Talk:Polytope

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Mathematics rating: B Class High Priority  Field: Geometry

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)

Contents

[edit] 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 Rk is defined to be the intersection of some finite number of half spaces in Rk. Bounded polyhedra are called polytopes. (A polytope can be definted equivalently as the convex hull of a finite point set in Rk).

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)

[edit] 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)


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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)