# Talk:Polytope

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## Mathematical nonsense

Does anyone know who the 'I' is that remarked at the bottom of this article? The history is lost with the change of software.

Rather than restrict ourselves to ASCII art, could someone please draw these figures in a graphics program and upload them? I would, but I know nothing about the subject and can't make heads or tails of the existing depictions. - user:Montrealais

Obviously needs an edit.

Charles Matthews 09:33 25 Jun 2003 (UTC)

On a closer inspection: is polytope just being used here for simplicial complex embedded in Euclidean space? Is there some condition too that makes it a manifold (or not)?

Charles Matthews 12:36 25 Jun 2003 (UTC)

Mathematical nonsense removed.

Roughly speaking this is the set of all possible weighted averages, with weights going from zero to one, of the points. These points turn out to be the vertices of their convex hull. When the points are in general position (are affinely independent, i.e., no s-plane contains more than s + 1 of them), this defines an r-simplex (where r is the number of points).

Mikkalai 08:30, 1 Mar 2004 (UTC)

The "mathematical nonsense" should be rewritten and put in an article on convex hulls, if it hasn't already

mike40033 11:20, 1 Mar 2004 (GMT+0800)

## half spaces & convex hulls

I think there's an error here:

One special kind of polytope is a convex polytope, which is the convex hull of a finite set of points.

Convex polytopes can also be represented as the intersection of half-spaces.

How can this be simultaneously true? Consider a single half-space: note that it is certainly convex. Of what finite set of points is this polytope the convex hull?

My understanding, from the reference given below, is that

A (convex) polyhedron in ${\displaystyle R^{k}}$ is defined to be the intersection of some finite number of half spaces in ${\displaystyle R^{k}}$. Bounded polyhedra are called polytopes. (A polytope can be definted equivalently as the convex hull of a finite point set in ${\displaystyle R^{k}}$).

Note that this agrees with Wikipedia's article on polyhedron.

If there are no arguments, I will edit to reflect this definition.

Reference: Dobkin, D. & Kirkpatrick, D., "A Linear Algorithm for Determining the Separation of Convex Polyhedra," Journal of Algorithms 6, 381-392 (1985).

-Alem

I don't know if the term 'polytope' consistently refers to bounded polytopes (which is the definition you have here). Some papers refer to "unbounded polytopes" where the bounding halfspaces enclose an unbounded region. I don't think the term "polyhedron" generally refers to arbitrary dimensions; it usually refers only to R3. But you're right, there's a hole in the current definition which needs to be addressed.—Tetracube 21:18, 13 September 2006 (UTC)
The usual custom these days is to say a "polytope" is bounded. The intersection of half-spaces may be unbounded; then it is not a polytope. "Polyhedron" has two uses: 3-dimensional polytope, and arbitrary, possibly unbounded, intersection of half-spaces. I've rewritten the article to correct the misleading impression about boundedness (and to fix other errors). Zaslav 18:34, 30 March 2007 (UTC)
```(Note that the intersection of arbitrary half-spaces need not be bounded; it is a convex polytope if and only if it is bounded.)
```

The intersection of arbitrary half spaces, if bounded, need not be the hull of a finite number of points. e.g., the disc. I am removing the if.MotherFunctor (talk) 18:37, 12 September 2008 (UTC)

That should be “the intersection of finitely many halfspaces, if bounded.” The disk is not an intersection of finitely many halfspaces, although it is an intersection of halfspaces. —David Eppstein (talk) 00:07, 26 October 2008 (UTC)

I just have a question about the inequality, Ax =< b (as written) is this consistent with the page on half-spaces (http://en.wikipedia.org/wiki/Half_space) where the inequality that reads a1x1 + a2x2 + ... + aNxN >= bn ? It seems to me that one of the > (or <) is around the wrong way? —Preceding unsigned comment added by 128.178.54.24 (talk) 09:49, 4 January 2008 (UTC)

## polytopic membrane protein

What does the word "polytopic" mean in the context of polytopic membrane proteins? Jeff Knaggs 22:17, 9 March 2007 (UTC)

Polytope is a greek compound meaning 'many' + 'places'. In biology, it refers to species that arise from "many places", rather than a single locale. Such might be co-breeding sub-species that remerged to form a new species. It is rather more the case of what's it doing replacing polyschema, the word Schläfli used to describe the thing. Wendy.krieger 07:15, 22 September 2007 (UTC)

## Image of face lattice

I have to signal an error in that image: there is no element "abcd" (last element of the second line from above), but there is instead "bcde" (the basis square of the pyramid). Somebody correct it! :-) —Preceding unsigned comment added by 213.140.11.138 (talk) 21:51, 11 September 2007 (UTC)

I second that. someone should correct this! 80.178.114.234 (talk) 16:41, 27 March 2008 (UTC)
Sorry about that. Fixed. It may take a while to propagate through wikimedia's caches. —David Eppstein (talk) 17:19, 27 March 2008 (UTC)
thanks! 80.178.114.234 (talk) 12:21, 2 April 2008 —Preceding unsigned comment added by 128.139.226.37 (talk)

## Several things are in a real mess

This page is a terrible mess. Most of the reasons behind this are understandable, but it still needs sorting out. Here are some highlights. -- Steelpillow (talk) 18:29, 8 March 2008 (UTC)

I must agree, and am adding {{confusing}} for these and other problems, in particular, very poor conformance to WP:MSM. Orange Knight of Passion (talk) 00:30, 14 September 2008 (UTC)

### Polytopes and convexity

There is also a terrible mess over the definition of "polytope". Historically it refers to the extension of the idea of polygons and polyhedra to higher dimensions, as closed geometric figures. Some years after their discovery, the term "polytope" was coined to describe them and soon became the established term. Over half a century later, Grunbaum published his seminal work "Convex polytopes". His definition omitted the word "convex", and having become the standard definition in this area of mathematics, people habitually take it out of context and hold that a "polytope" must by definition be convex. They fail to notice the elephant in the room - the title of Grunbaum's book! Along the way, they also developed a new definition for "polyhedron". Others of us, being more interested in pure geometry, continue to use the words with their original meaning. I have more than once been told by irate theoreticians that the "convex" definition is standard and that I should not mess with established mathematical definitions. Pointing out that they started it does not go down well. Somehow, we need to get all this explained tactfully. -- Steelpillow (talk) 18:29, 8 March 2008 (UTC)

It should be noted that rules exist to exclude potential hopefuls, [eg "No Dogs allowed"], rather than fanciful ones ["No Snakes Allowed"]. It then becomes best to describe the polytope definitions in terms of field of operation, and structural form.
I have spent some time considering what is going on with "polytope" etc, in that there is a large structure for which some parts and linkages represent "polytope" for different definitions. In the first level, there is an realm of figures which may or may not be polytopes (or things like cylinders, spheres etc). These are the "fields of operation".
Solids are things with surface and content, such as represented by models. The surface is a definite thing, since one can also consider nebular figures (like the gaussian distribution and wave-particles). Cylinders and Spheres are solids, for example. Much of the notion of 'convex polytopes' are to deal with solid polytopes without holes. (PG has this as 'glomous = sphere-like surtopes'. It is glomous multitopes for which
Starry figures allow the surface to cross. This makes a difference between the surface (gradiant of density), and the periphery (limit of points referred to). It should be noted here that replacing a figure by its periform (ie the shape of the perimeter as a solid), might loose the essential properties of the figure: the periform of the stellated dodecahedron is a kind of apiculated dodecahedron: the regularity of it is lost.
Blend figures are a representation of a hyper-space surface, flattened to fall in a lesser dimension. An example is a projection of a polytope. The Blend operator of Jonathan Bowers corresponds to finding missing cells of a blend figure: ie treating it as a flat polytope.
Tilings are also things made of polygons etc. If the space is infinite, so is the face-count.
Configurations are idealisms of points, lines etc, without necessarily having a realisation. The Desarges configuration is an example. It can be represented in terms of numbers where two=point, three=line, four=hedron &c. When one allows one=D-1, and zero = D-2, the thing becomes a dyadic polytope.
Wedges (or Norman Johnson "Polytope clusters"), are based on the drawing of a solid against a perpendicular space, so as to produce a wedge. In three dimensions, wedges occur in three forms (point, edge, face), where a triangle, line, and point are drawn into a vertical tip (triangle, line or face). The tip is used for punching holes of zero, one or two dimensions (pin, knife, press), while the "polytope cluster" is formed in how this receeds from the point to solidness.
Infinita, or figures with infinite regions, eg half-plane. There are some subdivisions, such as horospheric infinita (where the surface is Euclidean), vs hyperbolic. Norman Johnson's definition of polytope admits horospheric infinita, but not hyperbolic ones. One can inscribe surface features, such as the planotopes (which consist of a plane tiling, and half-space, such as the hexagonal tiling and the ground under it.
These are the structures.
(unnamed) A class of figures that have definite surface dimensionalities, but do not have dyadic descent. Examples include a sphere (1c+1h+1n), cylinder (3 faces, 2 edges, 0 vertices), cone (two faces, one edge, one vertex), etc.
Multitope is an mounting or joining of figures, with no definite top closure. That is, two squares joined by an edge is a multitope. Multitopes obey for example, Euler's Polyhedron rule, but can have more than one solid content.
Polytopes is an multitope with a top closure (that is, every element is part of the surface of a single higher-dimension element). For tilings and blend figures, one counts the space of the tiling as the higher incidence. Closure could, for example be the dyadic rule. Wedges (or polytope clusters) are also dyadic, except that the lower closure is not the usual nullitope of -1D, but the dimensionality of the tip of the wedge.
Many of the definitions for polytopes consist of selecting what kind of fields and structures shall apply, and trying to lumber these into a single sentence. Because there is so vast usage, it serves well to describe it in terms of variations.--Wendy.krieger (talk) 10:50, 23 December 2008 (UTC)

### Faces and things

Different parts of this article use "facet" or "face" for the same thing. Also, a 2-face is not usually called a "face" any more - I call it a "wall" but not everyone agrees. Again, we have have been refining our ideas in recent years and different naming schemes have come and gone - hence the present issues. I will try to come back later with some more detailed proposals. -- Steelpillow (talk) 18:29, 8 March 2008 (UTC)

The problem with these terms is that they were originally 3D specific, and there are several different ways of generalising the terms to higher dimensions. In 3D, face and facet mean the same thing. But in 4D, a 2-face serves an analogous role to an edge in 3D, so one way of disambiguating is to introduce the term "cell" to mean 3-face, while "face" remains 2-face. However, if one is to go to higher dimensions, eventually a more extensible terminology is needed, so in 5D, should we use "facet" while "cell", "face", and "edge" refer to lower dimensional elements? On the other hand, if one understands "face" to mean (n-1)-face, then a different set of terminology must necessarily arise, such as using "ridge" for 2-face in 4D: but in 5D and higher, should it refer to 2-face or (n-2)-face?
All of these schemes suffer from the problem of trying to retrofit terms after the fact, with various distinct concepts conflated into overloaded terms. Wendy Krieger's Polygloss may be better when it comees to a consistent, systematic terminology for discussing higher-dimensional polytopes (up to about 12D or so), but then it is far from standard.—Tetracube (talk) 04:48, 15 September 2008 (UTC)
It should be noted that the Polygloss is so far from the 'standard', because the standard is so far from natural meanings, while the Polygloss sides with the natural meanings. There is a particular danger in preserving the 'standard' terms, because the meanings are not the ones used outside the field. A cell, for example, is a solid tile in tiling: John Conway's "Game of Life" is cellular automation, where the cells are squares of a square-lattice. (ie a cell is a solid surtope in a tiling). Armies might face each other across a river (a line partially dividing areas). Cells have Walls and Sills, preserving the ll mnenotic.
Even newer meanings like 'ridge' suffer from the same problem. We are lead to suppose that the edges of a hexagonal tiles are 'ridges', even though the real implementation is a valley of grouter between ceranemic tiles. Of course, it supposes that ridges in four dimensions have a 2d crest, and a peak one-dimensional.
The Polygloss is as much a lingusitic work as a mathematical work. Words are carefully weighed outside the field, to see what the root meaning is. When I encounter new meanings, hiding under a hodgepodge of random idioms, a new word is made for it (cf hedrous / hedrid).
Terminology is indeed needed, but the Polygloss selects these by letting common words drift to the top (as in nature), rather than the bottom. It is this that makes it feel so far from the standard: the standard is so far from nature, and the polygloss corrects this.--Wendy.krieger (talk) 11:58, 20 December 2008 (UTC)

### Page organisation

The table of elements and names is generally applicable and should not be under the "Convex polytopes" heading. I haven't checked the main text content for similar issues. -- Steelpillow (talk) 18:29, 8 March 2008 (UTC)

### Cleanup attempt

Alright, I've decided to take a shot at cleaning up the article (specifically, the intro). Instead of trying to find a single formal definition that fits every possible usage of the term, I decided to go for a surveyor's approach, listing some of the attested uses of the term. (I think this may be the best approach; after all, we're writing an encyclopedia, not a math thesis). I've started a stub history section; hopefully someone (Steelpillow?) with more knowledge of the history of the term can expand on it. I'll work on rearranging some of the material in the main body of the article as well.—Tetracube (talk) 20:14, 30 September 2008 (UTC)

## Simplicial decomposition and convexity

The discussion of simplical decomposition applies primarily to convex polytopes, but in general not to star polytopes or other self-intersecting (non-simple) polytopes. Does anyone have any objections to moving this discussion to the Convex polytope article, and accordingly modifying the section on Different approaches to defining polytopes? -- Cheers, Steelpillow (Talk) 11:49, 25 February 2009 (UTC)

Done. -- Cheers, Steelpillow (Talk) 20:55, 26 February 2009 (UTC)
This was a bad idea. The simplicial decomposition section that was moved is not about convex polytopes, but about point sets that can be decomposed into simplices. This is a much more general family that is motivated by convexity but goes far beyond it. I think the changes should be reverted, but I'd welcome further discussion here first. —David Eppstein (talk) 21:05, 26 February 2009 (UTC)
I agree that simplicial decomposition appears to be much more general than convex polytopes. The definition it employs permits many objects that are clearly not convex.—Tetracube (talk) 21:33, 26 February 2009 (UTC)
I take your point that simplicial complexes are more general objects than convex polytopes. I moved the section away from the Polytopes article because polytopes are also more general than convex polytopes, but in a way which is not necessarily consistent with simplicial decomposition, for example in the matter of density (where simplices would overlap or be counted multiple times). I know little of the general ramifications of simplicial complexes, but they do at least appear to have greater affinity for specifically convex polytopes than for polytopes generally. For example Grünbaum addresses them in his Convex polytopes whereas he deliberately avoids discussion of non-convex polytopes. Perhaps someone who understands them better could edit down the section I moved to focus more exclusively on their arising from the decomposition of convex polytopes? Wherever it belongs, I do not think that it is here. -- Cheers, Steelpillow (Talk) 21:42, 26 February 2009 (UTC)
I think you are confused. In the decompositions under discussion, simplices are not allowed to “overlap or be counted multiple times” and polytopes are not equipped with additional density information. Rather, the point of this issue is the following: some people require a polytope to have a manifold boundary (as convex polytopes do) but the objects with simplicial decompositions are more general (for instance, the shape formed by gluing together two tetrahedra along a single shared edge has a simplicial decomposition, but does not have a manifold boundary). By attempting to merge a definition that is even more general than a commonly agreed on definition of polytopes (that is, simplicially decomposable sets) with a different definition that is less general (convex polytopes), you are creating a mess. I am going to revert. —David Eppstein (talk) 22:49, 26 February 2009 (UTC)
My point is not so much the suitability of the simplicial decomposition material for the convex polytope article, as its total unsuitability for this one. For example, as far as I can tell, the small stellated dodecahedron cannot be decomposed into a simplicial complex without creating new vertices, which AIUI is forbidden. The puative section begins, "Given a convex r-dimensional polytope P, a subset of its vertices containing (r+1) linearly independent points defines an r-simplex". Now if that is compatible with star polytopes and does not specifically restrict itself to convex polytopes, then I am truly confused. If there is any reason to retain the section in this article, please let us know what it is! Whether or not you find anything appropriate to the convex polytope article, I am happy to leave to you. -- Cheers, Steelpillow (Talk) 14:13, 27 February 2009 (UTC)
BTW, on another point I have posted a question on your talk page -- Cheers, Steelpillow (Talk) 14:24, 27 February 2009 (UTC)

Let's try again with a really really simple example. A bowtie shape formed by two triangles joined corner to corner (⋈) forms an example of a two-dimensional polytope under the simplicial decomposition definition. It is however, not even close to being convex. Therefore, your insistance that simplicial decomposition has something to do with convexity is baffling to me. The shapes that have simplicial decompositions are polytopes, but not in general convex polytopes; therefore, they belong here, not in the convex polytope article. By the way, nothing in the definition prohibits the introduction of extra vertices. The small stellated dodecahedron doesn't fit into this definition for a different reason: it has interpenetrating faces, which the simplicial complex definition doesn't really model. But if one thinks of a polygon as the shape enclosed by some piecewise-linear boundary, the small stellated dodecahedron doesn't count as a polyhedron. The point is that there are different incompatible definitions of nonconvex polyhedra, and your insistence on a single one is detrimental to the article. —David Eppstein (talk) 21:18, 28 February 2009 (UTC)

OK I misunderstood the niceties of why star polytopes cannot be decomposed, but I got the answer right. I have qualified the paragraph accordingly. -- Cheers, Steelpillow (Talk) 22:37, 28 February 2009 (UTC)
It's probably more important to understand that a polytope exists firstly in "simple space" (ie d=0 or 1). A cloud is something that has volume without surface (such as the sort of wave-particle thing, or the gaussian curve). A solid has a definate surface, made of solids of lesser dimension variously incident on the content of a solid (eg sphere, cone, cylinder). A solid is a solid bounded by flat faces. From this you can derive the dyadic rule, for convex polytopes, Euler's rule, etc.
When one carries polytopes into new kinds of spaces, or into new areas of mathematics, one preserves some of the meanings as continues to make sense. Allowing the surface to cross, creates two kinds of surface (surface + perimeter), the content still remains the moment of surface, but regions of space are counted several times (density). Surface, for intersecting itself, no longer separates inside (1) from outside (0): perimeter does this.
When one considers polytopes as a kind of incidence diagram, (vertex connects to edge etc), then one has configurations. These become 'polytopes' if there is a meaningful top incidence to the thing: some sort of 'closure' for example. Figures like the desarges or fano configurations have polytope-like incidences.
With the definition of content = moment of surface, it is possible to calculate the content of the complex polytopes (since moment and centre can be found), and so even though the surface does not divide (ie contain), there is content and density.
You get compounds vs composite. A compound is a composite of several polytopes taken as one. That is, there is meaning in the total that is not present in the sum. Of course, there are many places where one has to do a test or surtope-walk to decide if there is a compound or not, and some figures admit compound faces. Its marginal whether compounds are included or excluded in polytopes. They sure look like polytopes. --Wendy.krieger (talk) 07:55, 27 July 2009 (UTC)

## Origins

The History section currently states that, "The discovery of quaternions by William Rowan Hamilton in 1843 and octonians by Arthur Cayley in 1845 prompted the interpretation of ordered sequences of 4 or 8 numbers as Cartesian coordinates, thus opening up the possibility of geometry in higher dimensions than 3.". Is there any sound basis for this view? If these dates are correct, Schläfli described his higher polytopes before Cayley published his discovery of octonians. Meanwhile, quaternions involve three imaginary dimensions which behave differently from ordinary ones and it is not possible to construct ordinary polytopes in such spaces (complex polytopes, which inhabit only one imaginary dimension for each real one, were not discovered until the 20th century). OTOH, if Schläfli did not get the idea from quaternions, then where else? -- Cheers, Steelpillow (Talk) 12:06, 25 February 2009 (UTC)

Actually, I'm rather unclear about the true origins of higher-dimensional geometry, save that it happened sometime in (or possibly before) the early 19th century. When I wrote that sentence, it was merely to trace out a general outline of the development of the subject in the hopes that somebody with better knowledge would come along and fix it up.—Tetracube (talk) 16:33, 25 February 2009 (UTC)

OK, I have given it a makeover based on the historical information in Coxeter's "Regular polytopes". -- Cheers, Steelpillow (Talk) 10:31, 26 February 2009 (UTC)

## Table

What about include a table with a list of polytopes; like this one i made :

```Polytope families [replaced by link to save bandwidth -- Cheers, Steelpillow (Talk) 12:37, 13 February 2010 (UTC) ]
```

Mateus Zica (talk) 12:54, 6 January 2010 (UTC)

This table is a nice idea, but I am not sure that it is quite right yet. It only lists a few special families of polytope (though it will probably grow a few more more, such as the duals of some of the existing families). So:
1. It might be better in its own article, say Polytope families or similar.
2. I think the images should be smaller, to allow visitors with small screens (netbooks, etc) to see more columns.
3. If it grows a lot more types, say generic entries for regular, quasiregular, etc. families, then it mght be better to put dimensions along the top and families down the side.
-- Cheers, Steelpillow (Talk) 15:23, 6 January 2010 (UTC)
Hello Steelpillow!
I don't have a great knowledge in polytopes and all of its families. To tell you the truth, i made this table to help in my process of learning. Can you help me find the other families of polytopes ? I will include these new families with pleasure. Can you help me with possible mistakes and erros in this table? Thanks in advance!!!
--Mateus Zica (talk) 16:22, 6 January 2010 (UTC)
HI Mateus Zica! A very nice table. I helped get most of these graphs on Wikipedia, so I'm proud to see it! I'm also not sure where it might belong. The article List of regular polytopes is a good place too, only counts the first 3 families, but also included regular star polytopes, and tessellations. Petrie polygon also has all of these, I assume you noticed! Certainly they all belong under uniform polytope, and that article is underdeveloped, more on terminology. Tom Ruen (talk) 20:23, 6 January 2010 (UTC)
I don't think uniform polytope is the right place. Many of those have wacky symmetries (e.g. icosahedral) that have no equivalent in most dimensions, leaving big gaps in the table. Also, if we include duals or generic entries (i.e. a table entry for a sub-family of several polytopes), they may not all be uniform. It really needs a page creating for it. -- Cheers, Steelpillow (Talk) 22:07, 6 January 2010 (UTC)
I added the Coxeter-Dynkin diagram symbols, removed the infinite tessellations. The Coxeter-Dynkin diagram symbols are sufficient to generate nearly all uniform polytopes, except a small set of ones without reflection symmetry. Tom Ruen (talk) 21:19, 6 January 2010 (UTC)

### Templating, navboxes, etc

My broadband is beginning to struggle with the mass of images. Might be worth structuring it as a set of collapsible navboxes. That would avoid loading the mass of images all at once, as you would just open the box(es) you are interested in.

Darn. Still loads the lot, even when collapsed.

Could this table and the similar listing in Petrie polygon be made identical? If so, then perhaps it should be templated. We might be able to design a set of navbox templates that can be used to show individual blocks on other pages, or all together for this list.

What do people think? -- Cheers, Steelpillow (Talk) 22:02, 6 January 2010 (UTC)

I have changed the class of the table : from "collapsed" to "autocollapse". Please someone verify if it stills automatically loads the images even when its collapsed. (i can't do that because broadband is fast)

Thanks. Mateus Zica (talk) 23:23, 6 January 2010 (UTC)

No, it still loads all the graphics before collapsing. I tried using a navframe instead, but same problem. Looks like we're stuck with the graphics load. -- Cheers, Steelpillow (Talk) 14:17, 7 January 2010 (UTC)
Since this table contains all the families given in the Petrie polygon article, I replaced those family sections with this unified table there. There was minimal other content in those sections (which I wrote!). Tom Ruen (talk) 23:56, 6 January 2010 (UTC)
The family corresponds to (mostly) Gosset-Elte polytopes, and more specifically, the various "heads of symmetry" of the different groups. The remaining ones shown below: E7 i think is 2_31, 24_ch, 120_ch and 600ch are heads of other symmetries (along with the 3d icosa + dodeca), which would greatly extend the table.
The set as shown above are the "trigonal groups" (ie of unmarked branches only), with the tetragonal group (3...4) shown for simplicity. These groups are closely related (corresponding to the simplex, with a branch at positions 1, 2, 3,)
The remainder could be shown under "higher order groups", with the pentagon dividing into the icosahedron / 500ch, and to the dodecahedron / 120ch. --Wendy.krieger (talk) 07:28, 7 January 2010 (UTC)

#### Transcluded page

Well, for now I have created Polytope families for it, and transcluded it here and on the Petrie polygon page. -- Cheers, Steelpillow (Talk) 15:32, 8 January 2010 (UTC)

Discussion on renaming Polytope families moved to Talk:Polytope families. -- Cheers, Steelpillow (Talk) 17:56, 8 January 2010 (UTC)

### Unknown Family (for me)

Can this polytopes be included in this table?

Mateus Zica (talk) 01:31, 7 January 2010 (UTC)

Status report:
• 1 22 is already in the table, it is the 1k2 of dimension 6
• F4 appears to give the 24-cell, which does not seem to belong in any of the families listed.
• E7 is already in the table, as the 321 polytope: the k21 of dimension 7
• 24-cell is another image of the F4
• 120 cell has a different symmetry from any of the families listed
• 600 cell has the same symmetry as the 120 cell
Others with different symmetries include most regular polygons, the icosahedron and dodecahedron and many other semiregular polyhedra and semiregular polychora.
We must be careful what families we want, as there is not room enough for all of them.
-- Cheers, Steelpillow (Talk) 14:44, 7 January 2010 (UTC)

1_22 is included under E8, 1_k2 column. See [[1]] F4 (24-cell), H3,4 (dodecahedron/icosahedron/120-cell/600-celll) columns for these "special symmetries". They could be added. Tom Ruen (talk) 20:46, 8 January 2010 (UTC)

Okay, I expanded at Polytope families. I reduced the image sizes, but didn't help much due to wide CD diagrams. Tom Ruen (talk) 20:58, 8 January 2010 (UTC)

## Citations

This article has a "citations missing" tag. It has a sensible set of references, considering that many topics have their own main article. To my eye, the main content does not need any inline citations, as it is pretty uncontroversial stuff and findable in the existing references & links. Does anybody have any objections to removing the "citations missing" tag? -- Cheers, Steelpillow (Talk) 12:37, 13 February 2010 (UTC)

No, go ahead. Such an article on a broad, general area of science, in particular maths, does not need every point citing, as per WP:SCG. It looks well referenced to me by this standard.--JohnBlackburnewordsdeeds 13:01, 13 February 2010 (UTC)
Done -- Cheers, Steelpillow (Talk) 21:19, 18 February 2010 (UTC)

## Standard spaces

I think that most of the section on standard spaces, that has been recently added, is out of place in this article. It seems to have been copied from the other pages, and a one-liner with one or two links is all that is needed here. Also, no reference has been given for the term "standard space". Opinions? -- Cheers, Steelpillow (Talk) 21:06, 25 February 2010 (UTC)

Yes, it's not something I've seen before but thinking about it it's rather trivial: of course you can map e.g. a 7-dimensional simplex with 8 vertices to the cube in three dimensions, and as every two vertices are joined you also get every edge, every face etc.. With more complex polytopes even just in three dimensions you need ever higher dimensional and so more abstract simplexes; but the map is not one-to-one so you can't turn the polytope into a simplex. So it doesn't tell you anything about the polytope.--JohnBlackburnewordsdeeds 21:23, 25 February 2010 (UTC)

## Coxeter-Dynkin diagram graphics

If anyone feels able to contribute, please visit the discussion over SVG vs PNG formatting for these diagrams. We are trying to establish a consensus to end a reversion war, and there are literally hundreds of instances to sort out. 83.104.46.71 (talk) 19:05, 11 March 2010 (UTC)

## Triacontakaiditeron

I think Triacontakaiditeron should redirect to 5-orthoplex. --84.61.182.248 (talk) 17:16, 22 December 2010 (UTC)

And Hexacontitetrapeton should redirect to 6-orthoplex. --84.61.182.248 (talk) 18:18, 22 December 2010 (UTC)

## Naming polytopes in higher dimensions

Hi, I am the author of the note that Octahedron80 cited and Arthur Rubin felt is "not usable as a reference; appears to be the author's original research, without much indication of the author's identity or expertise".

First, I'd like to point out that wp:or applies to Wikipedia content not references - the whole point of many papers is to document the author's original research. That's why they get referenced - it's what references are for. Using wp:or to criticise such references is simply mistaken.

Second, my note acknowledges help from among others Prof. Norman Johnson (of Johnson solids fame among other things), and George Olshevsky who also needs no introduction.

My identity is simply exposed by following the link at the top of the note. Since the note is published on my website, I felt that confirmation enough. Ah, well, I guess that one man's obvious is another man's one-click-too-many-away.

My expertise must be left for others to judge, but again as evidence I offer my website, together with some of my peer-reviewed papers I reference and reproduce there.

I would not comment here, except that I feel the reversion of Octahedron80's well-meant edit should really have been discussed first - but then I have to declare an interest don't I ;-)

The edited page before reversion is here and its diff is here

— Cheers, Steelpillow (Talk) 21:46, 14 January 2011 (UTC)

Thanks Guy. Your writeup at [2] is excellent for showing the origins and development of the terminology for dimension-dependent polytopes. The link to your page was also added to 5-polytope, and equally reverted there. However it was already given in a intro citation note [3] there as well. Anyway for discussion the central section should at 5-polytope#A_note_on_generality_of_terms_for_n-polytopes_and_elements which I wrote up originally. These names are used exclusively in the context of uniform polytopes, although polychora may be used more widely, like hendecachoron. I've mostly tried to keep most usage within uniform polytope terminology more generic. The simplex family is where it is most used as a suffix: pentachoron (5-faceted polychoron, same as 5-cell and 4-simplex), hexateron (6-faceted polyteron, same as 5-simplex), etc, but n-simplex can always be used. But I have no printed sources, and Norman's book on uniform polytopes is yet to be published. Tom Ruen (talk) 22:34, 14 January 2011 (UTC)
I tried to find the the name of the author of the site, and got a 404 after 3 clicks from the page heading where I thought information about the author should be, and that page doesn't attribute the names to George. He may need no introduction in the field, but he does need an a reliable source naming him for the attribution to be listed. — Arthur Rubin (talk) 01:34, 15 January 2011 (UTC)
Hey, thanks for alerting me to the 404! I'll fix that when I get a moment. Since my note is getting so much attention, I'll also see if I can tidy up some of the issues you raise. Hope nobody's in a hurry, I have some more urgent stuff to do first. Meanwhile I am puzzled why you have a problem with attributability to Olshevsky. My note contains a paragraph which explains all this. Are you perhaps concerned that I did not provide a reference? That was not possible - I was giving the first coherent published account of it. — Cheers, Steelpillow (Talk) 16:19, 15 January 2011 (UTC)

I don't know what you are talking about. I just have seen some example out there that used the words (2006) BEFORE here words were tagged (2010). "Original research" means that a Wikipedian newly creates something which has never been seen somewhere before; SO it is not in the case. I am pointing out that I just refer to the secondary source. If you don't accept this policy, delete those words as well (in every related articles also). --Octra Bond (talk) 09:37, 15 January 2011 (UTC)

To begin with, I apologize for not recognizing "Guy" as being sufficient (self-)identification as a recognized expert. (It should be pointed out that Bowers is apparently not recognized by the project as an appropriate expert, at least for naming conventions.) Nonetheless, the article attributes two different naming systems, one to George, and one to Wendy, without indicating which is primary, although the table seems to be Wendy's. If the reference is considered reliable, both naming systems should be included, where they differ. — Arthur Rubin (talk) 16:11, 15 January 2011 (UTC)
No need for an apology over my name - I never wrote the note with formal referencing in mind. You remark on two other unrelated issues: whether Bowers' system should be used on Wikipedia, and whether my note suffers shortcomings of its own. If you feel that it is time to embrace Bowers, that is a different issue to be pursued under its own topic. If you feel obliged to peer-review my note, that again my be pursued elsewhere - I would be happy to correspond. — Cheers, Steelpillow (Talk) 16:47, 15 January 2011 (UTC)
No, I don't feel your note has shortcomings, actually. However, Octohedron used it to source George's naming system, while you seem to prefer Wendy's, even though it doesn't go beyond 9-topes. If your site is considered a valid reference by Wikipedia standards, we should include your preferred naming system. — Arthur Rubin (talk) 18:03, 15 January 2011 (UTC)
Didn't he use it to source Wendy's (as extended by me and used on Wikipedia)? Which reminds me I need to add polyyotton (or polyotton?) to my list, but I personally would get rid of polyxennon - "Xenna-" has not been ratified for SI and has several competitors, see Non-SI unit prefixes. When SI does get a 1027 prefix, we will probably have to discard polyxennon anyway. — Cheers, Steelpillow (Talk) 19:00, 15 January 2011 (UTC)
p.s. Richard Klitzing's enumerations of the uniform polytopes includes up to 10D (polyxennon) [4], and names up to 13D as polyudekon, following a reverse alphabetic sequence, similar to some proposed SI extentions. Tom Ruen (talk) 22:39, 15 January 2011 (UTC)
The earliest web reference to my names (teron to yotton) is [5], which is the 2003 version of the polygloss [6]. I suggested the use of metric prefixes like zetto- to get around the apposition of two numbers (eg twenty four-D vs 24-D). Metric prefixes are meant to be apposed to number.
The names were advanced and eventally adopted in a private list that a number of people in this list, such as Norman Johnson, George Olshevsky, and Guy all read. The polygloss was written one christmas day (2002), with the intent to simplify the revised notation that these form part of.
Since the names are advanced by others, rather than me, it is hardly original research, but rather a recommendation to use this rather than the more generic set of names. It's rather like the name of new elements, but we haven't set down to decide the 'winner' yet. The PG does list George's names under [7], along with the full, but never used, thousand-rule scheme. I have not seen Dr Klitzing's extensions past my naming, or even Guy Inchbald's referenced paper. My recollection is that George's list derives from mine.
Choron is from GO *chemera (room), simplified by NWJ. I had no reason to alter this name, so it is assumed without modification. However there were no names for the higher stuff.
W --Wendy.krieger (talk) 09:02, 16 January 2011 (UTC)
Thanks for the clarification, Wendy. It seems to me that the 2- to 9-dimensional names are constant across the various lists (mine is here), though there is the odd typo here and there. But the others are still in flux, so are best referred to as n-poyltopes, possibly with references to some names used by notable authors (and I think we need to be vary careful about who might be notable). I seem to recall someone introducing "polyxennon" just to tidy up the set of 10 dimensions deemed worthy of their own articles. But I find this artificial and unjustified by the facts. So: -
I have no idea why the 5-polytope article was left in such a state for so long, but I've done my best to correct it now. ~ Keiji (iNVERTED) (Talk) 17:13, 9 March 2011 (UTC)

### Proposed demotion of "polyxennon"

Since the term polyxennon is not widely accepted and breaks the "SI prefix" convention that does seem to be becoming accepted, I propose that we move the Uniform polyxennon article to Uniform 10-polytope and confine "polyxennon" to redirects and in-page references.

Votes
• For — Cheers, Steelpillow (Talk) 11:54, 16 January 2011 (UTC)
• Support the move. I don't see why we even need to have in-page references to the non-widely-used name. —David Eppstein (talk) 17:06, 16 January 2011 (UTC)
• I have no grounds for an objection. I did my best to set up the articles to not depend on any specific names. The uniform n-polytope listings were stopped at 10 simply as a round number. Originally it was under 10-polytope, but I moved because the contents were focused entirely on the uniform ones. Tom Ruen (talk) 23:40, 16 January 2011 (UTC)
• No Comment I had technical difficulties with this, since all of x, e, n are in use elsewhere. We have, from n=-1 to -8, -1n 0v 1e 2h 3c 4t 5p 6x 7z 8y. It would be acceptable if one puts -1w (wessian = existance), then 9n xenno- would indeed work. I suppose this would get around all those people who think of polytopes made of things as well. Eventually, we're going to head faster than the SI prefix rule, in any case, So we need to address issues at hand. --Wendy.krieger (talk) 07:06, 17 January 2011 (UTC)
• You write as if we're free to make up notation here as a way of promoting it elsewhere. We're not — Wikipedia is a follower, not a leader. See WP:NOR and WP:SOAP. —David Eppstein (talk) 07:52, 17 January 2011 (UTC)
• We're not actually making up new notation. Polyteron in my meaning is now the top six pages of google, except for a sole reference in spanish to the drug. The particular debate here is whether we should accept 'xennon' from list B (Bowers-Olshevsky) into list A (Krieger). Bowers-Olshevsky do follow my list as far as it goes, and then uses a different rule of number contraction to go higher. An alternative would be to use the rule of Wilberforce Mann, which goes past this dimension, and also meant to stand with numbers.
• Since the PG is as much about language use as it is about mathematics, one needs to consider the linguistic issues and clashes of meaning, when adopting new terminology. What I listed above is in the PG under 'chevn', 'thousand-rule' (on th/þ), as well 'polytope name'. The chevn-rule is about heading up columns with letters representing dimensions, eg c3 h2 e1 v0 n-1, or c4 e3 h2 v1 n0, according to the rules of multiplication.
• So my comment is more about what is needed to bring a list B name into list A. Note PG lists both A and B under thousand rule entries. --Wendy.krieger (talk) 07:59, 18 January 2011 (UTC)

Well, having thought about this a while I think there is a clear mandate to expunge "polyxennon" from article titles and if need be section titles. The term may be used in the text where its origin needs to be cited. If any wathcer of this article wishes to contest this, please confine your arguments to Wikipedaia policy/guidelines and post below or on my talk page. — Cheers, Steelpillow (Talk) 18:52, 24 May 2011 (UTC)

I added a move request at Talk:Uniform polyxennon. Tom Ruen (talk) 19:44, 24 May 2011 (UTC)

## polyteron, etc

Much of this new notation stems from my constructions over a decade ago, eg in the polygloss, and also in "Symmetry: Culture and Science" paper "Walls and Bridges: the view from six dimensions". While i doubt that 'google scholar' would show many results, the regular google search shows many results on this word. The whole point behind the polygloss was to make things easier for people to visualise the higher dimensions, by removing meanings of words that do not continue to have that force in the higher dimensions. In any case, the terminology has been adopted by other sites, for example, George Olshevsky.

--Wendy.krieger (talk) 07:23, 1 July 2011 (UTC)

Wendy, I wish this terminology could be recognized on wikipedia. I agree 'google scholar' ought not to be the singular online standard for geometric terminology. Norman Johnson suggested polychoron with George Olshevsky for 4-polytopes, but John Conway hasn't expressed anything. George Olshevsky has published information on the convex uniform polychorons, but nothing on his archived website for higher dimensional names, even if in private. Guy Inchbald has a good summary of the subject at [8], and Jonathan Bowers goes a bit wild with his fun attempt to systematic naming [9]. We have your list at Polygloss [10] which along with Jonathan, diverges from the SI prefixes to say polyectons for 7-polytopes. Who sets the standards and what prevents you from arbitrarily deciding next week to change the terminology again, and which version should wikipedia support (even assuming self-published sources were acceptable)? At least in the days of printed books, there was a fixed edition to reference, and even if corrections were later attempted, if usage takes off, authors lose control to "correct" the terminology they created and their first publishing. It's messy, but the real work I think must be to get active geometers to use the language and publish with it, and then we have something to cite. Tom Ruen (talk) 18:58, 1 July 2011 (UTC)
Have to support Tom's view here. Wikipedia is quite strict about this. Several of my ideas cannot be discussed here for the same reason. — Cheers, Steelpillow (Talk) 19:09, 1 July 2011 (UTC)

## Facets

The disambig page Facet (disambiguation), recently had some material cleaned out, and moved to Facet (geometry). I've now moved some remaining material there.

One item that was remaining at the disambig page, was the interwiki link to de:Polytop (Geometrie)#Nomenklatur - but that seems to be aimed here. Possibly this will help someone with ideas of further work, or pages that need to be merged or integrated. –Quiddity (talk) 19:38, 8 June 2013 (UTC)

## Body, facet, ridge

I notice Norman W. Johnson in his preprint Geometries and Transformations, defines the words scope, chamber, and wall in topological parallel to body, facet, and ridge respectively, in regards to n-honeycombs in n-space. I've not read previously on distinct terms for infinite vs finite forms. Tom Ruen (talk) 18:01, 27 February 2015 (UTC)

Just please don't start using them here until the wider mathematical community accepts his terminology. — Cheers, Steelpillow (Talk) 20:45, 27 February 2015 (UTC)
That's why I put it here, and perhaps these terms or others are referenced elsewhere as well. Tom Ruen (talk) 20:52, 27 February 2015 (UTC)

## Who coined the term?

We have a link to the person who "translated" the original term by simply adding an -e, but none to the person who actually coined the term in the first place. Unfortunately, I couldn't find a reference, and the German article de:Polytop isn't helpful, either. Does anyone know who this "Hoppe" (de:Hoppe) was? It's a common name, and there are a number of mathematicians with that name. — Sebastian 16:52, 19 August 2015 (UTC)

According to Coxeter's Regular Polytopes (Dover Edn. p. 317), Reinhold Hoppe is the one you are after. The Internet seems to think his biog. dates are 1816–1900. HTH — Cheers, Steelpillow (Talk) 17:59, 19 August 2015 (UTC)
We have articles on him in de:Reinhold Hoppe and nl:Reinhold Hoppe. As an important journal editor and member of Leopoldina he's clearly notable, so we should have one here too. The Dutch article says he's also known for proving that someone else's solution to the kissing number problem (how many unit spheres can touch a central unit sphere in 3d) was bogus, and then publishing his own solution which also turned out (many years later) to be bogus. Neither article mentions coining "polytope" but Coxeter can be used to source that as well. —David Eppstein (talk) 19:08, 19 August 2015 (UTC)
Coxeter's note on his coining the term is on Page iv of the Dover edition. — Cheers, Steelpillow (Talk) 19:23, 19 August 2015 (UTC)
That's weird, my copy (the 1973 edition) doesn't have a page iv, and p.317 is just the index page that has Hoppe's name on it. The pages (144 & 165) that the index page links to do mention Hoppe, but in unrelated contexts. Instead the note on the coinage is on page vi.—David Eppstein (talk) 19:30, 19 August 2015 (UTC)
Ok, new article created. —David Eppstein (talk) 20:11, 19 August 2015 (UTC)
Wow, thanks a lot, that was quick! — Sebastian 20:33, 19 August 2015 (UTC)

## Introduction picture

The picture in the introduction is the same as the picture for polygons. This can lead to confusion. These two pages have similar titles and talk about similar things. the polygon page also links here, and visa versa. A person could get easily confused and think they are on the wrong page.The introduction should really have a 3rd+ dimension picture. — Preceding unsigned comment added by IllQuill (talkcontribs) 00:58, 19 April 2016 (UTC)

An n-dimensional picture would be most appropriate though sadly impossible. But I agree that something better than a few polygons would be welcome. (By the way, we don't top-post here. A new discussion should go below the earlier ones.) — Cheers, Steelpillow (Talk) 08:06, 19 April 2016 (UTC)