Talk:Complex polytope

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Examples[edit]

Some examples would be nice mike40033 (talk) 07:13, 19 May 2008 (UTC)

I've added a simple one, with illustration. Sorry I'm not up to the prettier ones. -- Cheers, Steelpillow 20:59, 20 May 2008 (UTC)

Picture[edit]

The picture shows the complex polygon 4(4)2.

The complex edges can be presented by polygons. This is purely a convention to keep track of the vertices: the actual content of an edge of unit length (diameter) is v/2 (where v is the vertices). Here, the edges are presented as the eight squares that are undistorted in the projection.

Edges appear at the vertices rather like fan-blades approach the axle of the fan: they form the "up" strokes of a zigzag. The sixteen vertices are the corners of the squares, we see that there are two edges at each vertex (one diagonal, one straight).

4(4)2 stands then for edges with four vertices, and vertices with two edges at a vertex.

We could look eg at a 3(3)3 as another example.

This has eight vertices and eight tri-teelon (three-vertex) edges. The middle section of an 16choron is an octahedron, the other two vertices are in a perpendicular direction. Two edges appear as top and bottom of the octahedron (as it lies on its face), and the other six edges show as the six lines alternating between the top and bottom.

These edges are completed by connecting to the +w and -w axis (the hidden vertices), in alternate steps. Turning the thing around on the face, we see the rising edges connect to +w, and the falling to -w.

With 5(3)5, we take the 600choron.

One can take a girthing decagon of this figure. This decagon is the complex diameter of the figure. On this ring, there are also 10 pentagons halfway between pairs of vertices. These pentagons represent the edges of the figure. One can find on the surface of the 600ch twelve such decagons, that do not cross. Five go in a spiral around the first one, and five are further out, go in a matching slower spiral. Finally, there is a perpendicular, in the complete orthogonal plane. Each of these 12 decagons form edges in the same manner.

We then look at the vertex figure. The 600ch has an icosahedron as vertex figure. Taken from top to bottom, an axis of the icosahedron represents a diameter of the figure. This is the decagon around the polygon, here, giving three vertices (including the centre of the vertix figure), and two edges (representing pentagons perpendicular to the axis).

The ten triangles between these two pentagons can be formed into five rhombs, by joining these in pairs. The pentateela (5-vertex edges) are represented by the long diagonal of these rhombs, appearing like 5 fan blades coming together at the centre.


The 3(5)3 is again from the 600ch,

The vertex-figure of the 600ch is again the icosahedron, but we now stand it on its face. The top and bottom faces now represent two edges on the diagonal (through the centre-vertex), These don't intersect the vertex. What we do see is a shallow skew hexagon around the mid-height. Alternate edges of this represent the three edges that come into the vertex of 3(5)3.

One can place six such icosahedra, face to face, around the 600ch, this gives eg the faces of {3,5,5/2}. This means that the diagonal of 3{5}3 has six vertices and six edges on it.

--Wendy.krieger (talk) 11:10, 30 October 2008 (UTC)

We now list 3{4}3 and 4{3}4. Both of these derive from the 24ch {3,4,3}.

The 24ch has a cube as vertex-figure. The 4{3}4 is described in the same manner as the above. The cube stands on one of its faces. The top and bottom faces appear as edges in the 4{3}4. These edges fall on the diagonal that passes through the vertices. The fur edges at the vertex appear as diagonals (in the same direction) at the vertex.

The 3{4}3 gives an even construction, separate diagonals for the vertices and for the face centres (cf hexagon or square). The vertex figure is a cube again, three successive vertices appear as the long diagonal. The three incident edges appear as alternating edges of the zigzag not including the long diagonal. The 24ch has vertices on a hexagon, four of these make the diagonal on the 3(4)3.

The edge diagonal is thus derived: The 24ch has octahedra as faces. A stack of six of these wrap to form a band around the 24ch. The faces that these touch at form the six edges on the same axis.

--Wendy.krieger (talk) 08:05, 31 October 2008 (UTC)

Notation for regular examples[edit]

I have added Shephard's original "modified Schläfli" notation for the regular polgyons, except that I have used curly brackets in place of his round ones. This is probably not the modern form, for example I suspect that the first term is p0. Can anyone provide the modern notation, and explain any differences from Shephard's? -- Cheers, Steelpillow (Talk) 21:56, 2 November 2008 (UTC)

Quote from Coxeter + Moser p79[edit]

In Coxeter & Moser's Generators and Relations for Discrete Groups, much of §6.7 is spent discussing the presentations p[2q]r and p[q]p, and how these become equal to other groups.

quote:

These groups are important because of their occurence in the theory of regular complex polytopes (Shephard 1952 p. 92). In the complex affinite plane with a unitary metric, a reflection is a congurent transformation having invariant all points on a line; its period may be greater than 2. A regular complex polygon is a finite connected configuration of points (called vertices) and lines (called edges), invariant under two unitary reflections: one, say R, which cyclically permutates the vertices of one edge, and another, say S, which cyclically permutates the edges through one of these vertices. It follows that the group {R,S} of order g, say, is transitive on the vertices and on the edges. The polygon is said to be of the type

l(g)m

if R is of period l and S of period m, so there are l verticies on each edge, and m edges through each vertex.

...

In view of the discovery (Coxeter 1962b) that every finite group l[q]m is the symmetry group of a pair of recriprocal polygons

l(g)m and m(g)l

it is clearly desirable to replace Shephard's symbol l(g)m by l(q)m. ...

Wendy.krieger (talk) 01:11, 18 January 2009 (UTC)

Yes, I think the bones of this could usefully be added to the article. -- Cheers, Steelpillow (Talk) 11:12, 18 January 2009 (UTC)

Suggest immediate removal of this article[edit]

Unless there is someone who understands the concept of a complex polytope and is able to express that concept in clear English, this article should be removed; it is an embarrassment.Daqu (talk) 05:40, 27 March 2010 (UTC)

The concept is a mildly abstruse mathematical one. Unless you have sufficient knowledge (or determination) to understand the second para of the lead, "The 'imaginary' number i is defined as the square root of −1. A complex number, say (a + ib) where a is real and ib is imaginary, lies in a complex plane, which may be represented as a real Argand diagram. An n-dimensional unitary space comprises n such complex planes, all orthogonal to each other," the concept will be meaningless to you. You might like to follow up the references and see if they make things any clearer, or perhaps you could follow the links to articles on any terms you are unfamiliar with. No doubt this article is not perfect, but given your apparently uninformed and uncivil remark one has to wonder where the embarrassment really lies. -- Cheers, Steelpillow (Talk) 20:34, 27 March 2010 (UTC)
I'm quite familiar with finite-dimensional complex vector spaces, thank you, and have been for over 40 years. Unfortunately, repeating the most elementary definitions in complex analysis -- which belong in their own articles, not here -- tells the reader nothing about how the subject of this article, complex polytopes, are defined.
Yes, indeed, I am definitely following up the references, because this article will not be the place to find out even the most basic definition of a complex polytope. The only term I'm not familiar with is "complex polytope", and nothing in this article clarifies its definition.
Which is why this article should be immediately deleted.Daqu (talk) 01:20, 29 March 2010 (UTC)
Normal polytopes are dyadic, that is, for an element of N+1 and N-1 dimensions, there is incident on it exactly two elements of N dimensions: for example, at a vertex of a polygon, there are exactly two edges. This dyadic nature has been used to define various abstract polytopes.
What considers when one allows more than two elements here: for example, three edges incident on a vertex. Shephard considered the problem from an abstract point. Coxeter showed that these correspond to a symmetry p[2q]r, or p[q]p, defined as A^p = B^r = 1, ABA.. = BAB.. (q terms), and that such groups could indeed be implemented in unitary space (ie euclidean geometry with complex numbers).
The relevance of these polygons is that unitary n-space is a proper subset of real 2n-space, and that for example, there exists a condition of even-dimension-space, for which two points define a unique 2-space, 3 define a 4-space, and so forth. Also, things like the Poincare dodecahedron etc, are base groups of order 8. 24 and 120, that underly the symmetries of polygons like 3[5]3.
--Wendy.krieger (talk) 07:46, 29 March 2010 (UTC)

Thank you Daqu for expanding on your remarks. I took them to indicate that you did not understand the concept, so my apologies for misreading you. I agree that the definition is not as clear as it could be (and I do not think that subsequent edits are helping). Some relevant material appears in the section on Characteristics. However is seeking a more succinct definition I found that as defined by the original authors, a regular "complex polytope" is more akin to a configuration that to a polytope. The problem then arises, what then is a complex polytope if it is not regular (and therefore is not a configuration)? Rather than see the article deleted, I would prefer to see the definition improved upon.

Meanwhile I am unhappy about the use of "dyadic" in the recent edits, as it is not a well referenced mathematical term (although I believe that Johnson introduces it in his forthcoming book on Uniform Polytopes).

-- Cheers, Steelpillow (Talk) 17:09, 29 March 2010 (UTC)

By the way, is Johnson's book any more forthcoming now than it has been for several years? ;) —Tamfang (talk) 22:28, 29 March 2010 (UTC)
Wish I knew. -- Cheers, Steelpillow (Talk) 16:35, 30 March 2010 (UTC)
Just looked at the article today (April 13, 2010) and it still lacks anything remotely resembling a definition of its subject. Which means that whoever has been contributing to the article either hasn't the vaguest idea of what its subject is, or is entirely unable to express themselves clearly, or both. Which makes me overwhelmingly convinced the article should be removed immediately.
This is an encyclopedia, and not a place for people to demonstrate their lack of knowledge about a subject. If someone knows at minimum the definition of a subject, and preferably more than that, then it is appropriate for them to contribute to a Wikipedia article. Since that has obviously not been the case with either "complex polytopes" or "regular complex polytopes", the entire article should be deleted until someone comes along who actually knows something about these mathematical objects.
If my time permits I can replace the text of the article in a few weeks when I've had a chance to write a reasonable few paragraphs, since Shephard's paper is beginning to make sense to me.Daqu (talk) 01:49, 14 April 2010 (UTC)
Just be warned that Coxeter takes the subject further, and adapts Shephard's approach rather than simply building on it (his differing notation, as explained in this article, gives some clue to this aspect). So relying solely on Shephard is ill-advised. I do not have access to Coxeter at the moment, this is one reason for my still somewhat fuzzy rewrite. The other difficulty is that neither Shephard nor Coxeter properly defined the "Complex polytopes" of which they sought the regular variety - what Grünbaum famously called "the original sin in the theory of polyhedra". The situation is compounded yet further by the fact that complex "polytopes" are not really polytopes at all - the regular ones are projective configurations. If you do come up with a sufficiently precise and useful definition, you may well find yourself guilty of original research.
If you lean to the view that the article is best re-purposed as "Regular complex polytope" then I would hope that you can first check whether ideas such as "balanced complex polytopes" and "complex polytope norms" apply to non-regular forms - the maths of all this is way beyond my skills. Of course, adding some remarks on these to the article would be wonderful.
Meanwhile, the topic is the subject of many papers and the odd book, so it is worthy of inclusion in an encyclopedia such as this. The stark choice you suggest, between getting the definition perfect or deleting the article, misses the true state of things, which does need to be explained. If you can bear metaphor, the lack of a sensible definition is a cross we must bear, not a sword we must use to kill. Let us work together to make the article better. -- Cheers, Steelpillow (Talk) 10:03, 14 April 2010 (UTC)
[update] I just tinkered with the "definition" (last para of the lead), as a generalised arrangement. Not sure if this constitutes wp:or, so I just said "may be understood as". -- Cheers, Steelpillow (Talk) 10:19, 14 April 2010 (UTC)
I don't believe that "complex polytope" is a standard term, as evidenced by its almost complete absence in the MathSciNet database. (By the way, there isn't even one instance of polychoron or polychora in that database, strongly suggesting that despite the enthusiasm of the contributors to the Wikipedia article of that name, that term has a very long way to go before it even begins to be accepted as standard, and until then should be omitted from Wikipedia articles. It was coined by George Olshevsky, who is an expert on 4-dimensional polytopes, but that is no justification for using Wikipedia to propagate its coinage.)
Well, I don't know what you mean by "standard term" in the context of an encyclopedia. Enough papers have been published on the topic for it to be worthy of inclusion, however much we may disapprove of the term (although for different mathematical reasons). I agree with you about "polychoron", but lost that battle to energetic enthusiasts who felt that Olshevsky's web site was sufficiently authoritative - I didn't feel up to starting a large-scale war. -- Cheers, Steelpillow (Talk) 01:01, 16 April 2010 (UTC)
Since I don't believe that Coxeter changes Shephard's definition of regular complex polytope, I will mention Shephard's definition to the extent I understand it. The main thing I haven't grasped yet is why does Shephard require that Npq, defined as the number of q-dimensional complex affine subspaces (in the configuration that is the polytope) incident upon any p-dimensional one, must satisfy the inequality: Npq.136.152.209.65 (talk) 23:57, 14 April 2010 (UTC)
My best guess is that it is analogous to the requirement for a real polytope to have at least 3 edges to a face, 4 faces to a cell, 3 cells around an edge, n (n-1)-elements around a vertex and so on, in other words to avoid degenerate forms. -- Cheers, Steelpillow (Talk) 01:01, 16 April 2010 (UTC)

Lead needs rewriting again[edit]

I have some problems with the revised lead to this article:

  1. "dyadic" is not referenced, and to my knowledge does not (yet) appear in any reputable published source.
  2. "Shephard enumerated all of the regular complex polytopes in an abstract sense. Coxeter showed these belonged to a kind of euclidean space with complex numbers". This is not so. Shephard developed his theory in the context of unitary (complex Hilbert) spaces - the whole idea of an "edge" having more that two vertices only occurred to him through the properties of the complex line (aka Argand plane). Specifically, the non-dyadic property arises from the complex dimensions, and not vice versa. This is transparently clear from the first two paragraphs of his paper. Further, Coxeter did not write with such unusual phrasing as, "a kind of Euclidean space with complex numbers" - this is just complex space, and he wrote using normal mathematical language.

If nobody objects, I shall attempt a second a rewrite. -- Cheers, Steelpillow (Talk) 10:47, 3 April 2010 (UTC)

Done -- Cheers, Steelpillow (Talk) 13:46, 5 April 2010 (UTC)
The recent edits were mine, based on what i understood from Coxeter. Dyadic is indeed a term from NW Johnson, but in the sense here, is taken to mean that if M+1 is incidident on M-1, there are exactly two M incident on both (eg an edge of a polyhedron, there are exactly two faces).
Describing unitary space as 'a kind of Euclidean space with complex numbers', is more precise than 'complex space' since the latter description could equally apply to something where complex assumes its usual meaning, and not the special meaning of the sums of square roots of positive and negative numbers. --Wendy.krieger (talk) 12:03, 6 April 2010 (UTC)
Since "dyadic" is not found in any reputable source, it is against policy to use it here. Since "complex space" is commonly used among mathematicians, it is appropriate here. The job of am encyclopedia is to explain what these things mean, not to avoid them. The article on complex space is the appropriate place to explain that term - although at the moment, sadly, it does not. -- Cheers, Steelpillow (Talk) 18:30, 6 April 2010 (UTC)
Wendy, can you please explain how the "unitary space" you mention is different from a finite-dimensional vector space over the complex numbers, supplied with the usual Hermitian metric? Thanks.Daqu (talk) 01:01, 14 April 2010 (UTC)
I used in conversations on some list the expression CE1 etc (for complex-euclidean = euclidean with complex coordinates). Norman Johnson called this space the 'unitary space'. It's a silly term, but apparently the one used. The distinction of calling it CE = complex-euclidean is to differentiate it with other euclidean spaces that complex numbers appear in (eg Minkowski, which uses real coordinates, but an axis with a complex value (ie x,y,z,ict). The distance metrics in these spaces are different: in the CE, the distance is real and positive (ie R* conj(R)), where in the Minkowski (space-time), the distance can be positive or negative (ie R*R).
I don't understand the Hermitian metric, since i implement the space as CE (ie Euclidean + complex numbers). --Wendy.krieger (talk) 08:06, 14 April 2010 (UTC)
I'm no expert on it, though I can calculate with it if I need to. The group of rotations of R2n = Cn that preserve the Hermitian inner product on Cn is called the unitary group and denoted U(n). It turns out that Shephard's definition of a complex regular polytope is an arrangement of complex affine subspaces of Cn that is carried to itself by unitary transformations (plus an important inductive condition and a couple of technical ones). I have a long way to go before truly understanding these things, but they're just beginning to make a bit of sense.Daqu (talk) 03:26, 15 April 2010 (UTC)

Complex Euclidean[edit]

Regular complex polytopes are regular, because their flags are transitive on their symmetry. Because these exist in complex space, which has a real reflex in twice the dimension, understanding complex polytopes is a lead to understanding real symmetries in higher dimensions.

The complex polytopes are complex in the usual sense, that is, at a boundary between faces, more than faces might meet. In normal polytopes, this is not permitted, because faces form fragments of the surface, and the surface does not divide into several sheets. In complex euclidean space, that is, euclidean space with complex numbers, this is allowed.

A line in euclidean 2-space is given by Y = aX+b. All of these are allowed to be complex numbers. In Euclidean space, two planes define a point, if a and a' are different, that is, Y=aX+b, and Y=cX+d, will define a unique point, X,Y which solves both, where a <> c.

One might use a line as a mirror in complex space. Every point, for example, lies on one of a parallel set of lines, Y=aX+c. There is a line perpendicular to all of these, Y=-X/a or -aY=X.

A mirror corresponds to a 'rotation' in the plane Y=-X/a, of the order of w, where wn=1. The mirror-image is of the order n, over the plane X=-aY, is effected by moving P to P', where P falls at aX+b, then P' falls at aX. A mirror can thus create 2, 3, 4, ... images.

A kaleidoscope consists of several mirrors, with a region (tether) between the mirrors, that is reflected throughout the plane. In complex space, a tether can join any number of mirrors, however, three mirrors is enough to create all of the groups. A flag is a kind of tether, where the ends represent the vertex, edge-centre, etc of a polygon, polyhedron, etc. This can be replicated over the space.

A regular polygon has a mirror through its vertex, and another through the centre of its face. Since only two mirrors are used in a polygon, only those that need two or less mirrors are used.

A polygon like 3{4}5, has a flag or edge-vertex, with an order three mirror at one end, and an order-five mirror at the other end. The '4' represents an alternating walk (or angle between mirrors) that is equal in both directions. In group notation,

   AAA=1       represents the order-three mirror
   BBBBB=1     represents the order-five mirror
   ABAB=BABA   represents the alternating walk.

Any valid string of A's and B's represents a valid walk, but these can be reduced only by the equalities above. So, a walk of the form ABABAA = (BABA)AA = BABAAA = BAB(AAA) = BAB. There are 1800 possible walks of this group.

All of the mirrors, except the vertex mirror, is perpendicular to the flag. This means that the flag is in the same flat space as its image in all mirrors except for the vertex-mirror. In complex space, this means that one can have five separate images to the plane Y=aX+b, which fall as a single line.

THE REAL REPRESENTATION

The real representation of complex space of N dimensions, is a real space of 2N dimensions. The complex CE2 gives rise to a real E4. A line in Complex space comes across as the Argand diagram. One can think of such a diagram, as not just a 2d space, but one with a definite arrow around it, connecting 1, i, -1, -i in that order.

The propositions of straight lines, tells us that in 4d, when all 2d spaces are 'clifford-parallel' to each other, then two points define a 2-space, and 2 2-spaces must cross in a point (and no other way, like a line). Moreover, there is a definite angle between them.

The complex rotation around a point, which transforms X,Y to wX, wY forms a clifford parallel. The trace of X,Y as w progresses around the unit circle tells us that in every even dimension, it is possible to comb the hairy ball so there are no calm points.

The complex polygons gives rise to figures with real symmetries. Underpinning each of the complex polygons, is a group of poincare polyhedra (like the poincare dodecahedron), which represent a repetition group under clifford rotations. These figures have 8, 24, 48 and 120 faces, representing the real polyhedron groups with symmetries of 8, 24, 48 and 120.

Wendy.krieger (talk) 09:53, 17 November 2011 (UTC)

Some questions about complex polytopes. Comments really, in the form of complaints.[edit]

1. What is the difference between the "complex line" and the complex plane? If there is no difference, why are we suddenly using the term "complex line", which I've never heard before in my life?

2. Claiming that boundaries don't or can't exist in complex spaces (or can't even be defined) strikes me as patently idiotic. Complex spaces are just like regular spaces except with half the dimensions labeled "imaginary", no?

3. "Since bounding does not occur, we cannot think of a complex edge as a line segment, but as the whole line." Ah yes, one of those "lines" defined by more than two non-collinear points. It's almost as though it weren't a line at all, but rather a plane.

4. I'd like to note that I've also never heard the term argand diagram before reading this page. I'm glad I looked it up, though, it seems quite useful, it's a way of representing the complex line as a sort of "plane". It seems strangely familiar!

5. "In the Argand diagram, of the edge of a regular complex polytope, the vertex points lie at the vertices of a regular polygon centered on the origin." But wait, so now we're examining the complex plane [redirected from Argand diagram] of the (complex) EDGE of a regular "complex polytope", and we discover the vertex points lie at the vertices of a regular polygon. How is this an edge? It's a face. Why don't you just call it a face? What is wrong with you? Also, why is it centered on the origin? Polytopes don't usually have edges (or faces) which include the origin, did that happen because we decided to look at an edge in isolation and we gave it its own coordinate system, and if so, how is the location of the origin of that independent coordinate system relevant in the slightest?

6. "Given the general point x + iy in the complex plane, for an edge having p vertices, these lie at the p roots of the equation: x^{p}-1=0" Ok, so now YOU'RE calling it the complex plane. Or is this actually R4 or something? I have no idea, but luckily it doesn't seem to matter because after picking a "general point" in it we wander off in mid-sentence to talk about an apparently unrelated "edge" (face) having p vertices. Is the "general point" one of them? It would seem not, as their locations have nothing to do with y. Still, the math is simple enough. For instance, let's say the general point is 7 + i9000. From this, we can deduce that 7^{4}-1=0. This is false, so oompa loompas come out and put a red bucket with a frowny face on our heads.

7. "Two real projections of the same regular complex octagon with edges a,b,c,d,e,f,g,h are illustrated. It has 16 vertices, which for clarity have not been individually marked." Yes, because when you're using the word "edge" to mean "face" and "line" to mean "plane" and "polytope" to mean whatever these are, nothing clarifies things like an unmarked diagram.

8. "The sides of the square are not parts of the polygon - this is important to understand - but are drawn in purely to help visually relate the four vertices." The "edges" are NOT faces, however much they may look like them! IGNORE THEIR RESEMBLANCE TO FACES

9. "The edges are laid out symmetrically (coincidentally the diagram looks the same as a common projection of the hypercube, but in the case of the complex octagon the diamond shapes which can be traced are not parts of the structure)." You may think this "complex polygon" is really a "normal hypercube", but NO! EVEN THOUGH IT LOOKS EXACTLY LIKE ONE, IT'S TOTALLY DIFFERENT!! WHAT'S THE DIFFERENCE, YOU ASK? WELL, 2/3 OF THE FACES ARE MISSING. OOPS, I MEAN "EDGES". WE WOULD'VE COLOR-CODED THE "EDGES" THAT ARE PART OF THE "OCTAGON" SO AS TO DISTINGUISH THEM FROM THE FAKE ONES, BUT THAT WOULD'VE DESTROYED THE CLARITY OF THE DIAGRAM.

10. If it really is just a coincidence that a complex octagon looks exactly like a hypercube with most of the faces missing, WHY THE HELL DID YOU PICK IT AS THE ONLY ILLUSTRATED EXAMPLE?!

Sisterly harmer (talk) 00:45, 10 February 2014 (UTC)

Hi, I have made some corrections and clarifications surrounding the equation which troubles you. The key to understanding all this is that in complex spaces the terms "line" and "plane" have double the usual number of dimensions. For example a complex line has two dimensions, one real and one imaginary. We can represent this line in two real dimensions, the Argand diagram, and this often leads mathematicians to refer to it as the "complex plane." Thus the term "complex plane" is ambiguous, according to context referring either to the complex 2-space represented in the Argand diagram or to the 4-space of two real and two imaginary dimensions. Likewise the complex 2-space represented in the Argand diagram may according to context be referred to as a "complex line" or a "complex plane." If you think this is completely mental, I could not agree more. The lazy and complacent mathematicians who let this situation develop should be ashamed of themselves. But Wikipedia can do no more than document their insanity.
The "complex plane" (as in Argand diagram) does crop up a fair bit in engineering, especially analogue electronics and radio - classically in the Smith chart used to characterise broadband circuit behaviour.
The issue over boundaries is a deep one, but to illustrate it consider a complex edge having four vertices. The argument goes something like this:
  1. Draw these as the vertices of a square in the Argand diagram.
  2. Now consider the question, is any of these points "between" any two others? The answer is no, because the whole thing is symmetrical.
  3. Add a fifth vertex, the same distance as the others from the centre of the square and equidistant from two of them. At first sight it appears to lie "between" those two, though a little offset.
  4. Draw in the side of the square it lies closes to. It looks "outside" that line, doesn't it? But that is an illusion.
  5. Now, add three similar vertices between the other pairs of vertices, to complete a regular octagon. By symmetry, none of these vertices is "between" any of the others. That "between-ness" of the fifth vertex was a delusion.
  6. Without any idea of "between" it turns out that we cannot pin down the difference between "inside" and "outside."
  7. Consider the status of that side of the square you drew. It is not a vertex of the complex figure. But nor is it an edge, since it lies within a complex edge. It is just a dense point set in the complex plane.
  8. You might be tempted to say "No, I'll run with my square sides and stuff your silly sophistry." Do so and you discover that mathematically you have created a real square as a face of a real 4-polytope - what you have drawn is no longer a complex polytope, just a real polytope injected into a complex space.
One consequence of all this is that complex polytopes are not really polytopes at all! Rather, they are what mathematicians call configurations - certain structurally organised arrangements of intersecting points, lines and planes. Quite why Shephard and Coxeter though that "polytope" was a good name for these infinite arrangements is utterly beyond me. I suspect they had just been studying the math of higher polytopes and, in extending it to the complex realm, were too lazy and complacent to sanity-check their language.
I invite you to colour-code the eight overlapping squares in the diagram and see if it makes things any clearer. Bear in mind that the edges are infinite in extent and the coloured regions are not bounded in the actual polytope (which is really an infinitely-extending configuration and not a bounded polytope at all), so that would need to be explained.
I also invite you to present a more compelling and comprehensible example. That may be possible, but I no longer have Coxeter's book to hand. The present example at least has the benefit that our eyes can understand the diagonals as orthogonal dimensions seen in perspective.
Hope all this helps. — Cheers, Steelpillow (Talk) 11:26, 10 February 2014 (UTC)

Hi. I'm not really into complex polytopes – particularly since they are not really polytopes at all, having no boundaries – but I'll do my best to answer your questions. Consider this a reply from someone who originally had the same misconceptions as you do now, and is trying to show how they eventually realized what a complex space and a complex polytope really are.

Usually you will see the complex line called a complex plane, as it has similarities to R2. But you must understand that R2 requires two real coordinates to describe a location, but C1 requires only one complex coordinate to describe a location. While it may be a useful way of thinking about them, complex numbers are not simply ordered pairs of real numbers, but independent numbers in themselves. Unfortunately, we are historically stuck with the term "complex plane", which as you have noted results in a complete giving up of logic when we consider C2, which looks like R4 but only has two dimensions.

A complex space with n dimensions is not quite the same as a real space with 2n dimensions, though I can understand why you might think of it that way after much exposure to the (undoubtedly useful) Argand diagrams! But treating the real and imaginary parts of each complex coordinate defeats the whole purpose of using complex dimensions. In particular, what you think is a 2D plane is really a 1D complex line, but it shares the real plane's property of there being no good way to define a sense of "between" given two points. Is (1, 4) between (0, 3) and (2, 5)? Is it between (0, 3) and (2, 6)? While the line y = 3, for example, may be a boundary in R2, what is the equivalent in C1? The set of complex numbers with imaginary part 3i? But how can this be a boundary? After all, the set of real numbers with fractional part 0.3 isn't a boundary either, and just like you can describe any complex number z as x + iy where x and y are real numbers, you can describe any real number n as a + b where a is an integer and b is a real number between 0 and 1. (This isn't a complete analogy, because the real numbers are an ordered set and the complex numbers are not: but I hope it gets the point across.) And without boundaries like the facets of a real polytope, how then do you define what is inside or outside the complex polytope? Clearly there is no way to, and hence you must abandon all concepts of boundaries for complex polytopes – which is why they would honestly be better called "complex configurations".

The n-2n dimension distinction is the reason why you think the edges of a complex polytope are faces, because you're thinking of each complex dimension as two real dimensions. Sure, the set of complex numbers {1, −1, i, −i} defines a square in the Argand diagram, but a square is a 2D object and each of its points requires two coordinates to plot. These four complex numbers are simply points in the complex line. If this disturbs you, look at a colour wheel graph of a complex function: now each complex number is represented by a point and its colour. The points {1, −1, i, −i} suddenly don't look as though they were destined to be the vertices of a square anymore, do they? (And yes, we gave each complex edge its own coordinate system in which it's parallel to an axis, in which case you can simply define the axes so that the complex vertices are aligned to the roots of unity.) And yes, it was really bad to use x + iy as a general point and then use x in the next equation, and abruptly change the topic in the middle of the sentence. Luckily, it is fixed now, and the Oompa-Loompas are happy again. Double sharp (talk) 14:58, 17 April 2014 (UTC)

As for why the edges are not colour-coded, I can do no better than to quote Steelpillow's comment:

"I invite you to colour-code the eight overlapping squares in the diagram and see if it makes things any clearer. Bear in mind that the edges are infinite in extent and the coloured regions are not bounded in the actual polytope (which is really an infinitely-extending configuration and not a bounded polytope at all), so that would need to be explained."

The edges overlap in the diagram and so colour-coding would be difficult. Additionally they are infinite and therefore any colour-coding (perhaps of the real-square regions) would require another explanation. Now do you see the difficulty in making the diagram any clearer? It would be great to be able to put people into a universe with true complex dimensions, and I would love to see such a thing for myself, but unfortunately I doubt life as we know it would be able to survive in such a universe, which is quite a disappointment, if you ask me.

(Although yes, I do think some more explanation on where the complex vertices are in the diagram wouldn't hurt.) Double sharp (talk) 15:13, 17 April 2014 (UTC)

Just to emphasise one point in reply to the OP's question "2. ... Complex spaces are just like regular spaces except with half the dimensions labeled "imaginary", no?" No indeed, absolutely not. This is perhaps easiest to see when multiplying using polar coordinates. In the real plane these take the form (r, Θ). Multiplying two real points as such makes no proper sense because it involves four separate real numbers. We could treat each coordinate as a vector from the origin, in which case we can choose between the dot or cross-product, but that's not really the same thing. On the complex line (aka plane) the polar form of a number is r.e. We can multiply two such complex numbers easily by multiplying the two values of r and adding the two values of Θ modulo 2π. So mathematically, the real plane and the complex line behave very differently. — Cheers, Steelpillow (Talk) 15:36, 17 April 2014 (UTC)
I completely understand that the complex plane and the real plane have different COORDINATE SYSTEMS, but I still can't for the life of me see the relevance of that fact. If I form my own country and declare everything shall be measured using polar coordinates and quaternions in bijective negahexavigesimal, it doesn't make my house turn inside out, because it doesn't have anything to do with the actual topology or geometry of the space. Sisterly harmer (talk) 00:38, 13 July 2014 (UTC)
First, I assume you mean to relate the real plane to the complex line, as the complex plane has four dimensions (two real and two imaginary). Your remarks about coordinate systems are true of physical 3-space but not in general of mathematical spaces. Some coordinate systems, such as polar and cartesian, are indeed mathematically equivalent: multiply the coordinates of two given points together in either system and you will get the same answer. But the real plane and the complex line are not mathematically equivalent in this way: multiply the coordinates of two given points together and the two systems, real vs. complex, yield very different answers. So the fact that two coordinate systems may have the same number of dimensions does not make them equivalent. Analogous mathematical constructs in each space may turn out to have very different properties. We are then faced with the problem of what form of words to use - what labels to attach to these anomalous mathematical ideas. We might have decided to ascribe the term "complex plane" to the space of one real and one imaginary dimension and to say that a "complex line" was a line drawn on such a plane. But we then find that the complex algebra has other properties we need new names for, and it was found far more productive to talk about and generalise the algebra if we ascribe the term "complex line" to the space of one real and one imaginary dimension. So an issue like this is a mixture of often purely abstract mathematics combined with attempts to label these abstractions in meaningful ways. — Cheers, Steelpillow (Talk) 11:41, 13 July 2014 (UTC)
On the other hand, if you do indeed mean to compare the real plane with the complex plane (as opposed to the complex line), the complex plane has four dimensions. Similarly quaternionic 3-space has twelve dimensions and your solid three-dimensional house would indeed be as insubstantial as a reflection in still waters - you could wander through and around its ghostly walls at will. — Cheers, Steelpillow (Talk) 12:01, 13 July 2014 (UTC)
Oh, and you can indeed have "polytopes" in Hn, defined analogously to those in Cn: the loss of commutativity is not a barrier here. Here's a source. For example, edges in H1 can have the symmetries of polyhedra in R3. You can't do this in On, presumably because the nonassociativity (in general) means that you cannot have reflection groups. There seems to be enough literature about this that an article titled quaternionic polytope may be warranted. Double sharp (talk) 09:33, 5 April 2016 (UTC)
It's also worth adding that there is a sharp distinction between the global topology of a manifold or space and its geometry. Topologically, a given manifold has certain global characteristics to do with things like holes and twists. A blank sheet of paper is a bounded 2-manifold. On it, you can impose any number of coordinate systems, which define various metrics to yield an overall geometry which can be analysed algebraically. Yes, a real plane and a complex line can both have the same topology - be similar sheets of paper. But the metrics applied - roughly analogous to the markings turning it into graph paper - are very different, yielding distinct geometries. — Cheers, Steelpillow (Talk) 10:18, 5 April 2016 (UTC)

Notation[edit]

The description of Coxeter's notation appears to be garbled, as well as badly written. Is anybody in a position to correct and clarify it? — Cheers, Steelpillow (Talk) 11:37, 10 February 2014 (UTC)

An attempt to explain what is going on with complex polytopes, perhaps for inclusion[edit]

In a real regular polygon, each vertex is shared by two edges, and each edge connects two vertices. The two vertices on a real edge can, after some appropriate scaling and rotations, be identified with the points ±1 on the real line, the real square roots of unity. Hence we generalise a complex edge with n vertices to be part of a complex line passing through the nth roots of unity.

Here we must clear up a potential source of misconceptions. In an Argand diagram, these vertices of the complex edge in question look like the vertices of a regular real n-gon. However, one crucial point about a real polygon is that its sides are bounded. When you look at the real line segment with endpoints {+1, −1} in R1, we can see immediately if a point is inside or outside the line, because the real numbers are ordered. Thus we can have the inequalities that can be used as the definition of a real convex polytope. But if we consider the complex edge with four endpoints {+1, +i, −1, −i} in C1, then we face a difficulty. It makes no sense to say that i > 0 or i < 0 since the complex numbers are not ordered. [To show that there cannot be an ordering, consider: it is a property of an ordered field that a < b and c < d, taken together imply that ac < bd. Now if i > 0, then we would have i2 = −1 > 0, which is absurd, so we must have i < 0 (since i is obviously nonzero). Yet this runs into the very same difficulty: if i < 0, then −i > 0, and we run into the same problem since that is also a square root of −1. Hence no ordering is possible.] The inequalities cannot be written, and so we have no way of deciding what is between the two endpoints and what isn't. Hence, we have to take the whole line instead of some part of it. (Yes, I admit that I lied slightly in the previous paragraph to simplify things. Your forgiveness and understanding is requested.)

We can now go up a dimension. We can then write p1{q}p2 for a polygon in C2, such that each of its edges have p1 vertices, and each vertex is shared by p2 edges. (For real polygons injected into C2, p1 = p2 = 2.) Further, q is the length of the minimal cycle of vertices needed for every consecutive pair (but not triple) of vertices lie on an edge. (For real polygons injected into C2, q is simply the number of sides.) The dual of p1{q}p2 is clearly p2{q}p1. A striking example of a complex polygon is 3{3}3, which has the symmetry of a regular real tetrahedron, and whose eight vertices can be divided into two quadrilaterals, inscribed in each other (impossible in Rn).

The definition of a regular polytope (having a flag-transitive automorphism group) generalises accordingly from Rn to Cn and Hn. (Any vector space over a field or skew field will work, if I understand this correctly.) In all three cases, this automorphism group may be generated by reflections, and thus the regular real, complex, and quaternionic polytopes have been classified. Double sharp (talk) 21:54, 13 April 2016 (UTC)

Some comments, good and bad:
The distinction between a complex polytope in general and a regular complex polytope in particular is important and one must not mix discussions of the two together. The lack of ordering on the complex line is better demonstrated in that article rather than this one. These come together in your second paragraph, where you discuss the regular arrangement of points on a circle in the Argand plane in an attempt to prove a general property of complex polytopes. A more accurate intuition of complex "polytopes" is obtained if one thinks of them as complex configurations and this actually becomes necessary with the more developed theory.
The distinction between polytopes and configurations has historically been a hazy one, with many mathematicians regarding it as trivial - for example old text books will as happily describe a plane configuration of three infinite lines as a "triangle" example of a "polygon", as they will the figure having three finite sides. Much of the mathematics of symmetry and of projective geometry underlies both. But modern polytope theory has gone well beyond that common ground. Firstly, when one considers polytopes as bounded regions, of which the most important are the convex polytopes, then one is forced to abandon configurations. Secondly, the modern theory of abstract polytopes does not generally admit configurations, save for a small number as regular abstract polytopes. Shephard and Coxeter chose their term during the ambiguous era before convexity and abstract theories gained primacy and changed our preconceptions. Really, it is time we renamed their object of study "complex configurations", but as nobody has done that we are stuck with the present anachronistic mess.
I would prefer the understanding as configurations to be included in the lead because it helps to clear the head for what follows and save many of the agonies evident in your draft, but somebody edited that out a while ago, sigh. Other than that, I don't think the introductory material needs a significant rewrite.
Your later paragraphs are useful and represent the beginnings of a needed expansion of the article to clarify the application of the symbols and to discuss and perhaps enumerate the simpler examples. — Cheers, Steelpillow (Talk) 09:48, 14 April 2016 (UTC)
I apologise for my sloppiness on regularity! It is really easy to fall into this sort of trap when talking about polytopes. I now seem to have first-hand experience at creating the kind of sloppiness that goes back to the original sin here. As for the ordering on the complex line, since the argument is pretty short (squares must be positive in an ordered field) perhaps it would work well in a note, so as to keep readers from wandering off on a wiki walk without ever actually understanding what is going on with complex "polytopes".
To be more precise I think we would have to start with a general, possibly irregular, complex edge, whose vertices are just anywhere, and then note that the definition of regularity as applied here would then lead to what I said about the Argand diagrams (similarly to how a real line, given that treatment, ends up with endpoints {±1}). We can use that regular complex 4-edge with vertices {±1, ±i} as an example of the lack of ordering, as it has nice numbers.
You raise a good point that we should say, right from the start, that these things are more like configurations than polytopes, and the naming is a historical accident (similarly for the quaternionic case). Doing this would I suppose require a short definition as well. As for examples: I mentioned the Möbius–Kantor configuration (83), which is a regular complex polygon 3{3}3. We then need to explain how this notation, used by Shephard and Coxeter, extends to higher-dimensional cases. At the very least, we should enumerate the regular complex polytopes in low dimensions, perhaps up to 3 or 4, since we're probably not going to create a separate page for them. The same goes for the quaternionic polytope article. Double sharp (talk) 13:09, 14 April 2016 (UTC)
The better notation and (as far as I know) the one normally used is Coxeter's - Shephard's was slightly different. In fact Coxeter's book Regular complex polytopes is the essential starting point for any such discussion. Sadly I no longer have ready access to a copy. — Cheers, Steelpillow (Talk) 14:24, 14 April 2016 (UTC)
By the way, I do not know whether complex configurations exist which are not regular complex polytopes. Offhand I cannot see any distinction. It would be useful to clarify that. — Cheers, Steelpillow (Talk) 14:32, 14 April 2016 (UTC)
It seems like some version of the discussion above of (not necessarily regular) complex polyhedra became the lead. I am very unhappy about this, for the following reasons:
1) it has little or nothing to do with the body of the article, which is entirely about *regular* polyhedra
2) it is unsupported by sources
3) I am skeptical that it could be supported by sources; I think there is no theory of (not necessarily regular) complex polyhedra, and that this is OR
I haven't had time to see what the previous state of the lead was like, but I would like to begin the process of putting the lead in a good shape, so that it summarizes the body and is supported by the references in the body. I am not well-qualified to do this myself, and I wanted some consensus before I started mucking around; thoughts? --JBL (talk) 13:07, 19 June 2016 (UTC)
The article is structured badly because the lead should merely summarise the main content and not make statements unsupported by the main content and its accompanying citations. The more interesting issue is that while complex polytopes are defined in a general way, this has always been a precursor to identifying and discussing some regular examples. Shephard makes passing reference to the possibility of irregular ones in his original paper, and Coxeter's book title "Regular complex polytopes" begs the question as to the existence of the irregular variety, so we probably do need to preserve the distinction. Regular complex polytopes are unitary configurations, but I don't know of anybody who has studied those of the less regular unitary incidence complexes which may justifiably be described as irregular complex polytopes. The problem is, how do we present this half-theory? — Cheers, Steelpillow (Talk) 19:54, 19 June 2016 (UTC)
As far as I can tell, there are no sources about non-regular complex polytopes. I think, from the point of view of writing an encyclopedia article, this makes life very simple: we should just write about the regular ones. (And maybe this article should be titled regular complex polytope, with the redirect going the opposite way as at present.) As a first step, I am in favor of scrapping essentially all of the present lead. One thing that would be helpful is if someone with a copy of Coxeter's book could add a section that includes an actual definition of regular complex polytope; at present, there is none. --JBL (talk) 21:58, 24 June 2016 (UTC)
Example nonregular complex polygons
Complex polygon 2x5 stereographic3.png
{}×5{} has 10 vertices connected by 5 2-edges and 2 5-edges.
Complex polygon 3x4 stereographic3.png
3{}×4{} has 12 vertices connected by 3 4-edges and 4 3-edges
I have Coxeter's book and a few papers. The main nonregular polygon examples I know are prismatic forms, but I've not found any clear sourcing so far. The simplest real example is {}×{} or CDel node 1.pngCDel 2.pngCDel node 1.png, a rectangle or square if symmetry doubled to {4} or CDel node 1.pngCDel 4.pngCDel node.png, while 3{}×{} or CDel 3node 1.pngCDel 2.pngCDel node 1.png isn't regular, and p{}×q{} or CDel pnode 1.pngCDel 2.pngCDel qnode 1.png in general relates to the {p}×{q} duoprisms of real 4-space. And further CDel pnode 1.pngCDel r.pngCDel qnode 1.png isn't regular but can be called quasiregular, alternating p-edge and q-edges of CDel pnode 1.pngCDel r.pngCDel qnode.png and CDel pnode.pngCDel r.pngCDel qnode 1.png respectively.
For complex polyhedra, I found Coxeter writing CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 4.pngCDel node.png = CDel 3node.pngCDel 3.pngCDel 3node 1.pngCDel 3.pngCDel 3node.png, which is the same half symmetry relation as the real polyhedra CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png = CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png, i.e. octahedron, {3,4} having a tetratetrahedron, r{3,3} construction within tetrahedral symmetry, removing one mirror of octahedral symmetry. So anyway, CDel 3node.pngCDel 3.pngCDel 3node 1.pngCDel 3.pngCDel 3node.png is a nonregular construction of a regular polyhedron, suggesting other such "uniform polyhedra", but he didn't explain that there.
I know in general, with a Coxeter diagram, when all nodes are ringed (a generator point off all mirror planes), the resulting uniform polytope has exactly one vertex for each fundamental domain, so the vertex count equals the group order, but I've not found statements like that for the complex polytopes so far either. It would be great if he gave examples somewhere like CDel 3node 1.pngCDel 3.pngCDel 3node 1.pngCDel 3.pngCDel 3node 1.png for example will have 648 vertices, like the order of the group CDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png. Tom Ruen (talk) 02:45, 25 June 2016 (UTC)
There are actually many nonregular complex polytopes discussed, Coxeter calls these "almost regular polytopes", Chapter 14. Their Coxeter diagrams are like CDel node 1.pngCDel 4split1.pngCDel branch.pngCDel label4.png has symbol (1 1 114)4 and CDel node 1.pngCDel 3.pngCDel node.pngCDel 4split1.pngCDel branch.pngCDel label4.png has symbol (1 1 224)4. Like [1 1 1]4 represents a triangle diagram of 4-nodes, [1 1 1 1]4 represents a square diagram of 4-nodes, while [2 2 2]4 represents a triangle diagram with each node having tail branches, like CDel node.pngCDel 3.pngCDel node.pngCDel 4split1.pngCDel branch.pngCDel 3ab.pngCDel nodes.png, etc. Tom Ruen (talk) 20:23, 25 June 2016 (UTC)
Let me make my comment more precise: there appears to be no definition in any source of a "complex polytope." There are definitions in sources (but not this article!) of regular complex polytopes, and there are other related objects that some sources call complex polytopes, but there is no general definition that carves them out from other mathematical objects. Do you agree? --JBL (talk) 20:49, 25 June 2016 (UTC)
I do agree (unless Tom has a source with a clear definition). In fact the situation is worse than that because the literature embodies conflicting approaches. On the one hand we begin with the idea that a complex polytope is an extension of real polytopes to unitary spaces and on the other we have the dawning realisation, expressed by both Shephard and Coxeter, that the regular variety have more in common with real configurations than with polytopes. Yet nowhere do they stop to define what they are talking about. So we are left to deduce that Coxeter's "nearly-regular" examples (Shephard also alludes to non-regularity but gives no examples) are not symmetrical enough to be configurations, but unable to say so, never mind what they are instead, because that would be original research. It's a mess. But that is not Wikipedia's problem. The sources proclaim regular complex polytopes as regular examples of the more general set of complex polytopes, without defining them. Wikipedia follows the sources so we have to do the same. We must reflect the mess that is out there, it is not our job to clean it up. — Cheers, Steelpillow (Talk) 08:31, 26 June 2016 (UTC)
All of Coxeter's examples can be classified as vertex transitive complex polytopes (finite and infinite), a bit wider than regular complex polytopes, generated by complex reflection groups, and expressible in Coxeter diagrams, and nothing clearly beyond that, and probably no interest, thus no general definition. So we can't say what sort of collection of vertices, k-edges, construction in complex space is NOT a complex polygon. The only "topological" restriction I can see is a complex polygon needs at least 2 edges meeting at a vertex, and a complex polyhedron needs at least 3, etc. I'd assume regular complex polytopes could be distorted for different edge-sizes and lower symmetries which have degrees of freedom, like a rectangle vs square, but such examples don't seem to exist for a lack of interest. Tom Ruen (talk) 10:59, 26 June 2016 (UTC)
I have gone ahead and tried to capture this situation in a new, shorter, lead. I have also written a definition section. I would appreciate others looking them over! --JBL (talk) 16:25, 27 June 2016 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── I am afraid I found your version unacceptable. Firstly, it is simply not good enough to offer an unsupported definition in the lead, only to be forced to explain that there is no such definition. No source - no definition, see for example the WP:VERIFY policy. I have already explained all this and there are plenty more policies and guidelines along similar grounds. It is not negotiable. Secondly, there was a confusion over the significance of convexity theory, as if it was somehow the fount of all real polytopes. There is a large body of mathematics concerned with star polytopes, polytopes with non-spherical Euler numbers, wholly abstract polytopes and so forth, and the presentation needs to be intelligible in that context. I did not look further into it as it was too major an edit to untangle. — Cheers, Steelpillow (Talk) 17:10, 27 June 2016 (UTC)

You manage to be insulting and also completely unconstructive; wonderful! The sentence "A complex polytope may be understood as ..." (which is I assume the target of your first comment) is both correct (everything that is a complex polytope is such an arrangement), supported by sources, and present in the lead you restored (!) -- in fact it is one of the few parts of that lead that is actually supported by a concrete relationship to a formal definition (which I added, but which you of course reverted because who knows why). This all makes the following condescending sentences seem particularly ridiculous. About your second point, since convexity is also included in the present lead (to which you reverted), I am left wondering if you really thought very hard before making your reversion and comment. I would invite you to try again to read the two leads and make some attempt to engage thoughtfully and constructively next time. --JBL (talk) 22:00, 27 June 2016 (UTC)
I did not look deeply, as I said. I do now acknowledge that the flaws I highlighted are to some extent present in the earlier version, and I owe you an apology for assuming that you had introduced them and for any implied insult which you took from this. But wer were still in the middle of an unresolved discussion, so if you suddenly break off and make bold edits half way through, it is perhaps expected that other editors may hold their own strong views about that. Anyway, I find neither version wholly acceptable as yet, but it looks easier from here on to work on the outstanding issues as you present them. The main one is the comparison with convexity theory, which is fundamentally a narrow distraction from the broader theory of polytopes. — Cheers, Steelpillow (Talk) 08:10, 28 June 2016 (UTC)
Thank you for your response. I agree with you that the current version can be improved. I am not sure I agree with you generally about the importance of convexity, but in either form it was not supported by any sources and I think removing it was reasonable. --JBL (talk) 23:34, 28 June 2016 (UTC)
Some idea of the wider importance of non-convexity may be gathered from articles such as Star polytope, Projective polytope, Toroidal polyhedron and Euler characteristic. They are not very well written, but they give a flavour of it. Across the various mathematical disciplines which adopt terms such as "polyhedron" and "polytope", definitions differ widely - figures in complex spaces being just one example. Grûnbaum's standard reference work Convex polytopes uses explicitly three different definitions of "polyhedron" (for different reasons) in different chapters. He also writes at the beginning that when he uses the term "polytope" throughout the book, this should always be taken to mean the convex variety - a point which many readers miss. All this frequently-contradictory richness was one of the key motivations behind the modern set-based theory of abstract polytopes. The less one dips into the mess in an article like this one, the better! — Cheers, Steelpillow (Talk) 09:44, 29 June 2016 (UTC)
[The following should be understood as my input into an interesting conversation, not as an argument about what should be included in the article.] Yes, I am familiar with Grûnbaum's book. I think that it illustrates nicely the distinction that I see between the convex world and the other world. There is essentially only one notion of a convex polytope realized in R^n; the natural definitions are simple and equivalent. As soon as you give up convexity, to make a viable definition becomes very complicated. Whatever a complex polytope is going to be, there is no possibility that we can use convexity to define it, and this is one "explanation" of why there is no good theory of complex polytopes in general. All the best, JBL (talk) 16:03, 29 June 2016 (UTC)
I would be happy to continue this conversation but this is the wrong place. Drop me a personal email if you would like to continue it. — Cheers, Steelpillow (Talk) 16:46, 29 June 2016 (UTC)

Coxeter-Dynkin diagrams[edit]

Coxeter-Dynkin diagrams have recently been added, without sourcing. Is there a source for their use with complex polytopes? If not, this is original research (WP:OR) and needs to be reverted. — Cheers, Steelpillow (Talk) 09:29, 17 June 2016 (UTC)

The addition has also changed the meaning of the phrase "the modern notation" to apply to these diagrams instead of that attested in the sources. Whatever the case with the diagrams, the status of Coxeter's own original notation needs to be made clear. — Cheers, Steelpillow (Talk) 09:32, 17 June 2016 (UTC)

The diagrams come from Coxeter's "Regular complex polytopes", and an older paper in Kaleidoscopes — Selected Writings of H.S.M. Coxeter.. I'll get some page numbers and sourcing up momentarily. I'm really not sure what "modern" usage there is beyond Coxeter's books. I'm also unsure what advantage there is in the equivalent diagram and bracketing notations. I'm equally unsure if these diagrams are Coxeter groups in the same way that real regular polytopes have equivalences, i.e. CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = {3,3} = tetrahedron, and CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = [3,3] = tetrahedral symmetry. Tom Ruen (talk) 11:53, 17 June 2016 (UTC)
If I understand the question correctly, the symmetry group of a regular complex polyhedron is called a "Shephard group," a kind of complex reflection group. we don't have an article about them as far as I can see, maybe they deserve a section in the CRG article. --JBL (talk) 13:38, 17 June 2016 (UTC)
Wow, complex reflection group looks like a useful article I've not seen before! Here's a random paper that talks about "Shephard group," The Sign Representation for Shephard Groups. Tom Ruen (talk) 14:28, 17 June 2016 (UTC)
Thank you. I have edited the section to highlight the incorrectess of discussing Dynkin diagrams as if they were a modified Schlaefli form. I didn't know about Shephard Groups, yes it would be useful to see them explained somewhere. — Cheers, Steelpillow (Talk) 17:59, 17 June 2016 (UTC)
Steelpillow, I'm confused why you call 4{4}2 a regular complex octagon rather than a regular complex square. I'm also wondering if 4{}2, CDel 4node 1.png, exists in the C1 complex plane, and what it would be called. Maybe it is "too trivial"? Tom Ruen (talk) 02:33, 18 June 2016 (UTC)
p.s. I see on p.119 Coxeter says CDel pnode 1.png is a one-dimensional polytope, with p vertices. Its Argand diagram is an ordinary regular p-gon. But he doesn't write p{}, nor symmetry p[], CDel pnode.png. Tom Ruen (talk) 02:59, 18 June 2016 (UTC)
I changed complex octagon to complex polygon since I couldn't find any usage to support the word octagon. Tom Ruen (talk) 04:18, 18 June 2016 (UTC)
I based my wording on the definition of an octagon as a polygon having eight sides (by comparison to an octahedron which is a polyhedron having eight faces). A complex octagon is merely a complex polygon having eight (complex) sides, and I regarded this as a trivially convenient use of language. If this kind of "n-gon", "n-hedron", etc. usage does not occur in any sources then you may choose to argue that it is original research, but as it merely paraphrases the maths presented without saying anything new, I would find that hard to justify. — Cheers, Steelpillow (Talk) 19:25, 19 June 2016 (UTC)
I don't object to the presumed terminology, n-gon for n edges, only the confusion with the single example and diagram which both looks octagonal, Complex polygon 4-4-2.png, and you're calling an octagon for having 8 4-edges, while that's a conincidence, compared to the dual 2{4}4 while also looks octagonal in the same petrie projection, but has 16 2-edges (so a hexadecagon). Complex polygon 2-4-4.png. Tom Ruen (talk) 20:09, 19 June 2016 (UTC)
Unlike real polygons, a complex polygon does not necessarily have the same number of vertices as edges. In its dual these numbers are swapped round. Thus, in your example we have a complex octagon with 16 vertices dual to a complex 16-gon with 8 vertices and that is perfectly OK. Remember, they are not "polygons" in the usual sense but configurations in unitary 2-space, and consequently they do not behave like real polygons. There is a case for saying that the "n-gon" terminology is misleading, but Shephard and Coxeter dumped the terminology of polytopes on us and we have to make the best of it. — Cheers, Steelpillow (Talk) 07:54, 20 June 2016 (UTC)
I guess the primary confusion over n-gon is you can't say "regular n-gon" in general and know what you're talking about. Like there are two regular octagons 4{4}2, 3{3}3, and 4 regular hexadecagons 8{4}2, 2{4}4, 3{6}2, and 3{3}2. Tom Ruen (talk) 13:58, 20 June 2016 (UTC)

Visualizations[edit]

Seeing a stereographic projection (on a 4D duoprism wireframe), I realize edges CAN be drawn as filled squares as 4-edges, so the 4{4}2 complex polygon (octagon as you say) looks like this (would be slightly better with transparent faces). Or more specifically this coloring is 4{}×4{} which isn't regular, but the higher symmetry form is regular. Tom Ruen (talk) 12:46, 20 June 2016 (UTC)

Complex polygon 4-4-2-stereographic2.pngComplex polygon 4-4-2-stereographic3.pngComplex polygon 4-4-2-stereographic.png
The complex edges are NOT the filled squares but the entire plane for each square. Remember, a complex polygon is a configuration of points and complex lines, it has no concept of boundary or interior at any level. If you are colouring in the squares then it is better to leave out the lines, and vice versa. — Cheers, Steelpillow (Talk) 10:40, 21 June 2016 (UTC)
Here are 3 versions above. Tom Ruen (talk) 14:58, 21 June 2016 (UTC)
I think the style with planes alone is best, as it is least suggestive of any ordering around the rim of the square. Semi-transparency and shading are probably needed with drawings of intermediate "busy"-ness, but this one is just about OK without. I also doubt that these are true stereographic projections, as this generally leads to curved shapes in order to preserve angles, they are more like perspective drawings, which generally maintain straightness but change angles. — Cheers, Steelpillow (Talk) 16:18, 21 June 2016 (UTC)
Nor are these drawings Schlegel diagrams, because they are not projected inside a single 3-face - there are of course no 3-faces to project them inside of. As I have already said, they are perspective drawings. — Cheers, Steelpillow (Talk) 16:30, 22 June 2016 (UTC)
I don't think saying perspective is quite sufficient. They are "perspective" in the sense of mapping from 4D to 3D, and then skew orthogonal projections (or a different perspective projection) from 3D to 2D. I don't see a problem with calling them Schlegel diagram since the principle is identical, and the article on Schlegel diagram willl clarify more than an article on perspective drawing, which will be 3D to 2D. Tom Ruen (talk) 19:18, 22 June 2016 (UTC)
No the principle is NOT identical. Read the definition of a Schlegel diagram a bit more carefully. Then re-read what I wrote above. In geometry, a near-miss is as good as a mile (or π would equal 3). A perspective view is a projective transformation, a Schlegel diagram in general is not. Your diagrams are not simple perspective views, granted, but they are projective transformations. And they are drawings which involve perspective. Hence my choice of term "perspective drawing", which has no precise mathematical definition and is typically used to refer to the whole class of such things. One might alternatively call them "perspective sketches", which is even looser. — Cheers, Steelpillow (Talk) 20:14, 22 June 2016 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── I also prefer the picture that is currently in the middle (planes but no edges). --JBL (talk) 21:59, 24 June 2016 (UTC)

Table of n-dimensional groups generated by n unitary reflections[edit]

From Coxeter's paper, Finite Groups Generated by Unitary Reflections. Tom Ruen (talk) 14:15, 17 June 2016 (UTC)

Group Order Symbol or Position in Table VII of Shephard and Todd (1954)
CDel node.png, CDel node.pngCDel 3.pngCDel node.png, CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, …
[], [3], [3,3], …
(n + 1)!, n ≥ 1 [3n − 1] = G(1, 1, n)
CDel pnode.pngCDel 4.pngCDel node.png, CDel pnode.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, CDel pnode.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, …
p[4]2, p[4]2[3]2, p[4]2[3]2[3]2
pn (n + 1)!, p ≥ 2 G(p, 1, n)
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png, CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, …
[31,1,1], [32,1,1], [33,1,1], …
2n − 1 (n + 1)!, n ≥ 4 [3n − 1,1,1] = G(2, 2, n)
CDel branch.pngCDel labelq.png, CDel node.pngCDel 3split1.pngCDel branch.pngCDel labelq.png, CDel node.pngCDel 3.pngCDel node.pngCDel 3split1.pngCDel branch.pngCDel labelq.png, CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3split1.pngCDel branch.pngCDel labelq.png
[q], [1 1 n − 2 q]3
qn − 1 n!, q ≥ 3 G(q, q, n)
CDel node.pngCDel 3split1.pngCDel branch.pngCDel 3ab.pngCDel nodes.png, CDel node.pngCDel 3split1.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3a.pngCDel nodea.png 72·6!, 108·9! Nos. 33, 34, [1 2 2 q]3, [1 2 3 q]3
CDel node.pngCDel 4split1.pngCDel branch.pngCDel label4.png, CDel node.pngCDel 4split1.pngCDel branch.pngCDel label5.png, CDel node.pngCDel 4split1.pngCDel branch.pngCDel 3a.pngCDel nodea.png 14·4!, 3·6!, 64·5! Nos. 24, 27, 29
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png, CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[5,3], [3,4,3], [5,3,3]
5!, 2(4!)2, (5!)2 [5,3], [3,4,3], [5,3,3]
CDel nodea.pngCDel 3a.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, CDel nodea.pngCDel 3a.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, CDel nodea.pngCDel 3a.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, …
[32,2,1], [33,2,1], [34,2,1]
72·6!, 8·9!, 192·10! [32,2,1], [33,2,1], [34,2,1]
CDel 3node.pngCDel 3.pngCDel 3node.png, CDel 3node.pngCDel 4.pngCDel 3node.png, CDel 4node.pngCDel 4.pngCDel 3node.png, CDel 3node.pngCDel 5.pngCDel 3node.png, CDel 5node.pngCDel 3.pngCDel 5node.png
3[3]3, 3[4]3, 4[4]3, 3[5]3, 5[3]5
4!, 3·4!, 4·4!, 3·5!, 5·5! Nos. 4, 5, 8, 20, 16
CDel 4node.pngCDel 4.pngCDel 3node.png, CDel 5node.pngCDel 4.pngCDel 3node.png
4[4]3, 5[4]3
12·4!, 15·5! Nos. 10,18
CDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png, CDel 3node.pngCDel 3.pngCDel 3node.pngCDel 4.pngCDel node.png, CDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png
3[3]3[3]3, 3[3]3[4]2, 3[3]3[3]3[3]3
27·4!, 54·4!, 216·6! Nos. 25, 26, 32

Configurations[edit]

A recent sequence of edits states that all complex polytopes are configurations. In general this is untrue, only the regular variety are configurations. This is because the definition of a configuration demands the same high degree of symmetry that is seen only in the regular polytopes - the same number of lines at each point, the same number of points on each line, etc. These edits are now buried behind a further sequence, so this all needs unpicking again. — Cheers, Steelpillow (Talk) 20:10, 30 June 2016 (UTC)

Sorry, you're right (though actually the whole thing is awkward for other reasons, like the fact that the configurations article seems to want them always to live in the (real?) projective plane). I'll try to fix it later if someone else doesn't first. --JBL (talk) 20:21, 30 June 2016 (UTC)

Tables[edit]

The tables have got too complicated and in anything but the widest screens they begin to squash badly. The Dynkin symbols are the first suffer. It is worst with the widest tables, such as the list in five dimensions. A better approach is needed. Either the information for each entry must be drastically cut back, or the idea of listing them all in the root article abandoned. — Cheers, Steelpillow (Talk) 13:23, 10 July 2016 (UTC)

I'm okay with moving comprehensive tables to a different article at some point. Its all still in progress. Tom Ruen (talk) 13:37, 10 July 2016 (UTC)
Of course, the layout issue will follow the lists in their present form, they need to be simplified anyway. — Cheers, Steelpillow (Talk) 14:12, 10 July 2016 (UTC)

Example 4{4}2 images[edit]

Three views of regular complex polygon 4{4}2, CDel 4node 1.pngCDel 3.pngCDel 4.pngCDel 3.pngCDel node.png
ComplexOctagon2.svg
This complex polygon has 8 square edges, labeled as a..h, bounded by 16 (unlabeled) vertices. In the left image, the outlined squares are not elements of the polytope but are included merely to help identify vertices lying on the same complex line. In the middle image, each edge is seen end-on so it looks like a real line.
Complex polygon 4-4-2-perspective-labeled.png
This perspective view shows the same 16 vertices in black and 8 4-edges as squares in 2 orthogonal color sets. The green path represent the octagonal perimeter of the left image.

User:Steelpillow made the image Image:ComplexOctagon.svg, and I modified it for clarity filling the 8 squares with transparent colors, and adding black node circles, as seen in right set of images. Since User:Steelpillow complained about my editing his image and reverted so I made a "2" version for this article, moving here for discussion. I also added the right perspective image to help clarify the global topology of the polygon.

When I asked for clarity on the reverted changes, this was his reply.

I am saying, if you wish to upload a substantially altered version to the Commons, do not do so as an update to the original, but instead upload it as a new image. I posted here because your Commons talk page advised me to.
The image to be used in any Wikipedia article is different issue, but while I am here I can say that the warnings remain about drawing lines or colouring in plane regions which are not in themselves structural elements of the figure depicted: that has to be done as carefully and as minimally as possible in order for the supposed added "clarity" not to add to the common misconception instead. For example the criticism I made of your perspective projections applies - no drawing should emphasise both the lines and the plane regions as the end effect is to present a bounded real polygon to the eyeball, precisely the effect we need to avoid.
— Cheers, Steelpillow (Talk) 14:08, 10 July 2016 (UTC)

The primary issue I have with the original left-most image (uncolored and no node markings) is its less clear what the squares are, or how they're connected, or the ambiguous meaning of the intersecting lines. Here's a reproduced image from Coxeter, p.31, Figures 4.2B an 4.2C. So the 4-edges are "unfolded" into 4 line segments in each row and column. I added blue and red colors, and used Coxeter's vertex labeling. Tom Ruen (talk) 21:34, 10 July 2016 (UTC)

Complex polygon 4-4-2 two labeled.png
For the left-hand image, this latest is preferable to the full-colour version, for the reason I gave. For the right-hand image, I do not have Coxeter easily available but I seem to recall that he ended the grid lines short as you have done, while explaining somewhere in the text that they are in fact endless. I regard that as a poor choice of presentation (there is no error in the mathematics either way), so I would prefer to add my improvement in presentation - the partly-extended grid - to your latest version. That way we get both your and my improvements. — Cheers, Steelpillow (Talk) 08:06, 11 July 2016 (UTC)
I've not seen such a quote from Coxeter, but my impression is the lines are "endless" because it's a flat torus so you can repeat the 4x4 grid in the entire plane, but I'm not sure if that's what you mean. Tom Ruen (talk) 13:23, 11 July 2016 (UTC)
Here's all 3 in one. Tom Ruen (talk) 13:54, 11 July 2016 (UTC)
Complex polygon 4-4-2 three labeled.png
No, your impression is quite wrong and that is certainly not what either Coxeter or I mean. The figure does not fold round on itself like a torus, it is endless as in having indefinitely large extent, commonly referred to as "infinite". It is composed of complex points, which appear in your projections as real points, and complex lines, which appear in your projections as flat, unbounded planes. As has been endlessly repeated it is a configuration and all that entails - please do read the linked article and let it sink in.
May I ask whether your "five-barred" central diagram is direct from Coxeter? You will presumably be aware that the outer two lines in each orientation (upper/lower), (left/right) are in fact two images of the same line. I am wholly unclear of the benefit of doing this, it seems to me most misleading. — Cheers, Steelpillow (Talk) 15:29, 11 July 2016 (UTC)
Okay, then we're in disagreement and you're going to have to find text from Coxeter that supports your meaning. Yes, the image is from Coxeter, I just added the colors. That grid plane IS a FLAT TORUS, zero curvature plane section rolled up so left/right and top/bottom match up! I made one adjustment to the flat torus image above. Coxeter repeated vertices which is a bit confusing, so I shifted the center to cut the torus mid-edge. This is closer to what you seem to want, even if you're still in diagreement what it is! The row/column lines are the 4 sides of an unfolded square. Tom Ruen (talk) 15:56, 11 July 2016 (UTC)
The only thing Coxeter says about configurations is on p.117 "Since a regular polytope is a special kind of configuration (Veblen and Young 1910, p.38; Coxeter, 1963, p. 12), many of its numerical properties can be exhibited in a matrix.." Tom Ruen (talk) 16:38, 11 July 2016 (UTC)
Not from Coxeter on this occasion but from G.C. Shephard, "Regular complex polytopes", Proc. London Maths soc. 1952 Vol Series 3, Vol 2, pp82-97. On page 83; "It is from considerations such as these that we derive the notion of the interior of a polytope, and it will be seen that in unitary space where the numbers s cannot be so ordered such a concept of interior is impossible. [Para break] Hence ... we have to consider unitary polytopes as configurations." I can see one possible source of confusion between us: one can inject a finite part of the configuration either into a finite region of the Euclidean plane or into a flat torus and in this particular case the two images may look similar, is this what Coxeter says he is doing? can you give some context for what he does say he is doing here? An accurate reproduction of his own diagram would be useful. — Cheers, Steelpillow (Talk) 21:10, 11 July 2016 (UTC)
4.8 A FAMILY OF REGULAR COMPLEX POLYGONS
We saw, in 4.5 that the plane faces of the double prism {p}×{p} consists of p2 squares and 2p p-gons. (When p=4, this distinction can still be made, actually in 3 ways. One way is plainly visible in [2D projection] figure 4.5A where the 16 squares are foreshortened into rhombi while the 8 p-gons appear as undistorted squares.) Unlike the p2 squares which form {4,4|p}, the 2p p-gons do not form a polyhedral surface: their 2p2 edges are all distinct, and each of the p2 vertices only belong to just 2 of its edges, we naturally ask whether there is any sense in which our 2p precariously connected p-gons might be regarded as the 'edges' of some kind of generalized polygon: a regular complex polygon. A clue is provided by the complex coordinates u and v, in terms of which the vertices of each p-gon satisfies a linear equation: v=ε/sqrt(2), u=ε/sqrt(2) and thus lie on a line of the unitary plane. Let us call this line an edge of the complex polygon, even though it contains p vertices where p may exceed 2.
More precisely, let us call it a p-edge. The p vertices on it are cyclically permuted by the '1-dimensional unitary reflection' u'=ε2*u, ε=exp(π*i/p). Thus a p-edge is represented in the Argand plane by the real polygon {p}.
An appropriate Schlaefli symbol for the complex polygon is p{4}2,
where the p indicates that there are p vertices on each edge while the 2 indicates that there are 2 edges through each vertex. For a real polygon {q}, both these numbers are 2; thus the symbol {q} may be regarded an abbreviation for 2{q}2, in agreement with the fact that, when p=2, p{4}2 reduces to the ordinary square.
Omitting the nine triangles from {3}×{3} Figure 11.5A, we are left with 6 triangles representing the 6 3-edges of 3{4}2. Similarly, the 8 4-edges of 4{4}2 are represented by the undistorted squares in figure 4.5A. The analogous drawing for any p{4}2 consists of a peripheral {2p} belonging to a {p} whose opposite edge belongs to a concentric star polygon {2p/(p-1)}. When p is odd, p of the p2 vertices are projected together into the centre of the {2p}; but this awkwardness can be remedied by using, instead of the peripheral {2p}, an equilateral (but not equiangular) 2p-gon whose vertices lie alternately on two concentric circles.

Here's a quick retyping from pp. 46-47. On your questions of interior of a polytope, that is a different issue than the interior of an edge, or k-edge. A star polygon has no interior, but its edges can still be drawn with a solid interior. Coxeter's regular skew polyhedron in R4, {4,4| 4}, IS the a polygonal surface of a flat torus, the 16 squares existing in the tesseract faces, but missing from the 4-edges of 4{4}2. Tom Ruen (talk) 01:30, 12 July 2016 (UTC)

No mention of any "flat torus" there. Is there any such mention in the caption to his image?
On the matter of the interior, the lack of any such arises from the "fundamental" property "that complex numbers cannot be ordered" on the complex line. These quotes from Shephard set the context for the brief analysis which concludes in the passage I quoted above. Do you have his paper available or do I need to give you the whole passage before you will believe me?
It is difficult to unravel Coxeter's references to his illustrations, since you have embellished the linked examples by your own lights. For example in his discussion of the (real) double-prism {p} X {p}, your File:Complex polygon 4-4-2-perspective-labeled.png bears little apparent relation to his counts and descriptions of the various elements and cannot be taken as representative of his figure 4.5A as you suggest. However he is careful to talk of a complex p-edge as "represented by" or "reduced to" a (real) p-gon and not as actually being one. It should be obvious even to you that to treat an edge (1-polytope) in any way as a polygon (2-polytope) breaks every theory of polytopes ever conceived and Coxeter would never endorse such a mathematical understanding. It is foisted on us only because of the limitations of the real plane in which we embed our illustrations. All in all, I find nothing in Coxeter to support your position and everything to support mine.
Re. our most recent editing issue, note also that Coxeter writes, "A clue is provided by the complex coordinates u and v, in terms of which the vertices of each p-gon satisfies a linear equation: v=ε/sqrt(2), u=ε/sqrt(2) and thus lie on a line of the unitary plane. Let us call this line an edge of the complex polygon [my emphasis], even though it contains p vertices where p may exceed 2." Note here that he defines an edge as a line and I trust you are willing to accept that a unitary line is infinite. Coxeter does not need to use the word "infinite" here because it is implicit in the unitary line, just as it is implicit in the Euclidean line.
May I heartily recommend that you carefully read Shephard's original paper, which I cited, and perhaps study Coxeter's Regular Complex Polytopes a little more thoroughly and critically, before forming your final opinion on these matters. — Cheers, Steelpillow (Talk) 13:15, 12 July 2016 (UTC)
Steelpillow - "It should be obvious even to you that to treat an edge (1-polytope) in any way as a polygon (2-polytope) breaks every theory of polytopes" ... Garbage thinking. It should be obvious that a 1-polytope in is NOT a polygon but two points with a simple interior between, while a k-edge in IS a polygon, i.e. a regular k-gon, as Coxeter states, and with a simple interior! Tom Ruen (talk) 13:35, 12 July 2016 (UTC)
Can you please cite where Coxeter states that a k-edge is identifiable as a k-gon and not merely "represented by", "reduced to" or other such caveat? Can you also please cite where he states that a k-edge has a defined "interior"? I am keen to see these statements you claim to exist, as they explicitly contradict Shephard. Until then, Wikipedia must give credence to Shephard's peer-reviewed text over your unsourced interpretations of Coxeter." FYI that is why I persist in correcting your edits. No source - no deal. — Cheers, Steelpillow (Talk) 14:51, 12 July 2016 (UTC)
Complex 1-topes as k-edges.png
A complex polytope contains not only ordinary edges with 2 vertices, but can also contain p-edges, which are represented by a real polygon {p}. Vertices are drawn as black circles, and the edge-centers have red circles. The interiors of the p-edges are filled in yellow. The polygon sides show the path of sequential applications of a single complex reflection generator, ε=e2π/p.
Coxeter: Thus a p-edge is represented in the Argand plane by the real polygon {p}.
Here is the Argand plane, . I have drawn 3-edges, 4-edges, and 5-edges as regular polygons {3}, {4}, and {5}. What are you missing here? EVERY DIAGRAM is A REPRESENTATION. How do you draw imaginary things? Are we Bill Clinton lawyering the meaning of the word "IS"?! Tom Ruen (talk) 15:11, 12 July 2016 (UTC)
I tire of this. Every point of Shephard's which I confirm, you raise a new canard. Suffice to say there is a difference between a mathematical representation and a graphical one. A mathematical "representation" is not precisely defined in the way that say an injection, an embedding or an immersion is. Coxeter's point, therefore, is that by saying the figure is "represented" as in the mathematical Argand plane, this is not a precise mathematical image. What you are holding up to me are graphical representations of deliberately ill-defined mathematical representations. The lines you draw from vertex to vertex are a case in point. As Shephard pointed out, there is no natural ordering of complex points in the complex line. The correct projection of a regular k-edge onto an Argand plane (one real axis, one complex) has the k vertices distributed equally around a common centre (typically the origin) and nothing else. Drawing in the real lines between minimally-separated vertices imposes an ordering on them. This is quite wrong because, as Shephard explains, the original vertices cannot be so ordered. No ordering of the vertices means you can't draw those lines - they just do not and cannot exist as features of the complex figure. Consequently, it is impossible to define an interior. Shephard does use the term "bounding" in the sense that the entire line is a "bounding edge" and is equally explicit that this does not "enclose" anything. Part of your confusion may arise from the fact that a "polygon" is only sometimes treated as a bounded region and is also sometimes treated as a set of intersecting lines known as a configuration. See for example Lines' popular school textbook of the day; Solid geometry, Macmillan, 1934. Shephard specifically discusses this ambiguity and, as I have cited for your benefit, his analysis forcefully rejects the possibility you so tigerishly fight your corner for. But I am done with this discussion, you will only create yet more canards without regard to Shephard, why waste my time trying. — Cheers, Steelpillow (Talk) 16:18, 12 July 2016 (UTC)
11.5 Cayley diagrams for reflection groups, p. 108
Consider a nonstarry polygon p{2q}r and its group p[2q]r. Since [Complex reflection] R1 cyclically permutes the p vertices on an edge, it conjugate reflection: Rν=Sν-1R1S1-ν, ν=2,...r. cyclically permutes the p vertices on another of the r edges that radiate from the initial vertex. Since the p-edges appear, in the real representation, as simple p-gons (or just edges, if p=2), it is natural to use one of the r different colors for each of the r representative p-gons (or edges). Since the r reflections Rν generate a subgroup of p[2q]r, the colouring can be continued consistently over the whole of the real representation.

We conclude that this arrangement of 4s^2/pq coloured p-gons (or edges) meeting by r's at the 4s^2/pq points, provides a Cayley diagram (with r colours) for the group. Although the arrangement of points and p-gons appear first in an Euclidean 4-space, there is a conspicuous advantage (beyond the obvious advantage of making it visible) to be gained by projecting it onto a suitable plane: there is no need to mark arrows along the edges of the regular p-gons (p>2) provided these edges are understood to be directed in a positive sense round each p-gon.

This happy state of affairs becomes obvious when we first project the complex polygon onto a complex line () and then represent the resulting one-dimensional figure on the Argand plane.

Steelpillow: "Drawing in the real lines between minimally-separated vertices imposes an ordering on them. This is quite wrong because, as Shephard explains, the original vertices cannot be so ordered. No ordering of the vertices means you can't draw those lines - they just do not and cannot exist as features of the complex figure."
You are simply wrong here. A 1-dimensional complex polytope has ONE generator. Sequentially applying that generator creates the regular polygon. There may be different meanings in different contexts, but I'm simply following Coxeter's complex reflection generators. And I have plenty more canards if that what you need to call things you don't like to hear. Tom Ruen (talk) 19:05, 12 July 2016 (UTC)

one-dimensional[edit]

A one-dimensional real polytope is often regarded as a closed line segment - in Plato's words it is "solid". Only in certain theories is it regarded as a point pair, a bounding "1-surface". The term "body" is not in wide use and needs citing. If the treatment currently given here is to be updated, it needs some care in avoiding PoV bias. — Cheers, Steelpillow (Talk) 09:34, 4 August 2016 (UTC)

To clarify: it is somewhat invidious to define a real 1-polytope here when the polytope article does not. Nevertheless, the article on Edge (geometry) makes it clear that an edge includes both the end points and the interior, i.e. it is a closed line segment. The present article cannot differ from that treatment without rigorous referencing. — Cheers, Steelpillow (Talk) 14:30, 4 August 2016 (UTC)
That's fine. I assumed your objection was to "The body of p{} can be represented by the open region of a real regular p-gon." so I removed that. Feel free to discuss what better fits your understanding. Tom Ruen (talk) 14:37, 4 August 2016 (UTC)
It looks like body is only defined at Polytope#elements. Tom Ruen (talk) 14:40, 4 August 2016 (UTC)
Norman Johnson wrote this "A complex p-on [1-dimensional polytope] can be more fully described as a partially ordered set consisting of the empty set as its unique (-1)-dimensional element, p vertices lying on a chain, and a body (a piece of the complex line). When p = 2, the body is a real segment. When p > 2, the body is an open convex region, the interior of a real p-gon. When the vertices are equally spaced around the chain, the p-on [1-polytope] is regular (and uniform). All dions [p=2] are regular." Tom Ruen (talk) 14:47, 4 August 2016 (UTC)
Johnson is not a reliable source, as you well know. I have removed "body" from the article you highlighted, as it was otherwise unsourced. — Cheers, Steelpillow (Talk) 15:05, 4 August 2016 (UTC)