Talk:Pascal's triangle

Phrasing/Potential need to rewrite

Under Relation to geometry of polytopes, there are several instances where phrasing is weird and might benefit from a rewrite. I would appreciate any insight before making changes.

For example, at the end of the first subsection it says: "This new vertex is joined to every element in the original simplex to yield a new element of one higher dimension in the new simplex, and this is the origin of the pattern found to be identical to that seen in Pascal's triangle." This feels out of place when reading, and also somewhat repetitive to things stated prior.

Generally speaking, several points end in a way that feels abrupt and makes explanation difficult to follow. There is also a lot of restating that feels shoved in and not actually beneficial to material. EM 1NH3 (talk) 09:23, 10 February 2025 (UTC)

To arbitrary bases section ...

... is a total nightmare, largely incomprehensible, with dubious use of both sources and wikilinks. I see that it was added with a pointer to a talk-page discussion in which two editors (myself and Mgnbar) objected to it and no one but the proposer agreed. Can anyone do anything about it? --JBL (talk) 19:15, 29 July 2025 (UTC)

The 1st figure of this Article seems blank.

Hi. Currently, the 1st figure of this Article seems blank or empty. I'm using the dark mode of the Wikipedia, and it seems a complete white. This issue seems to be fixed. This issue also affects other articles using the same figure, such as Binomial Theorem. Goodphy (talk) 05:09, 7 November 2025 (UTC)

Hello! Why not to correct the three images in the same way as it was done at Binomial theorem. Let us fix the issue. ~2026-22039-85 (talk) 17:40, 5 June 2026 (UTC)

Al karaji

The encyclopedia brittancia article of al-karaji literally attributes him for the first description of Pascal Triangle but the wiki article literally says that he isn't the first for description of Pascal Triangle or is it maybe than the encyclopedia brittancia may talk about the Pascal Triangle within Islamic mathematics? and it has been already discussed on archive talk page discussion related to it Talk:Pascal's triangle/Archive 3#PingalaMyuoh kaka roi (talk) 16:26, 17 July 2026 (UTC)

The coverage at Binomial theorem#History is a bit clearer than here, probably. It's not entirely obvious what should count as "Pascal's triangle" per se, since the same content can appear in various different presentations. –jacobolus (t) 20:39, 17 July 2026 (UTC)

Again, because the writer is the same who wrote the AL-Karaji entry on helaine source; The archive's harsh critique of Indian mathematics by many, is overly strict and harsh(and judged through modern lens) and overlooks a common historical reality: many foundational works from antiquity are preserved only through later commentators who smetimes quotes their relavant work; or comment their work(this is quite common from ancient Indian tradition; Birch bark hardly survives). For instance, Al-Karaji's specific contributions are known primarily through 12th-century successors because his original texts are lost. Similarly, Pingala’s ancient Sanskrit works on combinatorics and binary numbers survive largely through the later commentary of Halayudha and other. This reliance on secondary transmission is a standard aspect of ancient mathematical history not unique but very common things! I have been researching from atleast 2 weeks nearby on this! — Preceding unsigned comment added by ~2026-41329-79 (talk) 05:53, 24 July 2026 (UTC)

Pascal's Triangle

Regarding this edit, its quite outdated, since the scholars such as Bag notes "Singh3 pointed out the existence of an identical triangular array of binomial coefficients, known as meru-prastara, in Pingala's Chandahsütra (200 B.C.), which received only passing notice in a footnote in Needham's admirable review4 with the remark, based on Luckey,5 that it 'has nothing to do with binomial coefficients. It is proposed to show that, apart from very early discovery in India of the triangular arrangement of binomial coefficients, the technique really appeared in association with the expansion of a binomial with positive integral exponents, posed by metrical problems." or even David Fowler who deemed Needham saying "but this surely is too restrictive a view..." source- 1 and 2

Again, for the following summary by the user "this is further confirmed by refrence five "Among those who considered the triangle before Pascal, we find: Al-Karaji (953-1029); Jia Xian (1010-1070), China; Al-Samawal al-Maghribi". One can clearly see Al-Karaji being the earliest. Anyway this was already discussed"

No, one cannot -because the author meant, besides from Pingala and Halayudha then followed by "...efore Pascal, we find: Al-Karaji , Jia Xian (1010-1070), China; .."

I will quote again what the author actually implied:

Known for more than a millenium in Asia and Europe (cf. Burton [Bur'07]), the origins of the Pascal triangle are lost in the mist of time. In Chandahsāstra, the Hindu scholar Pingala has classified meters (chandas) or rhythm of poems that are closely allied to music (Bag [Bag'66]). He enumerated and counted the meters of a given length n that have exactly r syllables of a kind. In doing this, he obtained Meruprastāra (the stairway to the mythical mountain Meru). Then Halāyudha (cca. 975) in Mṛtasañjīvanī, a text of commentaries on Pingala's Chandahsāstra, clearly described Meruprastāra as what is today known as the arithmetic triangle of Pascal.

Among those who considered the triangle before Pascal, we find: Al-Karaji (Jia Xian (1010-1070), China; Al-Karaji (953-1029); Al-Samawal al-Maghrib Iraq, Marocco, Iran in his treatise The brilliant in algebra ; Omar Khayyárn called Khayyám triangle in Iran; Yang Hui's (1238-1298) triangle in China; in

the Old Method by Chu Chijie in Siyuan yujian ..."

~2026-41329-79 (talk) 02:40, 24 July 2026 (UTC)

I think the first description of Pascal Triangle in Indian mathematics comes from halayuda as pingala is describing of binomial coefficient and encyclopedia brittancia article of al-karaji describe him as the founder of Pascal Triangle.I like to pin @Nø: and @Hu741f4: regarding this matter Myuoh kaka roi (talk) 07:40, 24 July 2026 (UTC)

Halayudha's description is precisely "Pascal's triangle" in essentially the modern form described in this article. He says you put 1 box on the top row, then 2 boxes below it, then 3 boxes below that, etc., filling the outside boxes of each row with 1s and filling each other box with the sum of the numbers in the two boxes above it. Halayudha attributes this to Pingala, but whether this is quite what Pingala himself was doing is not entirely clear to me. –jacobolus (t) 10:44, 24 July 2026 (UTC)

You need to read my above reply!

It's precisely because the author of Britannica and Al-Karaji entry to the Helaine source is same actually! Halayudha did actually write commentary on Pingala's treatise only!

Infact so called Al-Karaji's original work is also lost and have been known only through 12th century mathematician actually! ~2026-41329-79 (talk) 12:04, 24 July 2026 (UTC)

You may also want to look at Edwards's book as a source, https://books.google.com/books?id=CRmUDwAAQBAJ&pg=PA27 –jacobolus (t) 10:55, 24 July 2026 (UTC)

To say, I feel that this page is continously going through change by previous year itself, and infact whole Indian section itself was removed once; and till date it has been argued even though the description of the Pascala's triangle is crystal clear; nonetheless it is perfect at the place right now,

For example, beside what I quoted as a reference above, In the Asia, Europe, and the Emergence of Modern Science, historian George Gheverghese Joseph too mentions and noted this in page 41-42:

"Here we may

cite the fact that Pascal tried to establish the formula (1) cited above by using the so-called Pascal’s triangle for the binomial coefficients. This ....Chandasutra of Pingala and later in the tenth-century work of the commentator Halayudha. By the eleventh century it was also known to the Islamic

world and the Chinese.*"

(again sorry for inconvenience, it rendered ( ) after pasting and I have removed some unusual mark)"

Similarly, it is also noted by Plofker, pg-57]

(5)

"Finally, to find out how many of the 2" metrical patterns contain a specified number p of, say, heavy syllables, Pingala prescribes the construc-tion of what later commentators call a "Meru-extension" or mountain-shaped figure, which in fact is just what we know as Pascal's triangle. (The reader may like to try proving that the (n - p + 1) th entry in the (n+1)th row of Pas-cal's triangle does in fact equal the number of metrical patterns containing p heavy syllables.)"

Both of them are probably most cited historian in the field of Indian Mathematics.

Obviously there are more prominent historian including Radha Charan Gupta and other many sources stating so. But the above shall suffice I guess! — Preceding unsigned comment added by ~2026-41329-79 (talk) 08:57, 24 July 2026 (UTC)

This discussion started with the fact that the article asserted that al Karaji was the first known discoverer of Pascal's triangle just after the sourced assertion that Pingala discovered it much earlier. For avoiding endless discussions, the simpler is to remove "first" from the sentence.

Nevertheless Pascal's triangle has several aspects:

1. The construction method. Undoubtly to be credited to Pingala
2. The use for counting Combinations, also to be credited to Pingala
3. The use for computing binomial coefficients. I cannot decipher from above quoted sentence who discovered this application. Certainly not Pingala; possibly Al Karaji, but probably not because expanding binomial powers was not yet a problem at his time. So, it is possible that Pascal was the first discover of the relation between the triangle and the expansion of binomial powers.

Sources are needed for that. Without them, "first" must be removed from the article. D.Lazard (talk) 11:47, 24 July 2026 (UTC)

Yes, and also both sources actually comment to what Needham initially reacted.

David Fowler:

"...but this surely is too restrictive a view. It is true that the Indian interest is mainly in the

patterns that can arise in Sanskrit prosody (though not exclusively; for example

Brahmagupta (AD 629) did consider (a + b)^3), but the western studies of figured

numbers, binomial coefficients, and combinations are, ultimately, a search for the

same patterns and yields the same collections of numbers. (For another illustration

involving the Fibonacci numbers, see P. Singh, SCFN.) Here, for example, is

Halayudha's description of how to generate an enumeration of these syllabic

combination."

and Bag:

">..In addition to the method of computing the binomial terms b^4, b^3а, b^2a^2, etc., in the foregoing example, Pingala gave his general meru-prastāra rule for determining such binomial coefficients, which is exactly the same as Pascal's 'triangular array', or Chu Shih-Chieh's 'the old method chart of the seven multiplying squares'. The rule has been explained by Halayudha as follows.." ~2026-41329-79 (talk) 12:34, 24 July 2026 (UTC)

There are plenty of sources claiming "first" for this or that (including claiming al-Karaji was "first" to make a table of binomial coefficients), but in general I think such claims are problematic, (a) because they are essentially unprovable (only disprovable), being negative claims about whatever was prior, and (b) because often prior works are similar and later works are more developed, and there's typically not universal agreement about which features of an idea are required to claim priority for it. There are also sometimes problematic nationalist disputes that crop up on Wikipedia when someone wants to claim inventions or discoveries for "their" side. I prefer to just give a no-nonsense description of what was done and what evidence we have for that (e.g. there is an extant copy of a work from century Y which is a commentary on an earlier work from century X, and attributes ideas A, B to so-and-so), and then readers can come to their own conclusions. In this case, there were independent descriptions of this idea in around the 10th century if not earlier from India, Persia, and China. We should just briefly describe these. If someone feels motivated, this could be expanded in a more extensive article called History of binomial coefficients or similar. –jacobolus (t) 18:33, 24 July 2026 (UTC)

I agree with you (and so I think also with D.Lazard) that this is a good way of handling things. --JBL (talk) 00:39, 26 July 2026 (UTC)

Hmm! though I think later works are "more elaborated" or "more detailed" would be appropriate choice of word here. Also, many of the mansucript are already lost, only some of it is recovered or have been known through the later mathematician and the commentators. Offcourse, we can't also ignore the fact that Oral tradition was prominent in ancient Indian tradition due to their spiritual needs and also many of the materials that were written on Birch bark hardly had the tendency to survive for the long time. ~2026-41562-51 (talk) 03:50, 26 July 2026 (UTC)

• This may be of interest to the some editors here ; I found this researching back long periods.

Pingala Prastara Varna Matra Patakadi Yantrani 775 Gha Alm 4 Shlf 3 Devanagari Alankar Shastra from 775 AD.