Talk:Pascal's triangle/Archive 2

Archive 1

Archive 2

Archive 3

Vulgarization

Would it be possible to include a short paragraph that goes into creating or making Pascal's triangle in simple terms ?

The page is more than ample when it comes information but it's uncertain that it would be of value to someone trying to assimilate the concept. It is introduced in lower secondary school so summation would not be known by students.

I realize that Wikipedia's aim is not to provide learning material but there is that aim of diffusing knowledge to the greater number. --JamesPoulson (talk) 17:30, 8 March 2013 (UTC)

I moved this new post to the bottom of the talk page as we usually do. However, it's actually related to the very first post (at his time), "Why?".--Nø (talk) 18:03, 8 March 2013 (UTC)

Gray's Theory?

No relevant google hits on "Gray's theory". Is this actually a notable method? Seems maybe a bit OR to me. Staecker (talk) 19:14, 6 February 2013 (UTC)

Right, but rather than deleting the section I've removed the unsourced name ("Gray") and improved the presentation to use the same ${n \choose k}$ notation used elsewhere in the article. The material is not original as such, and I think the article benefits from having things of a fairly elementary nature in it too. But I wouldn't object strongly to the section being removed.--Nø (talk) 09:43, 20 March 2013 (UTC)

Re: Verification Needed

The last sentence of the second paragraph in "History" currently states:

"In 1068, four columns of the first sixteen rows were given by the mathematician Bhattotpala, who realized the combinatorial significance.[4][verification needed]"

The cited material simply reads:

"A further commentator Bhattotpala (1068) has given an explicit example involving combinations of sixteen things (Fig.11)."

Bhattotpala is referred to by Edwards as a "commentator," not a mathematician. Whether he made any realization is not clear.

GiantSteps (talk) 05:10, 11 June 2013 (UTC)

Added this to the history sub-section of this article - "Varāhamihira was among the first mathematicians to discover a version of what is now known as the Pascal's triangle. He used it to calculate the binomial coefficients for evaluating combinatorics.". Sources are cited. isoham (talk) 20:21, 15 September 2013 (UTC)

Failed to parse error

Am I the only one who sees a "failed to parse" error in the 'Binomial expansions' section? (starting after 'The two summations can be reorganized as follows:') Tropcho (talk) 17:46, 7 February 2014 (UTC)

Perhaps not, but I see the intended formula. I use Google Chrome - in what browser/version do you have the problem?--Nø (talk) 18:00, 7 February 2014 (UTC)

I see the problem both in Firefox 27 and Safari 6.0.5 on OS X Mountain Lion. Tropcho (talk) 19:15, 7 February 2014 (UTC)

Right, I see the same issue in Firefox on Windows 7. It reports that \begin is unknown. Looks to me like Chrome recognizes a larger subset of the TeX language used for formulae than Firefox. I don't know how to fix this, so I hope someone else do. Is there a way to draw the nattention of a wiki/tex expert to this issue?--Nø (talk) 10:39, 8 February 2014 (UTC)

Numbering of rows and columns

While I fully understand the awkwardness of it, the traditional manner of referring to the entries of the triangle is to say that C(n,k) appears in the nth row and kth position. The only way to make this work is to talk about the 0th row and 0th position in a row. This is what is done in most combinatorics texts and our job as editors is to report on what is in the literature and not try to make improvements on it. If you want to call the 0th row the "first row", then you will need to provide citations for that usage. I think a better approach would be to try to explain, in a more detailed manner, why mathematicians start the numbering with 0 in this case, so that the casual reader can see the reasoning behind this awkwardness. Bill Cherowitzo (talk) 20:26, 16 December 2014 (UTC)

Would it be acceptable to use "In the first row (n=0), ..." or should we use "In the zeroth row, ..."? I can see the latter being confusing but even more confusing is the present lack of consistency in the article. There are inconsistencies even in one sentence: "For example, the first number in the first row is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row." (emphasis mine)

I'm somewhat partial to using "first row (n=0)," because it seems clearer to me. Is there a way to ensure that a standard is agreed upon and made obvious to readers and editors alike? (Sorry, I am somewhat new to editing Wikipedia)

Ezrysm (talk) 19:32, 24 December 2014 (UTC)

I think it is possible to be clearer about this. For instance, I'd use phrases like, "In the top (or zeroth) row, ..." and "... the initial number in the first row is 1 (in the zeroth position)." Using the parenthetic phrases should alert the reader that something unusual is going on without over emphasizing the issue. Bill Cherowitzo (talk) 00:32, 26 December 2014 (UTC)

Usefulness of 1000th row image

1000th row of Pascal's triangle, arranged vertically, with grey-scale representations of decimal digits of the coefficients right-aligned. The left boundary of the dark segment-like shape corresponds roughly to the graph of the logarithm of the binomial coefficients, and illustrates that they form a log-concave sequence.

The 1000th row image (reproduced to the right) seems completely useless to me. I had to read the description several times before I really understood what it actually was (e.g., that the pixels represent decimal digits, or even the fact that it is "sideways" from the usual Pascal's triangle). There are no obvious patterns in the image, other than the symmetry along the middle. The only real thing that can be seen is the curved shape of the "length" of the numbers.

Since it is so confusing and doesn't really convey any useful information, I suggest removing it. I am going to remove it now, and have this discussion here if anyone wants to put it back.

asmeurer (talk | contribs) 21:23, 13 June 2014 (UTC)

I believe this might be the reference corresponding to this image (which I have removed from the main page). Bill Cherowitzo (talk) 21:59, 26 December 2014 (UTC)

Meeting: 1003, Atlanta, Georgia, SS 24A, AMS Special Session on Design Theory and Graph Theory, I 1003-06-607 Avery S. Zoch, Pascal’s Grey Scale

Notability

I have just removed the following new addition to the page:

New properties : Inequalities

In 2014 were discovered new properties involving inequalities, which are:

1- In all the infinite center column of the triangle in the figure below, the product of two of its elements is greater than the product of two elements belonging to the same center column, located symmetrically between them. For example , in the figure below : 1 x 20 > 2 x 6 , or 2 x 20 > 6 x 6 , or yet 1 x 6 > 2 x 2. This applies to the entire central column.

                               1
                          1       1
                      1       2       1
                  1       3       3      1
              1       4       6       4      1
          1       5      10      10      5       1
      1       6      15      20     15       6        1

2- Given two elements A and B in the infinite middle column, the product of these is greater than the product of two elements C and D belonging to the diagonal passing through A and B, which are located symmetrically in relation to A and B. For example, looking again the figure above: if a = 2 and B = 20 , then: 20 x 2 > 3 x 10 > 4 x 4 > 5 x 1 . If A = 1 and B = 20 , then: 20 x 1 > 1 x 10 > 1 x 4 > 1 x 1 .

There is a question of notability concerning this result. It is far too new to have received any reviews in the literature, so there are no secondary sources that attest to its notability. Wikipedia is not a place to announce new results. When this article has been vetted in the usual way, those secondary sources may be used as a basis for describing the result here. Even at that point, the above will need quite a bit of work to make it acceptable. Bill Cherowitzo (talk) 22:24, 7 April 2015 (UTC)

References

Wording, history

The history section begins:

The set of the numbers that form Pascal's triangle was known well before Pascal's time.

(Here, the second "the" has just been added by another editor.)

In a strict mathematical sense, the "set" of numbers in Pascal's triangle is all positive integers, which indeed were known... Am I nitpicking here, or should we rather have something like

The number patterns that make up Pascal's triangle were known well before Pascal's time.

Any views or better suggestions?--Nø (talk) 07:35, 5 October 2015 (UTC)

I like it and have made the change. --JBL (talk) 12:35, 5 October 2015 (UTC)

Martin Gardner column on Pascal's triangle

I am reverting the revert by User:Gamall Wednesday Ida.

I added the sentence, "Martin Gardner wrote a popular account of Pascal's triangle in his December 1966 Mathematical Games column in Scientific American."

User Gamall reverted it saying, "How in the world is that important enough to deserve mention as part of the triangle's history?"

reason for revert of the revert

At the time that Gardner wrote the article, he had over a million readers and the column probably did more to make people aware of the triangle than anything previously written about it. In support of this statement we have

• Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy dedicated their book Winning Ways for your Mathematical Plays, saying "To Martin Gardner, who has brought more mathematics to more millions than anyone else."
• [The column had] an audience of close to a million readers.
• Gardner was without doubt the best friend mathematics ever had—and it’s said that his column reached a million readers a month at his peak.

citations

books

Moreover, the column is included is included in several of Gardner's best selling books and continues to introduce the triangle to many readers to this day. All of this is certainly relevant to the history of the triangle in mathematics and human culture.

--Toploftical (talk) 18:20, 18 November 2016 (UTC)

@Toploftical: Yes, those refs show that the column itself is generally notable as a popular culture phenomenon (and a good one, for once), but not that this specific instance of the column on this specific topic warrants mention. Is there a reliable ref arguing that there was a measurable effect on the use of the triangle after this column? Unless that's the case, I don't see how it rises above the level of trivia. The section goes straight from Pascal's posthumous publication, to de Moivre giving it its modern name slightly less than three centuries ago,... and therefrom straight to its appearing in a popular recreational maths column -- with no documented consequence whatsoever. One of those items does not belong with the others, especially in a section called "History".

Now, I have no specific animus against Gardner, who was, as far as I can tell, an admirable human being, and I'd be perfectly ok with the material if it came with references establishing the importance of the event for Pascal's triangle. As it is, it does not, so I'm not ok with it. Are you going to go over the list of Gardner's columns and make similar mentions in every article on every topic his column might have touched on? Or at least, would you do so if you had unlimited time and patience? That would be the logical consequence of your stated reasons, given that they are not specific to Pascal's triangle. — Gamall Wednesday Ida (t · c) 19:39, 18 November 2016 (UTC)

I agree entirely with GWI, excepting the last three sentences. Some things Martin Gardner wrote about are notable and their notability and significance can be traced to the fact that Gardner wrote about them, and for those things discussing Gardner's role in their history is a good idea. But Pascal's triangle is definitely not in this category: it is several centuries old, was well studied and recognized as important long before Gardner was born, and the fact that Gardner wrote about it has had no discernable effect on its modern history. --JBL (talk) 20:59, 18 November 2016 (UTC)

Response to third opinion request:

I agree that the sources cited support the claim that Gardner's column was, in general, influential, and I don't think there's any dispute about that. But what we need here is a source that shows that his column on the particular subject of Pascal's Triangle was influential by itself, and what, specifically, that influence was. What changed as a result of the column, and what reliable sources describe that change? Anaxial (talk) 14:03, 22 November 2016 (UTC)

Merge from Pascal's pyramid

I have suggested for the article Pascal's pyramid to be merged into this article as they are very closely related; but this article is much more improved an has less errors compared to the other article. ^∞😃 Target360YT 😃^∞ (talk · contribs) 06:01, 10 October 2016 (UTC)

Oppose.: I don't see how merging would suddenly improve the other article's contents. This one is already cluttered enough as it is, no need to add related subjects to it. The solution is to improve Pascal's pyramid, not to merge it here. Gamall Wednesday Ida (talk) 07:02, 10 October 2016 (UTC)

Oppose.: I totally agree with Gamall Wednesday Ida. Although mathematically related, the two topics are quite distinct and merging them would result in a bloated and confusing article.--Toploftical (talk) 23:24, 18 November 2016 (UTC)

Oppose: Both article have more than enough content on there own and though related they are nevertheless different subjects.--Kmhkmh (talk) 23:32, 16 January 2017 (UTC)

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The properties

C(n,k) = C(n, n-k)

C(n+k, k) = C(n+k, n)

And, also, relation between Pascal's triangle and hypercubes holds

$m^{n}=\sum _{k=0}^{n}\sum _{j=0}^{k}{\binom {n}{k}}{\binom {k}{j}}(-1)^{k-j}m^{j}$

Reference at : https://arxiv.org/abs/1603.02468 --KolosovP (talk) 07:55, 11 January 2018 (UTC)

These elementary identities predate your manuscript by decades, if not millennia. --JBL (talk) 00:45, 12 January 2018 (UTC)

If so - find and show me exact identities in published reference - even in oies sequence wasn't that "so elementary identity" - they just forgot to put em. Okey ? By the way each identity here in article is elementary, thank you KolosovP (talk) 07:43, 17 January 2018 (UTC)

Literally the first combinatorics textbook that comes to hand is Agnarsson and Greenlaw's Graph Theory. On page 26 of the international edition (page 8 in the standard edition), we find the first identity (which was almost certainly known in ancient times). The second identity is a trivial change of variables from the first identity. The third identity is not written there, which is not surprising since enumerative combinatorics is only incidentally relevant to the content of the textbook, but it is just a special case of the multinomial theorem (expanding $(m+1-1)^{n}$ as a power of a trinomial, and writing the multinomial coefficient as a product of two binomial coefficients in the standard way). --JBL (talk) 14:53, 17 January 2018 (UTC)

Yes, Dear Doctor Lewis, they are not shown, its very easy to derive it, its very easy to proof it, but also i haven't found that identity, i found very near version in http://www.math.ucsd.edu/~jverstra/bijections.pdf on the page 5. And also identity about sum of the row is known thousand years, it is obvious, it is very easy proven by binomial theorem, so why is it included to the article then? By the way identity is generalization for section "Relation between Pascal Triangle and Hypercubes", Thank you for your reply and conversation, Sincerely Yours,

KolosovP (talk) 12:50, 20 January 2018 (UTC)

The symmetry identities and the row-sum identity are important; they appear in dozens of secondary sources (e.g. textbooks) and are taught to every student of combinatorics. This special case of the multinomial theorem is not important (and the same is the case for most of the infinitely many other identities that arise by specializing some variables in the multinomial theorem). --JBL (talk) 15:02, 20 January 2018 (UTC)

This identity could be derived as special case of binomial theorem, more info at https://kolosovpetro.github.io/pdf/Relation_between_Pascal_Triangle_and_Hypercubes_Theorem.pdf, also, may be you could be interested in the expansion https://kolosovpetro.github.io/pdf/Overview_of_preprint_1603.02468.pdf - here is general review and the main question, that is expansion of cube with binomial distribution of summ items, i hope it has connection with binomial theorem. Feel free to comment on and may be could you help with ? — Preceding unsigned comment added by KolosovP (talk • contribs) 16:52, 20 January 2018 (UTC)

Relation to Physics and Chemistry

Various sources mention, that chemical elements properties are closely tied with Pascal's Triangle and Fibonacci sequence. For example: http://oeis.org/search?q=1%2C+4%2C+6%2C+9&sort=&go=Search
http://knowledgebin.org/kb/entry/electron_configuration_atomic_structure_and_the_pascal_triangle_core_points_for_iitjee_and_aieee/
https://link.springer.com/chapter/10.1007/978-3-642-31977-8_1
http://theoryofeverything.org/theToE/2015/04/22/pascal-triangle-and-mod-2-9-sierpinski-maps/
http://theoryofeverything.org/theToE/2013/06/15/connecting-the-octonion-fano-plane-to-the-atomic-elements/

NikitaSadkov (talk) 18:41, 30 September 2018 (UTC)

Multiplication Table inside Pascal's Triangle

If you treat Pascal's Triangle as a side of a square pyramid, then the base of that pyramid would include multiplication table: http://lostmathlessons.blogspot.com/2015/03/pascals-pyramids.html

I'm sure I've seen an article on all hyperoperations being present in Pascal's Triangle, not just addition, multiplication and exponentiation.

NikitaSadkov (talk) 14:26, 1 October 2018 (UTC)

Animated binary

Each frame represents a row in Pascal's triangle. Each column of pixels is a number in binary with the least significant bit at the bottom. Light pixels represent ones and the dark pixels are zeroes.

This gif image was added recently - frankly I don't think it adds anything significant. Other opinions? --Nø (talk) 09:54, 16 February 2016 (UTC)

Nø: I don't see the point either, but nor do I see that it hurts the article. Since it takes no vertical space I'm for leaving it here. Gamall Wednesday Ida (talk) 07:07, 10 October 2016 (UTC)

Is that shape related to recent AMS article? http://www.ams.org/publicoutreach/feature-column/fcarc-normal Maybe you can cite them instead, because the animation alone is hard to understand. Pascal's Triangle is surprisingly obscure and unresearched, despite being a nice tool to teach children the basics of mathematics. I asked a mathematician with a background in geometry, who worked in AutoCAD and made several 3d packages in C++, and he never heard about it!!! --NikitaSadkov (talk) 18:17, 1 October 2018 (UTC)

Graph Theory and Pascal's Triangle

The graph theoretic structure of Pascal's Triangle is called Pyramid DAG (directed acyclic graph):

https://link.springer.com/chapter/10.1007/978-3-540-46642-0_38

http://tesi.cab.unipd.it/39775/1/Relazione_finale_De_Stefani_Lorenzo_621842.pdf

--NikitaSadkov (talk) 10:16, 2 October 2018 (UTC)

"Pascal sum-field"

Let every element of an infinte rectangular array (sort of like a spreadsheet) contain this formula: "Equal to the sum of two numbers in the row above, viz. one vertically above, and the other one column to the left." Assume boundary conditions so that all elements are zero. Now drop a "1" instead of the formula for one element, somewhere, and you get Pascal's triangle below and to the right of that element (in an assymmetrical layout).

Drop a number more, somewhere else, outside the triangle, and you get two triangles that will overlap and interfere.

So what am I driving at? No idea; probably not something that really belongs on a wikipedia talk page; sorry! I just have a feeling that the above is an interesting way of seeing Pascal's triangle, and that the interfering triangles might actually be useful or relevant to something. Any thoughts?--Nø (talk) 13:49, 5 October 2018 (UTC)

Why was Nilakantha Pi series discovery pulled

The Nilakantha Pi infinite series was discovered in Pascal's Triangle and has been sourced in a reputable journal, Mathematics Teacher, The National Council of Teachers of Mathematics. Tonyfoster46 (talk) 20:41, 17 April 2016 (UTC)

Wikipedia does not publish original research results (see WP:NOR). Assuming that you are the author of the journal article, according to your edit you are the originator of this material which is first published in this journal. That makes the journal article a primary source for this material. In general, Wikipedia does not publish material based solely on primary sources. Reliable secondary sources (see WP:RSS) are needed to determine the value of the contribution and whether or not it should be included in Wikipedia. The author of the material has a clear bias and should not take part in this decision (see WP:COI). If, and when, your contribution has been reviewed and vetted by the mathematical community and its significance has been evaluated, we will be able to reconsider its inclusion in this article. Bill Cherowitzo (talk) 21:34, 17 April 2016 (UTC)

Another articles says that relation with Pi was discovered already by Moivre: https://en.wikipedia.org/wiki/Normal_distribution#History
I'm sure you can take his formula, derive Pi from it. Why is that such a huge discovery and they don't mention Moivre? -NikitaSadkov (talk) 18:42, 5 October 2018 (UTC)

References