Talk:Figurate number

message: 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)
thread_title: Cleanup

message: You mention that you can get the value of \scriptstyle n^2\, by adding up the first \scriptstyle n\, odd positive integers, and that to get \scriptstyle 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 \scriptstyle n^a\, you sum the odd numbers from \scriptstyle n^{(a-1)}-(n-1)\, to \scriptstyle 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\mod2) 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)
replies: text: n^a = n n^{(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)
thread_title:  About sum of n consecutive odd numbers <math>n^a = \sum_{i = {n^{(a-1)}-n \over 2} + 1}^{n^{(a-1)}+n \over 2} {(2i-1)}\,</math> 

message: :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)
thread_title:  Wikipedia is not a textbook 

message: 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)
thread_title:  Question about the location of a paragraph 

message: 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)
replies: text: Right. There is already an article called Methods of computing square roots. PrimeHunter (talk) 12:28, 10 July 2008 (UTC)
thread_title:  Where should we put the shortcut? 

message: 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 (talk • contribs) 12:45, 15 February 2009 (UTC)
replies: text: 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)
replies: text: 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)
thread_title:   Starting to clean up this article. 

message: 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)
replies: text: 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)
thread_title:  Stop automatically undoing changes 

message: 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)
thread_title:  Figurates listed are not all the figurates 

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

message: The following Wikimedia Commons file used on this page has been nominated for deletion:
* 
Participate in the deletion discussion at the . —Community Tech bot (talk) 22:30, 9 June 2019 (UTC)
thread_title:  A Commons file used on this page has been nominated for deletion 

message: The introduction of this basically simple concept is unnecessarily obscure. Kdammers (talk) 04:51, 4 October 2020 (UTC)
thread_title:  Introduction