyond the conventional, then it may well be over-restrictive.

mr_e_man comments on projective spaces.
The double-cover of the projective plane is not in fact equivalent to the sphere; any figure in P2 will have an antipodal image, but this is not so in S2.
To respond to the conditional "if we treat a line as extending across infinity" by omitting the condition, as "we could forget about infinity", is not intelligible.
Projective spaces need no metric. For example P2 can be approached via either points or lines, there is an exact and universal duality between the two, and we can set up such a mapping between any point and line we wish. Such bundles and sheaves of lines are of profound importance in topology. We only require a metric if we want to make our space "flat" and "infinite", and then yes we do need an extra coordinate. In this form we can construct say P2 as a vector space, by treating each point as a ray from the origin. The natural coordinate system for this is polar coordinates, with the extra one acting as an inverse multiplier for r. Angles are then not affected. Dually, each line segment may be treated as an angle in some point and we enter the world of conformal geometries.

Here is the diagram of an incidence complex which is not monal, because the element A of the set {A,B,C} appears twice.

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    A
   /   \
  /     \
A      B
  \     /
   \   /
    C

Such monsters are allowed in set theory, hence the need to explicitly bar them here. But whether this diagram meets the definition specifically of a Hasse diagram, as opposed to some more general kind of incidence diagram, I do not know.

[mr_e_man]

The double-cover of the projective plane is not in fact equivalent to the sphere; any figure in P2 will have an antipodal image, but this is not so in S2.

Did you get that backwards? Any figure in S² has an antipodal image.

https://en.wikipedia.org/wiki/Real_proj ... e#Topology

The antipodal map on the n-sphere (the map sending x to -x) generates a Z₂ group action on Sⁿ. As mentioned above, the orbit space for this action is RPⁿ. This action is actually a covering space action giving Sⁿ as a double cover of RPⁿ.

To respond to the conditional "if we treat a line as extending across infinity" by omitting the condition, as "we could forget about infinity", is not intelligible.

Fair enough.

Here is the diagram of an incidence complex which is not monal, because the element A of the set {A,B,C} appears twice.

Code: Select all

    A
   /   \
  /     \
A      B
  \     /
   \   /
    C

Such monsters are allowed in set theory, hence the need to explicitly bar them here. But whether this diagram meets the definition specifically of a Hasse diagram, as opposed to some more general kind of incidence diagram, I do not know.

It seems to me that monality is not a property of a poset; it's a property of a diagram of a poset. (And I could argue that a correct diagram is necessarily monal.)

The set is {A, B, C}, which is the same as {A, A, B, C}; sets have no meaningful notion of duplication. Either an element is in the set, or it's not in the set.
The partial order is defined as follows: A≤A, not A≤B, not A≤C, B≤A, B≤B, not B≤C, C≤A, C≤B, C≤C.
This fully describes the partially ordered set. No diagram is needed. Is the poset monal?

[steelpillow]

The double-cover of the projective plane is not in fact equivalent to the sphere; any figure in P2 will have an antipodal image, but this is not so in S2.

Did you get that backwards? Any figure in S² has an antipodal image.

https://en.wikipedia.org/wiki/Real_proj ... e#Topology

The antipodal map on the n-sphere (the map sending x to -x) generates a Z₂ group action on Sⁿ. As mentioned above, the orbit space for this action is RPⁿ. This action is actually a covering space action giving Sⁿ as a double cover of RPⁿ.

"I read it on Wikipedia so it must support my argument!"

Note that it is the antipodal map which generates the group action and the double-covering. If you do not draw an antipodal map (for example if you are considering the inversive sphere) then you will not create the projective double-covering. Simples!

'Tis perhaps you who is arguing backwards: you should not read a page titled "Real projective space" and assume that it gives a complete account of anything else, real elliptic or inversive spaces say.

[mr_e_man]

"I read it on Wikipedia so it must support my argument!"

I considered finding other sources, or adding some explanation around the quote. But I didn't think it necessary.

The double-cover of the projective plane is not in fact equivalent to the sphere; any figure in P2 will have an antipodal image, but this is not so in S2.

Did you get that backwards? Any figure in S² has an antipodal image.

Oh! Now I see that the wording is ambiguous.
What I meant was: For any figure X on the sphere, the antipode -X makes sense, i.e. it could be drawn on the sphere, even if it isn't drawn.
I didn't realize that you, by "any figure will have an antipodal image", meant something different: If X is drawn then -X must also be drawn, i.e. -X = X. This is, of course, true in P² but not in S², just as you said.

However, I do maintain that the double-cover of P² is S².
A line through the origin in R³ represents a single point in P². A point x ∈ S² ⊂ R³ determines such a line, and the opposite point -x determines the same line. So two points in S² correspond to one point in P². The sphere covers the projective plane twice.

'Tis perhaps you who is arguing backwards: you should not read a page titled "Real projective space" and assume that it gives a complete account of anything else, real elliptic or inversive spaces say.

Elliptic space, in my mind, is projective space plus a metric.

Inversive space is not in my mind.

[steelpillow]

Elliptic space, in my mind, is projective space plus a metric.

Inversive space is not in my mind.

According to Johnson, establishing a certain polarity converts projective space into elliptic space. This is rather more general than applying a metric, and is not an identity.
As a corroborative factoid, Clifford parallels occur in elliptic spaces, but I am no aware that they do so in the projective.

Perhaps inversive spaces are relevant: the inversive sphere is, hardly surprisingly, a sphere with an inversive geometry. It is definitively not a projective geometry - with or without metric.

[wendy]

I tried to 'land' the abstrct polytopes described in the new chapter of RCP, it describes how to convert a group onto a hasse diagram ad leves you at that. You are left womdering where the polytope went. Any group can be turned into a strutured, set, even the Pano group.

There are given axioms for AP, which gives no clear path to things like density. The pretense that hgaving a set of axioms makes you master , in fact it's back to the postulate V when we thought we had euclidean geometry in the bag. a lot of the axiom are provable theorm, such as berin dyadic

[steelpillow]

Exactly as Wendy says. An AP is just a piece of pure maths. You have to wave your arms and "realize" it before you have a sensible polytope. This is why my morphic theory spends so much effort on making realization a bit more, er... realistic.