Talk:Dual polyhedron

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The statement "If a polyhedron has an element passing through the center of the sphere, it will have an infinite dual." makes no sense, since it there is no definition (or even explanation) what "an infinite dual" means. No definition of "polyhedron" admits vertices or edges or faces "at infinity.

Generalize to dual polytopes?[edit]

Many of the concepts described in this article also apply to n-dimensional polytopes. Would it make sense to extend this article so that it applies to both? There are quite a good number of articles on 4-dimensional polytopes (see polychora and uniform polychora), which would make good use of a more general description of duals.—Tetracube 22:14, 17 February 2006 (UTC)

Good to add something! Choices?

  1. Rename to Dual polytope and expand with divided sections by dimension? (starting with regular polygons as self-duals)
  2. Create Dual polychoron article and move/expand the 4D/honeycomb content there?
  3. Keep Dual polyhedron, add Dual polychoron, and add Dual polytope with dimensional article references?

Tom Ruen 23:10, 17 February 2006 (UTC)

I prefer to put them together, since otherwise there will be a lot of needless repetition. So option 1 sounds good to me.—Tetracube 01:54, 18 February 2006 (UTC)

Sounds good! I wonder who else is watching?

Incidentally, there's LOTS of links to this article. I've found it useful elsewhere to link expanded by article/headers. In this case like dual polyhedron and dual tiling and dual polychora and dual honeycomb for dimensional subsections. That works well as long as headers are not changed! Tom Ruen 04:08, 18 February 2006 (UTC)

As these articles pad out, I think this begins to make for rather large pages. For example, consider listing all the interesting dual pairs in every dimensionality all on a single page! <shudder!>.
I'd like to keep this page fairly simple, and add a Dual polytope page for the general theory, e.g. to do justice to the duality of abstract polytopes. Dual polychora? well, I'd keep it as part of Dual polytopes until it all gets too big, then float it off on its own. -- Steelpillow 21:26, 4 June 2007 (UTC)

Consensus to rename to Dual polytope?[edit]

YES

  1. Tom Ruen 02:43, 18 February 2006 (UTC)
  2. Dshin 19:39, 23 March 2006 (UTC)
  3. SLWoolf (talk) 04:50, 28 October 2008 (UTC)
  4. Tamfang (talk) 01:10, 29 October 2008 (UTC)
  5. Tetracube (talk) 04:57, 29 October 2008 (UTC) — we can use dual polyhedra to explain to concept first, and then lead on to polytopes. This should keep the article accessible to the general reader.

NO

  1. Steelpillow 21:26, 4 June 2007 (UTC)

Please, no[edit]

There is much in this article at an elementary level, such as the important Dorman Luke construction. To bury this in a discussion of polytopes generally is IMHO unhelpful to the majority of students who wish only to find out about dual polyhedra. If you guys move it, I will have to pull the elementary stuff back here - so by all means move or copy across the relevant stuff, but please save me the trouble of recreating this article over. -- Cheers, Steelpillow (Talk) 20:30, 29 October 2008 (UTC)

What about putting the polytopes stuff (including any polytope-specific generalizations) at the end of the article, after the elementary stuff? The concept of dual polytopes is a generalization of dual polyhedra, after all. We can add a section, maybe entitled "Generalization to higher dimensions", and put the polytope-related material under it.—Tetracube (talk) 20:38, 29 October 2008 (UTC)
The difficulty here is that the theory of topological and abstract dualities is primarily worked out for arbitrary dimensionality, i.e. for polytopes generally. Here, it would be better to treat polyhedra as a follow-on topic. There is a Wikipedia policy which allows some repetition of material, where it enables each article to stand on its own. In the present case, the treatment of the common material would tend to diverge in the two articles - the one becoming a little simpler, and the other more advanced. -- Cheers, Steelpillow (Talk) 20:56, 29 October 2008 (UTC)
In passing, I'm amused at the term polytope-specific, since polytope is the general case and polyhedron is specific. —Tamfang (talk) 07:24, 30 October 2008 (UTC)

Generalization of the Dorman Luke construction to n dimensions[edit]

The Dorman Luke construction of the dual essentially makes use of the fact that the facets of the dual polytope are precisely the dual of the vertex figures suitably enlarged. For polyhedra, it does not really matter whether it's the vertex figure or its dual, at least not for regular/uniform polyhedra, since regular polygons are self-dual. In higher dimensions, however, this quickly becomes obvious (e.g., the vertex figure of a 24-cell is a cube, yet its dual has octahedral cells; similarly, the vertex figure of the 600-cell is an icosahedron, but its dual has dodecahedral cells). Even in 3D, however, we do see a subtle hint that it's not simply the vertex figures, but the dual of the vertex figures, that form the dual polyhedron: take the vertex figures of the cube, for example. They are triangles, but oriented in the dual position to the orientation of the faces of the octahedron. Taking their duals gives us the faces of the octahedron in the correct orientation.

By this, it should be obvious that the Dorman Luke construction of the polyhedral dual is easily generalized to higher dimensions: given an n-polytope, finding its dual amounts to finding the duals of its vertex figures. Since vertex figures are (n-1)-dimensional, we simply recursively apply this process until we reach the trivial case (polygons).

I'd add this to the article, except that I'm not sure if this violates original research or not. :-)—Tetracube (talk) 20:36, 6 August 2008 (UTC)

None of this is OR. However, one needs to be very careful in generalising - Luke's construction was originally described only for highly symmetrical polyhedra. Wenninger generalised it to some extent, but it does not apply to polyhedra in general. It is in fact a special case of the more general reciprocity (also called polarity or, incorrectly in this instance, projective duality) of any polyhedron about any sphere. Yes the Dorman Luke construction can be generalised to higher dimensions, but at this level of difficulty it is more productive to generalise polar reciprocation. FYI, see my essay on Vertex Figures. -- Cheers, Steelpillow (Talk) 21:21, 7 August 2008 (UTC)

Transverse Homonym[edit]

Has anyone noticed that all references, citations, and Wiki links to my transverse homonym are missing from these articles? Even his/her/their biography(ies) is (are) missing from WikipediA. Laburke (talk) 19:48, 1 November 2011 (UTC)

I guess that's because he was (IIRC) an otherwise undistinguished school teacher, whose construction was only ever published by other authors. — Cheers, Steelpillow (Talk) 20:54, 1 November 2011 (UTC)
Too bad, but people do seem to go on about his "generalization". Was he so undistinguished that they don't want to honor him by raising it to a "conjecture"? Perhaps his own schools were too undistinguished for that ;) As for school teachers in general, Eriugena was a distinguished philosopher yet he was killed by the quills of his students. Thanks for answering so quickly and BTW, I "see" polyhedra as central point(s) joining apices. It's a chemist sort of thing. Laburke (talk) 03:33, 2 November 2011 (UTC)
The generalisation of his construction seems to have been first described by Wenninger. Both Luke's original and Wenninger's generalisation are simplifications of construction methods in projective geometry that had been known for many years - specifically the polarisation of a polygon in a conic section (here a circle). The appealing feature of these simplifications is that they are suited to the classroom, not that they have any theoretical novelty. BTW, Socrates was a distinguished philosopher but killed himself because he was condemned to do so by his fellow citizens. — Cheers, Steelpillow (Talk) 22:42, 2 November 2011 (UTC)

self and not[edit]

A sentence in Square pyramid prompts me to wonder: What's the simplest convex polyhedron which has V=F but is not self-dual? —Tamfang (talk) 05:33, 21 May 2010 (UTC)

Gyrobifastigium looks like a good candidate. Tom Ruen (talk) 21:25, 1 November 2011 (UTC)
First off, I realised that the solutions must come in pairs - dualising any solution obtains a second solution (because V and F dualise to each other). I then asked around. Someone pointed out that in the table of heptahedra, the 4th from the left in row 5 and the 2nd from the left in the last row are dual solutions. I checked the rest, then the simpler hexahedra, etc, and his solution appears to be correct. — Cheers, Steelpillow (Talk) 22:30, 2 November 2011 (UTC)

Duality[edit]

I was quite surprised about the sentence "regular polygons are geometrically self-dual". The dual of a cube is a octahedron and I wouldn't consider these to be congruent figures - which is required in the definition of self-duality. Can anyone explain how congruence of polyhedra is defined in this case, so that it makes sense at all? — Preceding unsigned comment added by 128.176.180.52 (talk) 11:15, 12 February 2013 (UTC)

Regular polygons are self-dual, most regular polyhedra are not (only the regular tetrahedron is). Hope this helps. — Cheers, Steelpillow (Talk) 20:49, 12 February 2013 (UTC)

r12 . r22 = r02?[edit]

Under Polar reciprocation it says:

If r0 is the radius of the sphere, and r1 and r2 respectively the distances from its centre to the pole and its polar, then:
r12 . r22 = r02

Shouldn't the formula read: r1 . r2 = r02 in accordance with Inversive geometry#Circle inversion?

--Episcophagus (talk) 17:25, 2 December 2014 (UTC)

Yes indeed. Now fixed. — Cheers, Steelpillow (Talk) 17:46, 2 December 2014 (UTC)