# Talk:Figurate number

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## Cleanup

I have cleaned up the page somewhat, removing irrelevant material, adding some material on terminology, history, and so on, and supplying references or asking for them where I don't know enough. The content is still quite incomplete, with no specifics on polygonal numbers past squares, polyhedral and polytopic numbers, and the formulas for calculating figurate numbers or using them in other formulas other than one each for Fermat and Euler. I am working on a sequence of Turtle Art lessons on figurate numbers on the Sugar Labs Wiki, and wanted to be able to send readers here for useful information. Sugar Labs started as the software arm of One Laptop Per Child, and has branched out to Replacing Textbooks (with Open Education Resources).--Cherlin (talk) 23:44, 11 July 2011 (UTC)

## About sum of n consecutive odd numbers $n^{a}=\sum _{i={n^{(a-1)}-n \over 2}+1}^{n^{(a-1)}+n \over 2}{(2i-1)}\,$ You mention that you can get the value of $n^{2}\,$ by adding up the first $n\,$ odd positive integers, and that to get $n^{3}\,$ you use something similar to a moving average... Actually, it seems that there is a simple formula that can tell you at which number to start and at which number to end, for any power.

To get $n^{a}\,$ you sum the odd numbers from $n^{(a-1)}-(n-1)\,$ to $n^{(a-1)}+(n-1)\,$ (the 2 bounds being always odd,) i.e.:

$n^{a}=\sum _{i=n^{(a-1)}-(n-1)}^{n^{(a-1)}+(n-1)}{(i\mod 2)i},$ or equivalently:
$n^{a}=\sum _{i={n^{(a-1)}-n \over 2}+1}^{n^{(a-1)}+n \over 2}{(2i-1)},$ For example:
3^2
start at: 3^(2-1)-(3-1)=3-2=1
end at: 3^(2-1)+(3-1)=3+2=5
So 3^2 = 1+3+5=9

4^5
start at: 4^(5-1)-(4-1)=256-3=253
end at: 4^(5-1)+(4-1)=256+3=259
so 4^5 = 253+255+257+259=1024

This method has worked with every test I tried, but I could've missed something. 64.223.238.101 17:29, 5 Mar 2004 (UTC)

What happens is that you end up adding $n\,$ odd terms with mean equal to $n^{(a-1)}\,$ , so you get:

$n^{a}=nn^{(a-1)}\,$ This is easily proven by using Euler's method:

If n is even:

n^{(a-1)}-(n-1)} + ... (n odd terms) ... + n^{(a-1)}+(n-1)}
+   n^{(a-1)}+(n-1)} + ... (n odd terms) ... + n^{(a-1)}-(n-1)}
-----------------------------------------------------------
2n^{(a-1)} + ...      (n pairs)       ... + 2n^{(a-1)}

then: (1/2) [n (2n^{(a-1)})] = n n^{(a-1)} = n^a

If n is odd:

(central        n^{(a-1)}-(n-1)} + ... ((n-1) odd terms) ... + n^{(a-1)}+(n-1)}
term)     +   n^{(a-1)}+(n-1)} + ... ((n-1) odd terms) ... + n^{(a-1)}-(n-1)}
---------------------------------------------------------------
n^{(a-1)} +        2n^{(a-1)} + ...      ((n-1) pairs)      ... + 2n^{(a-1)}

then: n^{(a-1)} + (1/2) [(n-1) 2n^{(a-1)}] = (1 + (n-1)) n^{(a-1)} = n n^{(a-1)} = n^a



Tentacles (talk) 15:43, 1 June 2010 (UTC)

## Wikipedia is not a textbook

Is there a shortcut? Can S be manipulated to generate fourth powers?

Wikipedia is an encyclopedia. It should not present exercises for the reader to solve; it should present the solution and explain it in detail. --Eequor 15:21, 27 May 2004 (UTC)

## A topic related site

Greetings, I am the moderator for

[]

where the topic of figurate numbers is open for discussion with hopes of finding answers to puzzles and practical applications for this fascinating field.

## Question about the location of a paragraph

The paragraph "This procedure (taking many words to explain, but quickly executed) is not restricted to calculating square roots of natural numbers or positive integers. It can even be applied toward calculating the irrational square root of 2, to any number of decimal places." appears at the end of the section on "Cubes and cube roots" but it appears to belong to the discussion in the section on "Square roots". Is this intentional for some reason that I have not grasped?

You have a good point there. In fact, if they are going to talk about square roots and cube roots, they should also include the roots of triangles, pentagons, and if possible, tetrahedrons, etc; Aside from the use applied by children in grade school, others need to mention additional potential uses for figurate numbers. --R3hall (talk) 22:37, 4 May 2008 (UTC)R3hall —Preceding unsigned comment added by R3hall (talkcontribs) 13:22, 25 March 2008 (UTC)

## Where should we put the shortcut?

I am not sure that Figurate number is the right place to detail the manner of calculating square roots. I don't frankly see what connection the shortcut has with figurate numbers per se... -- Cimon Avaro; on a pogostick. (talk) 04:26, 10 July 2008 (UTC)

Right. There is already an article called Methods of computing square roots. PrimeHunter (talk) 12:28, 10 July 2008 (UTC)

## Reasons for deletion

I am going to so nominate it, I hope I do it correctly. There are so many reasons that I will list them here. 1) A small web search reveals that in most places (excluding those lifting from this page) figurate number means polygonal number or at least a plane figure. It is true that the reputable site Wolfram Mathworld has a more expansive definition (indeed some of this text may be lifted from there). 2) the one outside link is to a 2001 conference paper from Korea which I can believe is mathematically correct and well written but not published in any peer reviewed setting. I have not checked the references from MathWorld (Guy, Conway Kraitchik) which MIGHT anchor a reasonable wikipedia entry. If desired I can do a MATHSCINET search on Monday to see how big a splash there is in the research math world. The one reference is to a specialized book. 3) Most of the article concerns square roots and cube roots in a somewhat idiosyncratic manner. 4) The reference to Picks theorem is not appropriate —Preceding unsigned comment added by Gentlemath (talkcontribs) 01:06, 15 February 2009 (UTC)

I have deprodded it. It's the main article in Category:Figurate numbers and would leave a hole. If you also want the category deleted then that would require a discussion at WP:CFD.
1) MathWorld is a large and important source for Wikipedia. And Encyclopædia Britannica says in : "Polygonal numbers constitute a subdivision of a class of numbers known as figurate numbers. ... Polygonal number series can also be added to form threedimensional figurate numbers".
2) Even if the name "Figurate number" was deemed inappropriate, the article might be renamed or merged. Some of the content looks OK.
3) I agree the square roots and cube roots stuff is problematic to have here. The square roots got no support in the above section. Feel free to remove it. Some of the content being bad is a reason to improve the article and not to delete it.
4) Pick's theorem is a see also link and not a reference. PrimeHunter (talk) 02:18, 15 February 2009 (UTC)

Thank you. You are correct that figurate number needs to be here unless many other articles are changed. Some of the content is OK, but maybe less than one might think (I could argue that the title, the first and third sentences, and the last few words of the second, are about it) BUT (to get the less important comments out of the way first)

4) Picks Theorem is not really an appropriate "see also" since it deals with the area of arbitrary non intersecting polygons which have all vertices from the square lattice while polygonal numbers correspond to patterns of dots which are regular in one way or another but in most cases (n-gon for n=5,7,9,11,13,14,15) not subsets of any planar lattices
1) (part 2) Britanica does say that. It is hard to explore Britanica without a subscription but it appear that Britanica may have figurate number only in that place (number games) and never defines them except by examples (which are not consistent)
arrangement of points representing numbers into series of geometrical figures. Such numbers, known as figurate or polygonal numbers [so are they numbers or arrangements? and are figurate and polygonal synonyms?]
Other examples include arithmetic progressions like 1,2,3,...,r (so all integers are figurate.. of course n is the first non-trivial n-gonal number.
You can make pyramids too
1) (part 1.. alas) I am going to get a little wordy here because I feel something important needs to be said but I don't want to seem petty. Disclaimer: I immensely value Wikipedia and I immensely value MathWorld. If I need to figure out something about Math I usually go to them in that order (that's where Google takes me..) and I am well served the majority of the time. However I usually go on to a more primary source (which I am aided in finding by starting at W or MW). So both Wikipedia and MathWorld are highly useful encyclopedeae, each is a labor of love supported by passionate contributors. In the case of MathWorld the vast majority of entries are written by Eric Weisstein (according to his author bio.)
OK, now that I said that, I think it is a mistake to use MathWorld as a source for Wikipedia (but a great idea to use it as a source of leads FOR source material such as the book by Conway and Guy.. I leave aside the question if that book is too much original research). two reasons that it is a bad idea . One is that in some case MathWorld is non-standard or even misguided and then Wikipedia (using MathWorld as a source) repeats the usage and then parasite web sites echo the usage making it seem widespread. A second reason is that in other cases chunks of text are just lifted out of MathWorld verbatim which is plagarism. Sometime both things happen at once. An example is in this article (regular p-topic number) which I hope to correct. --Gentlemath (talk) 10:07, 15 February 2009 (UTC)

I should add that perhaps PlanetMath (which I am not so familiar with) is a more robust source (although again not enough in and of itself). PlanetMath says figurate number is another name for polygonal number. The usage is just not that standard and Wikipedia should not favor one thing over another. I suppose oblong number should link to figurate number. The greek name for composite integer was rectangular number which is not a stub but maybe should be. The one place I found that it does show up is in Hexagonal number with the somewhat silly remark that the hexagonal number $h_{n}=n(2n-1)\,\!$ can be rearranged into the rectangular number n long and 2n−1 tall (or vice versa). (Ignoring the fact that 1 is not really a rectangular number) --Gentlemath (talk) 10:47, 15 February 2009 (UTC)

The term polytopic previously used here is very problematic both for Wikipedia and the web at large. As you can see, it links to the article polytope (which is noted as a confusing and unclear article) but that article never uses the word polytopic. It says that a polytope is a pretty generic generalization of polygons (including tesselations etc.)

The article prism uses the term "polytopic prism" and goes on to mention "0-polytopic prism" and "1-polytopic prism". A Google search turns up maybe one usage of "2-polytic". There are 188 uses of "polytopic prism" but they all look to be more or less automatically copied from the prism article.

The article on Figurate numbers in MathWorld contains a certain passage beginning

The nth regular r-topic number is given by

and ending

rising factorial

It seems plausible that polytopic number is a neolgism coined by the author of (most of) the articles in MathWorld. It appears 977 times in Google but only 74 of those are without the word regular and most of those 74 seem to still be lifted from this page. And "r-topic" shows up 217 times in Google. All of them also include the phrase "given by the pattern." Or perhaps it was coined by E.J. Dickson in his classic book. Not used much until MathWorld and then pow. I could be wrong about that history however I will assert that the meaning is completely unclear in this article so it should go unless someone wants to fix it.

The passage quoted from MathWorld includes equations with binomial coefficients, rising factorials and multichoose numbers. This Wikipedia article quotes this in a somewhat garbled way. I don't know if the garbledness gets around the command:" Do not copy text from other websites without a GFDL-compatible license."

At any rate the formula ${n\times {{n+r}! \over {n+1}!} \over {r!}}$ is a disaster because n+r! would usually mean n+(r!) rather than (n+r)! as intended here. More serious is the fact that even with proper parenthesis this does not equal the binomial coefficient on the other side of the equal sign.

Evidently the author of MathWorld intends regular r-polytopic to mean something having to do with a simplex and there we find that: The regular simplex family is the first of three regular polytope families, labeled by Coxeter as αn SO if it is really that valuable then someone should explain it better than is done in MathWorld.

Until polytopic is fixed the examples for r=2,3,4 are in limbo. I won't touch the Gnommon bit for now but I see no dots! —Preceding unsigned comment added by Gentlemath (talkcontribs) 12:45, 15 February 2009 (UTC)

I see you removed most of the article in 3 edits  and then Melab-1 brought it all back. I think some but not all of it should go. This is the main article in Category:Figurate numbers and it seems OK to duplicate some content from other articles there like Gnomon (figure), although a shortened overview would be better in most cases. Google Scholar has some hits on "polytope number" but I haven't read them. Not everything has a precise unique definition and Wikipedia may have to reflect that, although exact definitions are usually more appealing to mathematicians. I support removal of the stuff about square roots and cube roots which doesn't belong here. PrimeHunter (talk) 03:17, 17 February 2009 (UTC)
I do also support removing some of the cube roots stuff, but not the formulas. The formulas are neccesary for a good description of "Figurate number". Other math articles contain formulas to define their subject or properties in their subject. Keep the header section the way it is. And, I don't think the article should be ranamed to "Figured number". That sounds like someone figured out a number and "-ate" suffix is needed because it is a form not an adjective. --19:27, 23 February 2009 (UTC)

I don't know if it is enough to say that not so much should go. If I provided careful reasoning for what I removed then I think that before restoring it, my comments should be adressed.

On the positive side I now have Leonard Eugene Dickson's history of thetheory of numbers and T. L. Heath's History of Greek Mathematics so Imay try to build things up again.I'll suffice for now with removing the square and cube root stuff togetherwith the tragically flawed formula. But please respond to the following orI will feel justified in removing the corresponding stuff.

1) Why keep polytopic? Are all figurate numbers polytopic numbers?

2) A math research paper from 2003 defined the term "polytope numbers". Do we endorse primary research articles?

3) The expression I said was wrong is still wrong.

4) In 1999 Midhat J. Gazale wrote a book (now out of print) where heextended the idea of gnomon to things like the chambers of spiral seashell. He goes on to relate this to Fractals and the golden ratio.Why is that relevant as an outside link on this page? And why should the entire article gnomon(figure) be repeated here?

5) To be a "see also" Picks Theorem should have **something** to do withfigurate numbers. Does it?

1. The word polytopic appears 17 times in Wikipedia (not including talk and user pages). At most two are not either on this page, a redirect to this page or a repeat of the first section of the page in the article Number. There is no definition so what does it mean and why do we need it?

2. The conference paper linked to at the bottom of the page by Hyun Kwang Kim actually does appear in PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETYVolume 131, Number 1, Pages 65-75 That is a very good journal. The paper may never have been referenced. In it (according to Math Reviews MR1929024) the author constructs (regular)polytope analogues of Euler’s polygonal numbers. If we are going to link to the preprint of that paper, explain why.

3. The formula $P_{r}(n)={{n+r-1} \choose r}={n^{(r)} \over {r!}}{\mbox{ }}{\mbox{for}}\ n\geq 1{\mbox{ }}{\mbox{or}}{\mbox{ }}{n\times {{(n+r)}! \over {(n+1)}!} \over {r!}}\quad {\mbox{for}}\ n\geq 0$ is a little better than last time but still wrong. According to it $P_{2}(n)={\mbox{ }}{n\times {{(n+2)}! \over {(n+1)}!} \over {2!}}={{n(n+2)} \over 2}$ but that is not even an integer when n is odd. I think that this was lifted from MathWorld where it is rather garbled anyway. Actually this just says that binomial coefficients are figurate numbers for generalized triangles. This well known observation appears in the article for Pascals Triangle so I'll just link there.

• A final challange. I may propose changing this to Figured Number. Who beside MathWorld and Wikipedia clones uses Figurate Number in this way? This is the term I have found in the two books above. Dickson uses Figurate number for generalized triangular numbers (i.e. binomial coefficients).Many sources use figurate number for polygonal number. Comments?--Gentlemath (talk) 06:38, 17 February 2009 (UTC)

## Stop automatically undoing changes

I have given detailed justifications for removing things. In fact I first discussed, then waited a while, then deleted. It is fine to disagree but important to say why. Please discuss but do not just delete.

Case in point: I disliked the usage figurate but gave notice. No response but I did find more uses of it in that sense so I am now satisfied to leave the title as figurate numbers. I will need to add an explanation that figurate means only polygonal in some places (Planet Math and many scholarly books)and only triangular, tetrahedral etc in others (Dickson's History of the theory of numbers).

Some of this stuff is flat out mathematically wrong and I explained why. Please, let's create great articles but lets do it together. --Gentlemath (talk) 20:34, 23 February 2009 (UTC)

OK, but I just think that for now the formulas should be put back in ust without the part I added. Thoughts, I'd like the formula put back in. I admit that maybe there's something wrong with the last formula. --Melab±1 21:07, 23 February 2009 (UTC)

## Figurates listed are not all the figurates

The figurate numbers listed in this article are not complete; the list is missing one that the Pyagorians knew about. These are the oblong or oblate numbers. Like the square numbers, they are rectangular, but unlike them, that have exactly one more column than rows:

          7  7  7  7  7  7  7  7
7  6  6  6  6  6  6  6
7  6  5  5  5  5  5  5           number of columns, 8
7  6  5  4  4  4  4  4           number of rows, 7
7  6  5  4  3  3  3  3
7  6  5  4  3  2  2  2
7  6  5  4  3  2  1  1


The difference between the oblates and the squares is the seed which is 'cornered,' that is:

           oblate seed                         square seed
-----------                         -----------

             1  1                                   1


           oblate cornered once             square cornered once
--------------------             --------------------

             2  2  2                               2  2
2  1  1                               2  1

           oblate cornered twice           square cornered twice
---------------------           ---------------------

            3  3  3  3                           3  3  3
3  2  2  2                           3  2  2
3  2  1  1                           3  2  1


While the general formula for a square is n(n - 0), the formula for an oblate is n(n - 1). — Preceding unsigned comment added by 184.96.3.105 (talk) 18:56, 10 April 2014 (UTC)

## Related AfD

Please see Wikipedia:Articles for deletion/Linear number. —David Eppstein (talk) 18:54, 2 July 2016 (UTC)

## A Commons file used on this page has been nominated for deletion

The following Wikimedia Commons file used on this page has been nominated for deletion:

Participate in the deletion discussion at the nomination page. —Community Tech bot (talk) 22:30, 9 June 2019 (UTC)