Talk:Polyhedron

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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

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

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)

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)

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)

I have seen the term "Euclidean cone" used to describe stitching together finitely many sectors of the plane. The origin of each sector is identified as one point in the Euclidean cone, called the "cone point". The "angle" is the sum of the angles of the sectors. I'm not sure if this is standard usage, but it is the terminology we used in a topics course on surfaces that I took. Mark.howison 06:33, 17 December 2005 (UTC)

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)

What should we do with this line?:

Snub dodecahedron or

Why is the AKA commented out? Anton Sherwood 01:19, 27 December 2005 (UTC)

1 Technical level
2 Classical polyhedron and Uniform polyhedra
3 Relation with graphs
4 Strange polyhedron
5 Non-convex quasi-regular polyhedra
6 Quasi-regular and Semi-regular
7 New article and template
8 not a translation
9 Dual polyhedron
10 Geometric solid terminology. New pill bottle.
11 Bowers style acronym
12 Quasi-regular

[edit] Technical level

This article seems too dense, certainly initially. Regular polyhedron redirects here. I came looking for a place to redirect both regular solid and five regular solids; but the article as it stands would not be very helpful to the general reader needing those. Charles Matthews 07:17, 5 September 2005 (UTC)

I created a new subsection Regular polyhedra. I think Regular polyhedron can redirect there. (Regular polyhedra consists of 5 Platonic solids and 4 Kepler-Poinsot solids.) --Perfecto 07:13, 9 September 2005 (UTC)

[edit] Classical polyhedron and Uniform polyhedra

There. I've given the Classical section a major rewrite, hereby classifying all 75 uniform polyhedra. Most of the subsections should now be candidates to be stubs of new articles. The version I began with has several random musings. Uniform Polyhedra should be clear of those now, and should be easier to read than Mathworld. --Perfecto 07:07, 9 September 2005 (UTC)

[edit] Relation with graphs

With regard to the recent changes:

what is meant by isometric, what metric of the graph does this refer to?
to be restored: only tetrahedra have a complete graph: K4
there are many kinds of octahedra, e.g. a 6-sided prism, so without "regular" something else is needed
inserted "i.e." is not correct, because what follows is sufficient, not necessary

Patrick 21:51, 20 September 2005 (UTC)

Sorry the term should be isomorphic. I corrected it.
Is restore really necessary? Any tetrahedron is isomorphic to $K 4$ (and (topologically) to each other). Or do you mean the following?

Only the tetrahedron is congruent to a complete graph, which is called $K 4$ .

You're right. I've restored "regular".
If so, then "i.e., adjacent nodes have two common neighbors, and non-adjacent nodes have four common neighbors" is a weak. Tis stronger to write there WHY, by definition, a octahedron's graph is strongly regular. Compare:

A bee is an insect; It has three pairs of legs. and

A bee is an insect; i.e., a member of Class Insecta.

--Perfecto 23:55, 21 September 2005 (UTC)

A polyhedron has more structure than just the graph, so "isomorphic" does not seem correct.--Patrick 01:20, 22 September 2005 (UTC)

That is what the concept of "isomorphic" means. By mapping dog-leg to table-leg (a bijective mapping), a dog is isomorphic to a dining table. By mapping dog-leg to line-segment (this is also a bijective mapping), a dog is isomorphic to a square. A dog is more complex than a square, a square has corners, a dog has a left-front leg, but dog-leg:line-segment is a valid one-to-one mapping, so dog and squares are isomorphically related. I hope this clarifies the concept to you. I look forward to your comment on my other replies. --Perfecto 01:03, 23 September 2005 (UTC)

Tomo changed it, agreeing that the formulation was not right.--Patrick 21:58, 8 October 2005 (UTC)

[edit] Strange polyhedron

Hi all, someone just sent an image of an unusual non-uniform polyhedron. . It is formed by truncating the edges of a cube so that they form hexagons. Its not vertex regular as some vertices have one square and two hexagon, other vertices have three hexagons. Anyone know a name for it? Are their other such beast? --Pfafrich 12:10, 10 January 2006 (UTC)

I've not seen this before. 3 hexagons at a vertex means 3 faces will be co-planar! Looks like 6 squares, 12 hexagons: F=18, E=48, V=32.

Probably you can call this is a Uniform-2 polyhedron because it has 2 vertex types, like a square pyramid, but this one isn't a johnson solid because neighboring faces are coplanar. Tom Ruen 12:26, 10 January 2006 (UTC)

I don't think this polyhedra can be built with planar faces! A 4.6.6 vertex is a truncated octahedron. Once you have one square surrounded by 4 hexagons, this shape tried to put another hexagon between two existing ones. It won't work without bending them. Tom Ruen

This can be seen as a rhombic dodecahedron truncated at its acute vertices.

By the way, to emphasize that most polyhedra are not regular, we could work in some mention of [1] and [2].

--Anton Sherwood 17:33, 10 January 2006 (UTC)

Three hexagons cannot form a polyhedral vertex. See Platonic solid#Limited number of Platonic polyhedras for the proof. What your net forms is an imperfect rhombicuboctahedron. I appreciate your interest in the subject though. -- Perfecto 18:04, 10 January 2006 (UTC)

Three regular hexagons cannot meet and be strictly convex. (I'm inclined to be more liberal about coplanar faces, but that's beside the point.) Here, where three hexagons meet, their angle is 2 atan(√2) ≅ 109°; where two hexagons meet a square, the hexagons' angle is (π/2)+atan(1/√2) ≅ 125°. Assuming that the net is drawn accurately. --Anton Sherwood 01:49, 11 January 2006 (UTC)

It seems like it is a Truncated Rhombic Dodecahedron which has a few hits in google. The hexagonal faces are not pure hexagons one of the angle being aprox 110 deg. --Pfafrich 18:20, 10 January 2006 (UTC)

Indeed! VRML model for Truncated Rhombic Dodecahedron at:

http://www.georgehart.com/virtual-polyhedra/vrml/zono-7_from_cube.wrl

Looks like it can be "edge-uniform" at least, and hexagons are flattened in George Hart's model, while this net image LOOKS like regular hexagons.

Tom Ruen 22:02, 10 January 2006 (UTC)

Not even edge uniform, theres are square-hexagon and hexagon-hexagon edges. It is a zonohedron. --Pfafrich 22:29, 10 January 2006 (UTC)

You're right, I misused the term. I meant equal-length edges. Tom Ruen 00:02, 11 January 2006 (UTC)

P.S. Where did you get the "net" image. Was it meant only to be symbolic or really built?

I made a quick stub article for this polyhedron, and linked under zonohedron:

Truncated_rhombic_dodecahedron Tom Ruen 02:02, 11 January 2006 (UTC)

[edit] Non-convex quasi-regular polyhedra

This section currently says there are 13 such forms, and lists 14 of them -- but I count 15 in Wenninger (67, 68, 70, 73, 78, 80, 87, 89, 90, 91, 94, 100, 102, 106, 107; Coxeter # 36, 37, 39, 45, 51, 53, 61, 63, 64, 65, 70, 78, 81, 85, 86). Since the nomenclature differs I can't tell which one is missing, or have I goofed? —Anton Sherwood 19:57, 11 January 2006 (UTC)

90 not quasi-regular their are two different types of edges. Seems like we can't count, I'll fix the numbering. p.s. most has now moved to Uniform polyhedron. --Pfafrich 13:09, 12 January 2006 (UTC)

Ah thanks, I see now. I was misled by the vertex figure, drawn with the hex sides parallel when they ought not to be. --Anton Sherwood 17:39, 12 January 2006 (UTC)

Yes, W90 isn't quasiregular, but vertex figure looks correct - not symmetric. Image:Small dodecicosahedron vertfig.png. There a few of these weird ones (W74, W99, W101, W103, W109), vertex configurations go two steps forward, two steps back, ending with a figure-8 vertex figure polygon of ambiguous radius as computable by the vertex face arrangements alone. Tom Ruen 21:18, 12 January 2006 (UTC)

Yes, that's how the vf of W90 ought to look, a skew bowtie like those others you cite; but Wenninger shows it with four-way symmetry. (Perhaps it is corrected in later printings; I bought my copy in 1981 at the very latest, probably before 1976.) —Anton Sherwood 00:38, 13 January 2006 (UTC)

WOW! You're right. My book is wrong too (reprinted 1996!) I'll have to tape in a corrected image on the page! Another demonstration that independent generation is valuable! (Of course I never cross-compared my vertex figures systematically to the book diagrams, just figured they can't be wrong if mine actually generate the correct polyhedron!) Tom Ruen 01:58, 13 January 2006 (UTC)

[edit] Quasi-regular and Semi-regular

ON DEFINITIONS:

Quasi-regular if it is vertex-uniform and edge-uniform but not face-uniform, and every face is a regular polygon
Semi-regular if it is vertex-uniform but neither edge-uniform nor face-uniform, and every face is a regular polygon

I've never heard of Quasi and Semi regular being disjoint categories. I've always heard of all 13 Archimedean solids as semiregular.

http://mathworld.wolfram.com/SemiregularPolyhedron.html

A polyhedron or plane tessellation is called semiregular if its faces are all regular polygons and its corners are alike (Walsh 1972; Coxeter 1973, pp. 4 and 58; Holden 1991, p. 41). The usual name for a semiregular polyhedron is an Archimedean solid, of which there are exactly 13.

Tom Ruen 21:48, 12 January 2006 (UTC)

On Mathworlds uniform poly page they have a slightly more resrtictive def of semi-regular based on the vertex figure, rather than edge uniformity. See http://mathworld.wolfram.com/UniformPolyhedron.html By that def but the hemi-hedra are now versi-regular not semi-reg. --Pfafrich 03:23, 13 January 2006 (UTC)

I wasn't sure who tried making the strange categories. And he table with "nonconvex" semi-regular polyhedron counts also was an issue for me.

The Mathworld described categories look worthy.

Looks like I should get a copy of the book: (Hmmmm... can't even find it listed on bn.com or amazon.com!)

Johnson, N. W. Uniform Polytopes. Cambridge, England: Cambridge University Press, 2000.

Tom Ruen 04:12, 13 January 2006 (UTC)

For now I've moved the strange table on uniform polyhedron below. I don't think it is a clear or effective or even correct break down.

REMOVED FROM UNIFORM POLYHEDRA SECTION

Excluding the prisms and antiprisms the Uniform polyhedra can be organized as follows:

	Convex	Non-convex	total
regular	5 (Platonic solids)	4 (Kepler-Poinsot solids)	9
quasi-regular (excluding regular)	2 (some Archimedean solids)	14	16
semi-regular (excluding semi-regular)	11 (most Archimedean solids)	39	50
total	18	57	75

now corrected! --Pfafrich 09:36, 13 January 2006 (UTC)

[edit] New article and template

Hi all, I've just created a new article List of uniform polyhedra by vertex figure which shows some relations amoung the uniform polyherdron. It seems a slightly more systematic preseentation to me.

I've also created and infobox Template:Infobox Polyhedron Types which shows the different classes of polyhedron. We could posibly uses this on other pages as a navigation aid. --Salix alba (talk) 16:13, 17 January 2006 (UTC)

[edit] not a translation

The NL article linked -- nl:Afgeknotte hexa-octaëder -- appears to be much more specific than this. Why is it linked? —Tamfang 20:17, 1 February 2006 (UTC)

[edit] Dual polyhedron

Over on Wikipedia talk:WikiProject Mathematics they are discussing cron vs hedron

I wonder if these two Greek suffixes mean the same or almost the same thing. Then, the following redirects may make sense:

* Great dirhombicosidodecacron --------> Great dirhombicosidodecahedron
* Great dodecahemicosacron --------> Great dodecahemicosahedron
* Great dodecahemidodecacron --------> Great dodecahemidodecahedron
* Great icosihemidodecacron --------> Great icosihemidodecahedron
* Small dodecahemicosacron --------> Small dodecahemicosahedron
* Small dodecahemidodecacron --------> Small dodecahemidodecahedron
* Small icosihemidodecacron --------> Small icosihemidodecahedron

I stumbled into them at the Missing science project, and don't know what to do about them. Thanks. Oleg Alexandrov (talk) 18:55, 22 April 2006 (UTC)

I think the ones on the left are duals of the ones on the right or something. They should be given seperate articles. -- 127.*.*.1 20:33, 22 April 2006 (UTC)

Seems like we are missing much a lot of dual polyhedra. --Salix alba (talk) 22:37, 22 April 2006 (UTC)

Yes, probably ALL the nonconvex vertex-uniform polyhedron duals are missing. I don't know the source of the -cron naming, but see the value in it to show their relatedness to the vertex-uniform forms. The Catalan solid are the convex uniform duals, well along with infinite sets:Trapezohedron and Bipyramid. They're listed well at [3],[4] and [5].

I'd certain support an article Uniform polyhedron duals parallel to Uniform polyhedron, but must refrain myself for now.

I can generate the nonchiral uniform polyhedron duals with simple graphics. I've asked Rob Webb (who made the current wiki uniform polyhedron images by my request with his Stella software) if he'd make the duals as well, and might be persuaded to help. The duals are face uniform so only really can be singled colored. There some uncertainty of representation of the hemi duals which have faces that pass through the origin, so the duals have vertices at infinity which have different ways to draw something but aren't practically very friendly models.

Sorry I can't help more now, but I agree this stuff show be on Wikipedia! Tom Ruen 02:11, 23 April 2006 (UTC)

To answer Oleg's question, hedron means face (literally seat) and acron means vertex (literally high). —Tamfang 04:05, 23 April 2006 (UTC)

[edit] Geometric solid terminology. New pill bottle.

Q
What is the name of the geometric solid that is the new pill bottle used by Target Pharmacy?...
--dsaklad@zurich.csail.mit.edu 14:38, 9 July 2006 (UTC)
Image, click on... See what's new >
at
http://sites.target.com/site/en/spot/page.jsp?title=pharmacy_home
See also
http://www.google.com/search?q=%22target+pharmacy%22+%22deborah+adler%22

Um, well, if not for the rounded corners I'd call it a wedge. Why? —Tamfang 05:08, 9 July 2006 (UTC)

Q
Thank you! It's a topic of conversation. If that were a flat side, not curved and so six sided
what would the geometric solid be called that is the new pill bottle?...
--dsaklad@zurich.csail.mit.edu

an irregular hexahedron —Tamfang 18:09, 9 July 2006 (UTC)

[edit] Bowers style acronym

Both Wikipedia:Articles for deletion/Bowers style acronym and Wikipedia:Articles for deletion/Jonathan Bowers are up for deletion. --Salix alba (talk) 19:07, 14 July 2006 (UTC)

[edit] Quasi-regular

Could someone please improve this article? Towards the begining it states that quasi-regular polyhedra are edge- and vertex-uniform, but then later it makes a statement about how the icosidodecahedron cuboctahedron are quasi-uniform with the additional property of being edge-uniform. This seems contradictory. Please help. superscienceman 18:26, 3 October 2006 (UTC)

I changed the quasiregular definition to only edge-uniform, since there's two classes vertex-uniform and face-uniform polyhedra, which are actually duals of each other. I agree it needs further work. Tom Ruen 22:28, 3 October 2006 (UTC)

I also removed contraint that semiregular can't be edge-uniform, since all of the Archimedean solids, prisms, antiprisms can be considered semiregular. Tom Ruen 22:33, 3 October 2006 (UTC)

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