Talk:Polytope

message: Does anyone know who the 'I' is that remarked at the bottom of this article?
The history is lost with the change of software.
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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
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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)
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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)
thread_title: Mathematical nonsense

message: 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 R^k is defined to be the intersection of some finite number of half spaces in R^k.  Bounded polyhedra are called polytopes.  (A polytope can be definted equivalently as the convex hull of a finite point set in 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)
replies: text: 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)

text: 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)
thread_title:  half spaces & convex hulls 

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

message: 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)
thread_title: Page organisation

message: 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)
thread_title: Cleanup attempt

message: 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)
replies: text: 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)
thread_title: Origins

message: 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)
replies: text: Discussion on renaming Polytope families moved to Talk:Polytope families. -- Cheers, Steelpillow (Talk) 17:56, 8 January 2010 (UTC)
thread_title: Transcluded page

message: 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)
replies: text: 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)
replies: text: Done -- Cheers, Steelpillow (Talk) 21:19, 18 February 2010 (UTC)
thread_title: Citations

message: 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)
replies: text: 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)
thread_title: Standard spaces

message: 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)
thread_title: Coxeter-Dynkin diagram graphics

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

message: 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)
replies: text: I added a move request at Talk:Uniform polyxennon. Tom Ruen (talk) 19:44, 24 May 2011 (UTC)
thread_title: Proposed demotion of "polyxennon"

message: 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)
replies: text: 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 , and Jonathan Bowers goes a bit wild with his fun attempt to systematic naming . We have your list at Polygloss  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)

text: 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)
thread_title:  polyteron, etc 

message: 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)
thread_title:  Facets 

message: 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)
replies: text: 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)

text: 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)
thread_title:  Body, facet, ridge 

message: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)

text: Ok, new article created. —David Eppstein (talk) 20:11, 19 August 2015 (UTC)
replies: text:  Wow, thanks a lot, that was quick! — Sebastian 20:33, 19 August 2015 (UTC)
thread_title:  Who coined the term? 

message: 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 (talk • contribs) 00:58, 19 April 2016 (UTC)
replies: text: 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)
thread_title: Introduction picture

message: It seems the intro describes a polytope as a polytope. Quite clarifying. Madyno (talk) 09:37, 31 July 2017 (UTC)
replies: text: Yes, recursion is powerful, although knowing when to stop is the funny part. Tom Ruen (talk) 15:08, 31 July 2017 (UTC)
thread_title:  Description