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Some examples would be nice mike40033 ( talk) 07:13, 19 May 2008 (UTC)
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)
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)
In Coxeter & Moser's Generators and Relations for Discrete Groups, much of §6.7 is spent discussing the presentations p[2qr and pqp, 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
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
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)
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)
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)
I have some problems with the revised lead to this article:
If nobody objects, I shall attempt a second a rewrite. -- Cheers, Steelpillow ( Talk) 10:47, 3 April 2010 (UTC)
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)
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'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)
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)
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)
![]() {}×5{} has 10 vertices connected by 5 2-edges and 2 5-edges. |
![]() 3{}×4{} has 12 vertices connected by 3 4-edges and 4 3-edges |
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)
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)
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)
I also prefer the picture that is currently in the middle (planes but no edges). -- JBL ( talk) 21:59, 24 June 2016 (UTC)
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) |
---|---|---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [], [3], [3,3], … |
(n + 1)!, n ≥ 1 | [3n − 1] = G(1, 1, n) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() p[4]2, p[4]2[3]2, p[4]2[3]2[3]2 |
pn (n + 1)!, p ≥ 2 | G(p, 1, n) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [31,1,1], [32,1,1], [33,1,1], … |
2n − 1 (n + 1)!, n ≥ 4 | [3n − 1,1,1] = G(2, 2, n) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() q], [1 1 n − 2 q3 |
qn − 1 n!, q ≥ 3 | G(q, q, n) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
72·6!, 108·9! | Nos. 33, 34, [1 2 2 q3, [1 2 3 q3 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14·4!, 3·6!, 64·5! | Nos. 24, 27, 29 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [5,3], [3,4,3], [5,3,3] |
5!, 2(4!)2, (5!)2 | [5,3], [3,4,3], [5,3,3] |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [32,2,1], [33,2,1], [34,2,1] |
72·6!, 8·9!, 192·10! | [32,2,1], [33,2,1], [34,2,1] |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() 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 |
![]() ![]() ![]() ![]() ![]() ![]() 4[4]3, 5[4]3 |
12·4!, 15·5! | Nos. 10,18 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() 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 |
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)
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)
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.
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)
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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)
Consider a nonstarry polygon p{2q}r and its group p[2qr. 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[2qr, 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. |
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)
There are several complex apeirotopes with generators of infinite order, e.g. ∞{} and ∞{4}2. However, the symmetry groups of these objects aren't Shephard groups, since it is a requirement that the generating mirrors of a Shephard group be of finite order. These are not listed in Coxeter's "complete" enumeration, and don't meet the definition given by McMullen and Schulte. Where are these from? What definition is this article using? AquitaneHungerForce ( talk) 02:25, 8 March 2024 (UTC)
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Some examples would be nice mike40033 ( talk) 07:13, 19 May 2008 (UTC)
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)
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)
In Coxeter & Moser's Generators and Relations for Discrete Groups, much of §6.7 is spent discussing the presentations p[2qr and pqp, 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
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
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)
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)
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)
I have some problems with the revised lead to this article:
If nobody objects, I shall attempt a second a rewrite. -- Cheers, Steelpillow ( Talk) 10:47, 3 April 2010 (UTC)
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)
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'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)
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)
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)
![]() {}×5{} has 10 vertices connected by 5 2-edges and 2 5-edges. |
![]() 3{}×4{} has 12 vertices connected by 3 4-edges and 4 3-edges |
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)
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)
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)
I also prefer the picture that is currently in the middle (planes but no edges). -- JBL ( talk) 21:59, 24 June 2016 (UTC)
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) |
---|---|---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [], [3], [3,3], … |
(n + 1)!, n ≥ 1 | [3n − 1] = G(1, 1, n) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() p[4]2, p[4]2[3]2, p[4]2[3]2[3]2 |
pn (n + 1)!, p ≥ 2 | G(p, 1, n) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [31,1,1], [32,1,1], [33,1,1], … |
2n − 1 (n + 1)!, n ≥ 4 | [3n − 1,1,1] = G(2, 2, n) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() q], [1 1 n − 2 q3 |
qn − 1 n!, q ≥ 3 | G(q, q, n) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
72·6!, 108·9! | Nos. 33, 34, [1 2 2 q3, [1 2 3 q3 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14·4!, 3·6!, 64·5! | Nos. 24, 27, 29 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [5,3], [3,4,3], [5,3,3] |
5!, 2(4!)2, (5!)2 | [5,3], [3,4,3], [5,3,3] |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() [32,2,1], [33,2,1], [34,2,1] |
72·6!, 8·9!, 192·10! | [32,2,1], [33,2,1], [34,2,1] |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() 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 |
![]() ![]() ![]() ![]() ![]() ![]() 4[4]3, 5[4]3 |
12·4!, 15·5! | Nos. 10,18 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() 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 |
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)
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)
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.
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)
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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)
Consider a nonstarry polygon p{2q}r and its group p[2qr. 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[2qr, 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. |
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)
There are several complex apeirotopes with generators of infinite order, e.g. ∞{} and ∞{4}2. However, the symmetry groups of these objects aren't Shephard groups, since it is a requirement that the generating mirrors of a Shephard group be of finite order. These are not listed in Coxeter's "complete" enumeration, and don't meet the definition given by McMullen and Schulte. Where are these from? What definition is this article using? AquitaneHungerForce ( talk) 02:25, 8 March 2024 (UTC)