Talk:Centered polyhedral number

Sources

I can find no use of this phrase on ZMATH or Google Scholar. Is there a reliable source for it? Spectral sequence (talk) 06:40, 28 July 2013 (UTC)

I added a source. That source doesn't use the term centered polyhedral number explicitly. Chapter 2.6 in the book is titled Centered space figurate numbers, while for the regular cases the term centered regular polyhedron number is used. So I think the title of the article is probably okay. -- Toshio Yamaguchi 08:02, 28 July 2013 (UTC)

I don't see why you think it's OK. We still have no source using this phrase and on your own showing you concocted the phrase yourself out of two other phrases, which seems to be original research by way of synthesis. Let me ask another, related, question. Is there a reliable source that dicusses this concept, namely "figurate numbers, each formed by a central dot, surrounded by polyhedral layers with a constant number of edges", under any name at all? Or is it a concept that you invented and thought of a name for? Spectral sequence (talk) 08:56, 28 July 2013 (UTC)

Chapter 2.6 in the book begins

"In this section we consider space figurate numbers, formed by a central dot, surrounded by successive polyhedral layers ...."

which is the concept described in this article. Also, I do not think that WP:SYNTH applies to this article, because I see no combination of material implying a conclusion that is not in the source. The issue here is just the title of the article. That title is simply the result of applying a well established naming scheme for polytopes to those numbers. I don't have any doubt that the term polyhedron is the established term for three-dimensional polytopes. So simply applying that term to those numbers is just logical. Anyway I have yet to see a policy that says WP:SYNTH applies to article titles. -- Toshio Yamaguchi 09:25, 28 July 2013 (UTC)

If the sources call these things "space figurate numbers" then that is what we should call them. Wikipedia:Article titles policy mandates Article titles are based on how reliable English-language sources refer to the article's subject. Spectral sequence (talk) 10:45, 28 July 2013 (UTC)

True, but I think Centered polyhedral number is compliant with WP:NAMINGCRITERIA. It is recognizable, natural, precise, concise and consistent. Yes, the policy also says "Article titles are based on how reliable English-language sources refer to the article's subject", but in this case I don't see a good reason for choosing a perhaps less natural and consistent name just because the source refers to them by that name. -- Toshio Yamaguchi 12:13, 28 July 2013 (UTC)

The "good reason" is that Wikipedia:Article titles is a policy, which should be followed unless there a good reason and consensus otherwise. The only reason you have given is that you personally prefer a name that, as far I can tell, no source has ever used; and it is clear that, so far at least, there is no consensus supporting this choice. Spectral sequence (talk) 15:18, 28 July 2013 (UTC)

I could not find that exact phrasing either. There do exist polyhedral numbers, for instance, POLYHEDRAL NUMBERS, HOWARD HEMMERLY, The Mathematics Teacher, Vol. 66, No. 4 (APRIL 1973), pp. 356-362 and the MathWorld category PolyhedralNumbers. Perhaps this article could turn into a Polyhedral number WP:DABCONCEPT article with vertex polyhedral numbers and centered polyhedral numbers as separate sections. --Mark viking (talk) 19:43, 28 July 2013 (UTC)

As I said, the book by Deza & Deza uses this name, but uses polyhedron instead of polyhedral, ie centered polyhedron number, so it seems the article should simply be moved to centered polyhedron number. -- Toshio Yamaguchi 20:34, 28 July 2013 (UTC)

Neither of the terms "centered polyhedron number" or "centered polyhedral number" appear in the index of that book. There is an entry for "centered regular polyhedron number" pointing to p.120. Perhaps it should be "centred space figurate number"? Spectral sequence (talk) 20:48, 28 July 2013 (UTC)

As you say, the term centered regular polyhedron number appears on page 120, so I support moving the page to that title. -- Toshio Yamaguchi 22:47, 28 July 2013 (UTC)

… surrounded by polyhedral layers with a constant number of edges

First of all, the article has to define a polyhedral layer. Otherwise I’d prefer to see it redirected or de-articled in some other way. Incnis Mrsi (talk) 16:49, 28 July 2013 (UTC)

I agree that this could perhaps be formulated in a more understandable way. A polyhedral layer here is a set of points all lying on the edges of the defining polyhedron. -- Toshio Yamaguchi 20:18, 28 July 2013 (UTC)

Really on the edges only? How can you explain a cubic growth of centered tetrahedral numbers then? Incnis Mrsi (talk) 20:47, 28 July 2013 (UTC)

You are correct, it is not the edges only. For example, the the fourth centered tetrahedral number is 35, because there is a central point, a layer with 4 points, a layer with 10 points and a layer with 20 points. In the layer with 20 points, there are 4 points lying at the center of the faces of the tetrahedron.

I think a face of the layer with 20 points looks like this:

   x
  x x
 x x x
x x x x

In each additional layer, a new row of Xs (points) is added at the bottom of the triangle, containing one point more than the row above it. So the correct formulation probably should have been A polyhedral layer is the set of all points forming the surface of the tetrahedron. -- Toshio Yamaguchi 22:32, 28 July 2013 (UTC)