Talk:Pascal's triangle/Archive 2

This is an archive of past discussions about Pascal's triangle. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

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

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

Wording, history

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."^[1]
[The column had] an audience of close to a million readers.^[2]
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.^[3]

citations

[1]
Berlekamp, Elwyn R., John H. Conway, and Richard K. Guy (1982). Winning Ways for your Mathematical Plays p. v
[2]
Undiluted Hocus-Pocus The Autobiography of Martin Gardner p. xii
[3]
Math’s Best Friend, Martin Gardner, Scientific American, October 29, 2013

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

External links modified

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

Multiplication Table inside Pascal's Triangle

Animated binary

Graph Theory and Pascal's Triangle

"Pascal sum-field"

Why was Nilakantha Pi series discovery pulled

Deterministic Galton Board

Re recent edit: "Removed Britannica, unreliable"

Really? Only non-Western (or whatever (ad-hoc) qualification) sources reliable?

I honestly do not know about the relevant reliability, but I am embarrassed by the en-passent declaration of Britannica being unreliable. Purgy (talk) 07:40, 9 January 2019 (UTC)

I agree with Purgy. While I'm not saying that Britannica gets everything right, it is generally considered a very reliable source and should not be dismissed in this off-handed way. If there are problems with the entry, they should have been discussed on this talk page. --Bill Cherowitzo (talk) 19:24, 9 January 2019 (UTC)

@Purgy Purgatorio and Wcherowi: Hi to both of you guys and happy new year to you and your loved ones. As to Britannica, please take a look at this. Cheers.---Wikaviani ^{(talk) (contribs)} 19:35, 9 January 2019 (UTC)

@Wikaviani: Thanks for that discussion, but I am not convinced. It is a general screed against all tertiary sources and the only take-away from it that I get is a warning to use these sources with care. I would still like to see some specific concerns with this entry and not a blanket condemnation of the source. --Bill Cherowitzo (talk) 20:01, 9 January 2019 (UTC)

I did not participate to the discussion on Doug’s talk page. I think that tertiary sources are sometimes very good, like Encyclopedia of Islam (for Islam-related topics, of course) or Encyclopedia Iranica. I removed Britannica because Doug Weller and several other experienced editors say it’s not reliable, nothing more nothing less. I think you should share your concerns with Doug about this. Best regards.---Wikaviani ^{(talk) (contribs)} 20:16, 9 January 2019 (UTC)

The question whether the removed referral to Britannica should stay removed or should be reinstated, both until further discussions give a result, remains open?Purgy (talk) 11:22, 10 January 2019 (UTC)

(edit conflict)I think we should avoid it unless we can tell it's clearly reliable and NPOV (articles written by one author can always be POV). In this specific case I can see that it hasn't had fringe added to it and is by a lecturer at the École Polytechnique Fédérale de Lausanne, so it's probably ok. I'm not joking about fringe. Look at their article on the Mona Lisa.. Must be ok, it's written by the Editors, right? Wrong. It has unsourced material about the sitter, for instance. And that came from sockpuppet who failed to add it to our article. If you click on history you'll see material was added from a "Roni Kempler"- and although it was approved by an editor (who never answered my email asking about it), I still don't trust it without checking it. See Wikipedia:Sockpuppet investigations/Relpmek/Archive. Of course that's a specific anecdote and not typical. It can also be out of date (a general problem with sources). Doug Weller talk 11:41, 10 January 2019 (UTC)

Agreed, on my end, i try to remove Britannica as often as possible, another example here. However, i don't agree with questioning all tertiary sources' reliability like here, since for example, what source is better than Encyclopedia of Islam for Islam related topics ? and Encyclopedia of Islam is a tertiary source (same goes for Encyclopedia Iranica). As to Britannica, i think it should be used as a source only when we don't have better ones. Sorry Doug, i added a space between your message and Purgy's one for better readability. Cheers.---Wikaviani ^{(talk) (contribs)} 12:37, 10 January 2019 (UTC)

Just saw an edit at Timeline of the history of the region of Palestine and discovered that one very pov claim in the timeline is backed by the Britannica which states as fact that "After the Israelite conquest of Canaan, the Tabernacle and the Ark of the Covenant were installed in Shiloh until the Ark was captured by the Philistines (c. 1050 bc)". Considering that the mainstream archaeological opinion is becoming very dubious about such a conquest, this is bad. And it says "written by the editors". Doug Weller talk 14:15, 11 January 2019 (UTC)

New use for the triangle

Khayyam-Pascal's triangle?

Any evidence that this is the "internationally recognized name" as stated at the beginning of the article? I grew up in Soviet Union, where it was the good old Pascal triangle, and the same in Europe and US. Oleg Alexandrov 21:37, 5 October 2005 (UTC)

Its always been Pascal's triangle in my part of the world. Paul August ☎ 23:19, 5 October 2005 (UTC)

Together with Charles at Wikipedia talk:WikiProject Mathematics#.5B.5BKhayyam-Pascal.27s_triangle.5D.5D, that makes three well-respected mathematicians, to which I add my own voice. Regardless of which name is more historically correct, we should use the name that is actually used. So I renamed it back to Pascal's triangle. I also put the history section in chronological order again, without removing the new contents. -- Jitse Niesen (talk) 23:26, 5 October 2005 (UTC)

In italian schools it is usually nameded Tartaglia's triangle.

The following is copied from my talk page:--Niels Ø 16:23, 9 October 2006 (UTC) Dear Noe I'm sending you this message to prevent Edit war. I added "or Kayyam-Pascal triangle " because by seeing this page at the first time I fell in doubt if it was that. I'm from Iran and we always call this triangle "Khayyam triangle" or "Khayyam-Pascal triangle". As you can see this in Persian(Farsi) Wikipedia and tajik one . And the other revalant languages(Such as Urdu, Pashto,Azeri,Kurdi,Uzbek,...) don't have this article yet. You also can watch Keeper Movie (The legend of Omar Khayyam). In addition you can have a look at its invention history. Khayyam had invented it sevral centuries before Pascal and applied it for the other applications. Pascal just showed new applications of this triangle in the absence of the knowing about khayyam invension. I think "or Kayyam-Pascal triangle " is strongly needed for removing ~~dis~~ambiguation for hundred millions people. ~~--81.31.160.22 07:29, 9 October 2006 (UTC)~~--Soroush ^{☺talk | ☼Contributions} 07:32, 9 October 2006 (UTC)

My reply to User:Soroush_Mesry: I still disagree. This is an encyclopedia in English. The name of the triangle in English is Pascal's triangle. I suggest you create the relevant article (as a stub, for a start) in wikipedias in other relevant languages, and use interwiki links to connect things.

As for the history, I do not know whether Omar Khayyam invented the triangle. I don't think Pascal did. What we know for sure is that others invented it before any of them was born. So both names are misnomers, or at least give credit to the wrong people, but it's not our job to fix that - I mean, the name. What we can and should fix is giving credit to the right people when describing the history of the triangle, including the roles played by Khayyam, Pascal and others.

If names used in other languages must be listed at the top, the one used where you live is not the only one to be added.

PS. I am a bit of a Khayyam fan, but that does not affect the above.--Niels Ø 16:23, 9 October 2006 (UTC)

As can be gathered from my revert, I fully agree with Niels. -- Jitse Niesen (talk) 03:59, 10 October 2006 (UTC)

Agree too. Oleg Alexandrov (talk) 15:36, 10 October 2006 (UTC)

(See also the section Talk:Pascal's triangle/Archive 1#Recent changes to the lead further down on this talk page.)--Noe (talk) 13:26, 17 April 2008 (UTC)

I do not why you do not want to accept the reality and the fact.i accept that pascal's triangle is more popular than other names but we must respect the scientists and the one who was the first .

ya this page is an English page in wikipedia but it is not a good accuse for destroying the history and a great scientist.last time i added these terms to this page khayyam-pascal triangle and Tartaglia's triangle for respecting the scientist and the nations but again, you refuse to accept it and i got depressed because you as a mathematician do not respt the father's of mathematician and mathematics. —Preceding unsigned comment added by Roohollah1988 (talk • contribs) 21:41, 16 January 2010 (UTC)

It's still not our job to change the name, only to give credit to the right people, which we do in the lead (briefly) and in the history section (in more detail).--Nø (talk) 22:03, 16 January 2010 (UTC)

A very high proportion of theorems, perhaps most, in mathematics are named after the wrong person. However it is not Wikipedia's place to set the world right. What Wikipedia does is report on the world as as best it can from WP:reliable sources.. It is an encyclopaedia. It is not a place to argue over who first discovered something and then change the names given to things. If somebody does that elsewhere somehow the certainly Wikipedia can report it but not before. This is all part of WP:verifiability. Dmcq (talk) 22:14, 16 January 2010 (UTC)

It is well probable that Pascal was aware of Khayyam's work. i.e. is it a practice of plagiarism by Pascal? Anyway naming the triangle after Pascal is a misnomer and should be corrected. --Nevit (talk) 19:48, 22 April 2010 (UTC)

I'm agree that the Wikipedia should not change a famous name .But I remember that the English is a global language and also the Khayam's triangle and Khayam-Pascal's triangle are two other famous names in the world . I think these names should be added as the other names in this article. Parham23 (talk) 21:47, 8 October 2019 (UTC)

@Parham23: If you read the (excellent) history section of the article, you will quickly understand the problem: it is obvious that the article should not begin "Pascal's triangle (also known as Staircase of Mount Meru, Khayyam triangle, Yang Hui's triangle (杨辉三角; 楊輝三角), and Tartaglia's triangle) is ..." and there is no particular reason to weight the Persian name over any of the others. By contrast, there is a very good reason to given the name used in English special prominence, namely, this is an English-language encyclopedia. I would expect the Farsi Wikipedia to have an article on Khayyam's triangle, the Chinese Wikipedia to have an article on Yang Hui's triangle, the Italian Wikipedia to have an article on Tartaglia's triangle, etc. --JBL (talk) 00:23, 9 October 2019 (UTC)

Extending Pascal's triangle

The article currently has a section on "Extensions", but it has some problems.

First, it only covers two of the (infinite) possible extensions (which are mirror images of each other). These two may actually be the most common extensions (I have no idea how to verify that, but it seems reasonable), but they're hardly the only ones. Many other extensions have been studied, and have different useful properties. (For example, extending either line of 1s with "1/2 1/4 1/8 …" gives you a symmetrical extension, where the upper right and upper left have useful patterns of dyadic fractions). In fact, there are even extensions that "cheat" (e.g., the one discussed in https://www.sciencedirect.com/science/article/pii/089812219190119O which extends both lines of 1s, leading to a single node that implies that "1 = 1 + 1") that are still interesting and studied, and so probably worth mentioning.

The section also doesn't mention what happens to the rest of the plane under the two extensions. Since you're forced to fill everything else with zero, there's not much useful information there—but it's still useful to understand how choosing this ray of 1s forces the rest of the plane. More generally, it should be explained how picking any arbitrary rule for a single ray or jagged ray forces the rest of the plane, rather than just how one particular rule forces one sextant of the plane.

The section also doesn't explain why you'd want to extend the triangle, how any extension breaks the plane into six sections, why people explore different extensions (symmetry, integers vs. dyadic fractions, …), which extensions are commonly used or studied, …

Unfortunately, I'm not aware of any sources for the general idea of extending the triangle in the first place. Nor do I know of any surveys that compare a variety of different extensions and explain what they're useful for. The only sources I know of cover a single extension, and assume you already know the general idea. So, I have no idea how to improve this in a way that wouldn't be blatant original research or synthesis. But hopefully someone else either knows of some good sources, or has a better solution. --157.131.201.206 (talk) 20:09, 18 March 2019 (UTC)

Also, "extending Pascal's triangle" doesn't have to mean just covering (more of) the plane this way. Various things have been called "extensions to Pascal's triangle", such as triangles built by multiplication rather than addition, Pascal's pyramid or other non-triangle/plane analogues, the whole variety of "Pascal-like triangles" discussed by Atanassov in the series in _Notes on Number Theory and Discrete Mathematics_, etc.

Meanwhile, I found a book (https://www.fq.math.ca/pascal.html) called _Generalized Pascal Triangles and Pyramids: Their Fractals, Graphs, and Applications_ published by The Fibonacci Association (who are at least notable enough to have an article on Wikipedia) that looks like it might provide the useful survey, but I'll have to read a bit of it to see. --157.131.201.206 (talk) 00:14, 19 March 2019 (UTC)

I agree that, at present, the section is in poor shape; probably, it is the result of someone adding their own OR. Real sources would be excellent. --JBL (talk) 21:02, 19 March 2019 (UTC)

Agree.---Wikaviani ^{(talk) (contribs)} 21:08, 19 March 2019 (UTC)

Came here looking for more info on extensions. Glad to see others with the same thought, disappointed that it's been three years. Pascal's Triangle, or whatever people choose to call it, imho is one of the most amazing things in mathematics, both widely accessible and simple to understand the basics of, while also deeply complex and widely relevant to other parts of math. — Preceding unsigned comment added by 137.118.202.248 (talk) 02:40, 25 January 2022 (UTC)