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Oct 2012### Pascal’s Triangle and Chinese

This is one of those blog posts where I take two seemingly very different topics and connect them to China or Chinese. This time it’s about Pascal’s Triangle, one of my favorite mathematical concepts. In case you’re unfamiliar with Pascal’s Triangle, here are some images from Wikimedia Commons that nicely illustrate the principle:

Here’s the China connection (via Wikipedia):

> The set of numbers that form Pascal’s triangle were known before Pascal. However, Pascal developed many uses of it and was the first one to organize all the information together in his treatise, *Traité du triangle arithmétique* (1653). The numbers originally arose from Hindu studies of combinatorics and binomial numbers and the Greeks’ study of figurate numbers.

> […]

> In 13th century, Yang Hui [杨辉] (1238–1298) presented the arithmetic triangle that is the same as Pascal’s triangle. Pascal’s triangle is called Yang Hui’s triangle in China. The “Yang Hui’s triangle” was known in China in the early 11th century by the Chinese mathematician Jia Xian [贾宪] (1010–1070).

Yang Hui’s diagram contains some interesting-looking numbers. Check it out:

Compare that to Pascal’s triangle above. What’s up with these Chinese numbers? You can follow the upper-right to lower-left diagonal (one row in) to follow the numbers 1-8. You get this:

1. 一

2. 二

3. 三

4. 亖

5. [no Unicode symbol for this one; it’s just 亖 + 一 (vertical)]
6. ᅡ

7. ᅣ

8. [no Unicode symbol for this one; it’s just ᅵ + 三 (horizontal)]

You can gather that 10 is 으 [a symbol I borrowed from Korean Hangul for the purposes of this post], which also looks like “10” turned sideways. 20, though, is 〇二 [except with the 〇 sitting on top of the 二], and so on.

I’ve written before on Chinese number character variants, but these are different from those. The numbers look similar to Suzhou numerals and Shang oracle numerals, but are still a bit different from both. I’m curious if anyone out there know more about these numbers? The diagram supposedly dates to 1303 (more info on Wikimedia Commons).

There’s another personal connection between me and Pascal’s Triangle. As part of my research for AllSet Learning, I make use of basic set theory and higher-level Venn diagrams. Considering that in a Venn diagram, by definition, all possible logical relations between sets *must* be represented, it can get quite tricky to draw these things when you delve into Venn diagrams with higher numbers of sets (more than 3). But how do you know how many overlapping regions there are in the Venn diagrams as the numbers of sets increase? Pascal’s triangle.

(BTW, some of the research we’re doing now at AllSet Learning could make use of interns with a foundation in statistics, mathematics, or computer science. If that’s you, get in touch! More on AllSet Learning’s interns here.)