Talk:Polyhedron

The terminology problems mentioned in the first paragraph should be moved to the second paragraph. The first paragraph should contain the concise definition used in this encyclopedia. Right now, it is not explained at all what a polyhedron is. --AxelBoldt

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I came on this with it saying only "convex lenses are cool", so I grabbed the earlier form from the diff and restored it as best I could. The table didn't seem to work very well, so I redid it using html table markup and it seems to me to work better this way. Feel free to tinker with it. -- User:Blain

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I need help about the English term shape. In this article it is written: A Polyhedron is a shape... What a shape really means? Since English is not my native and since I've started an article geometric shape I understand a shape to mean something planar, but here and also in many places a shape is used for bodies or solids. Slovene -- my native -- distinguishes between a shape (lik) and a body or a solid (telo or trdno telo). What term should we use here? I know from CADs there are solids. Is a non planar surface also a shape of some sort? Basically as I've learned geometry from my early days I know for geometric shapes (e.g. triangle, square, rectangle, ...) and I couln't find this term here, so I've made an article on it. Someone might think it is too trivial, but anyway. How can we talk about mathematical realities, if we still do not know how many Fermat primes are there -- and such. So I guess we still have to look back on simple roots of pure mathematical objects. And as professor Dragan Marušič recently stole words from my lips that we just discover anew things which are there already from the eternity. I think a solid might be just fine, since it is used commonly in geometry (and CADs) and a body is used for example much more in astronomy, physiology and in related fields. Best regards. --XJamRastafire 01:44 Jan 24, 2003 (UTC)

Shape can be either planar or three dimemsional. -- User:Karl Palmen 3 Oct 2003 (UTC)

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I think we are in deep trouble here with the definitions. The Kepler solids are supposed to be regular polyhedra, but that only works if the terms "face" and "vertex" are properly understood. Not all intersections of edges apparently are vertices. How should one define polyhedra so that both the "normal" polyhedra and the Kepler solids fit the bill? AxelBoldt 18:23, 2 Oct 2003 (UTC)

Agreed. The trouble is that of having two viewpoints with regards to classical polyhedra which generalise in different directions:

• You can say that a polyhedron is a bounded solid body such that its boundary is made up of planar facets (e.g. the boundary is contained within a finite union of affine subspaces of codimension 1.) Then the "general polyhedron" definition is fine - something in the algebra of sets generated by half-spaces. (Actually there is still a slight problem, in that "has flat sides" stated in the article does not imply the finiteness condition.)
• You can say that a polyhedron is an arrangement of vertices, edges and faces in space with some combinatorial relationships, such as a a 2-to-1 surjection (ends of edges) -> (vertices of faces) and a 2-to-1 surjection (edges of faces) -> (edges of polyhedron). (Sorry, this suboptimal --- off the top of my head, but I was trying to remember how to set up simplicial geometry and then generalise it.) This case can be generalised to include polygrams (stars) as the sides and thus fit in the other regular and uniform polyhedra as generalisations of the platonic and archimedian polyhedra.

So yes, some modification of the definition to include two viewpoints would be useful. Something for my todo or some other brave soul.

Also I think the section on "Topological polyhedra" sucks somewhat. WTF is being defined here? I can't tell sufficiently well even to fix it up. Is it referring to a construction like a simplicial complex?? If so, I think "Topological polyhedron" is not standard nomenclature, but it is not my field.

-- Andrew Kepert 08:52, 6 Aug 2004 (UTC)

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We have a big problem in cartography. A flat piece of paper cannot be curved to cover a sphere exactly without some stretching or wrinkling.

Is there a general term for the sorts of shapes that paper *can* cover without stretching or wrinkling ?

In other words, I'm looking for terms to fill in these blanks:

• A polyhedron (such as the pentagonal pyramid) is made of flat plates (facets) (in this case, triangles and a pentagon) stitched together.
• A __________ (such as the quonset hut) is made of constant-curvature cylinders or planes (in this case, 2 half circles and 2 rectangles) stiched together.
• A __________ (such as the cone) is make of ____ surfaces ( developable surfaces ?) (in this case, a circle with a sector cut out, and another circle) stitched together.
• A __________ (such as the sphere) is made of ___ surfaces (such as Nonuniform rational B-splines) stitched together.

--DavidCary 20:29, 25 Jan 2005 (UTC)

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This article seems to imply that polyhedra are only used in some obscure, archaic branch of mathematics. It also focuses on "regular polyhedra" and "convex polyhedra", which are idealized shapes that have few practical applications. Most polyhedra are irregular and concave.

I want it to say more about how polyhedra are used all the time in Computer-aided design (in particular, Solid modelling), video games, etc.

I supposed all I want to say can be summarized as

The visual appearance of any physical object can be duplicated by a sufficiently detailed polyhedron.

How can I emphasize how important this is?

--DavidCary 20:29, 25 Jan 2005 (UTC)