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.