Ersby’s Triangle: An Update

A few weeks ago, I decided to revisit my post about Ersby’s Triangle – a semi-serious look at the multiples of square numbers needed to create Pascal’s Triangle.

I wanted to see if there were any formulae to describe the numbers along a particular diagonal. At first, I tried my original method of pen and paper and simply messing around with sequences of different powers of numbers, but nothing seemed to work. I put it to one side and forgot about it.

Then today I discovered The On-Line Encyclopedia of Integer Sequences. So I typed in a few of the diagonals (from top left to bottom right) from Ersby’s Triangle.

I soon came up with some interesting results as the first two diagonals I looked at (after the diagonal with 1, 2, 3...) both linked to the largest number of pieces you could get from a shape with each cut. The first diagonal referred to two-dimensional shapes and the next one referred to three-dimensional shapes.

So I was excited to see if that continued, and if the next diagonal referred to four-dimensional shapes.

It didn’t. But, undeterred, I kept searching.

Then I noticed that a particular paper kept being cited in the search results I was getting. I followed the link to Catalan Triangle Numbers and Binomial Coefficients by Kyu-Hwan Lee and Se-Jin Oh from the University of Connecticut. And there, tucked away on page 14, was the closest I’ve ever come to being referenced in a mathematics paper.

Amazing. If only I knew what the paper actually meant.

References:

Lee, K.H., Oh, S.J., (2016) "Catalan Triangle Numbers and Binomial Coefficients," arXiv:1601.06685v

Skeptical of mathematics

Almost every new endeavour, in fields such as science, art, or music, has gone through a period of mistrust from more established experts. It may seem strange to us now, but even mathematics went through a similar phase.

Roger Bacon, circa 1267, complained of writers who put maths as one of the seven Black Arts, due to it’s links with astrology. Despite the release in 1542 of the first comprehensive practical arithmetic in English (“The Grounde of Artes” by Dr Robert Recorde), mathematics remained linked to prognostics.

In 1624, William Monson wrote

“It is a question whether a man shall attain to better knowledge by experience or by learning? And many times you have controversies arise between a scholar and a mariner upon that point. The scholar accounts the other no better than a brute beast, that has no learning but have experience to maintain the art he proposes. The mariner accounts the scholar but verbal, and that he is more able to speak than act.”

In 1666 John Wallis wrote

“Mathematicks at the time, with us, were scarce looked upon as Academical Studies, but rather Mechanical; as the business of Traders, Merchants, Seaman, Carpenters, Surveyors of Lands, or the like, and perhaps some Almanack Makers in London... For the Study of Mathematicks was at that time more cultivated in London than in the Universities.”

And in 1701, as the tide had already turned in favour of studying mathematics, J Arbuthnot summarised those arguments against:

“The great objection that is made against the Necessity of Mathematics in the great affairs of Navigation, the Military Arts, etc., is that we see those affairs carry'd on and managed by those who are not great mathematicians: as Seamen, Engineers, Surveyors, Gaugers, Clock-makers, Glass-grinders etc., and that Mathematicians are commonly speculative, Retir'd, Studious Men, that are not for an active Life and Business but content themselves to sit in their Studies and pore over a Scheme or Calculation.”

References

Arbuthnot, J., (1701) “An essay of the usefulness of mathematical learning in a letter from a gentleman”
Taylor, E.G.R., (1954) “The Mathematical Practitioners of Tudor and Stuart England,” Cambridge University Press
Monson, W., (1624) “Naval Tracts”

Everyone gets a share of the pi

While reading about the history of mathematics in Japan I found a mathematical sequence which, for reasons I do not know, happens to contain a lot of the estimates for pi that have been put forward throughout history.

This sequence was discovered by Seki Kōwa and was published in 1712 after his death in a book called Katsuyō Sanpō. It runs as follows:

Start with the fraction 3/1. If this number is lower than pi, then increase the upper number by 4. If it is greater, then increase it by three. Then increase the lower number by one. And continue like this.

If you do this, you get a sequence of numbers that includes the estimates of pi from different people from different times in history. In this list (mostly Chinese mathematicians, since up until then Japanese maths was heavily influenced by Chinese textbooks), Kōwa finds the values from Chih (25/8 or 3.125), T'ung Ling (63/20, or 3.15), the “old Japanese” value (79/25 or 3.16), Liu Chi (142/45 or 3.155), Hui (157/50 or 3.14) and also 355/113 (which is very close to the actual value of pi at 3.14159292) but this isn't credited to anyone, so Kōwa must've known about it's use as a value of pi, but not known that Tsu Ch'ung-chih had found it.

I like mathematical sequences that seem to have a narrative, although I doubt it has any use. I don't think the series converges on the true value of pi (I haven't checked) but it does occur to me that this isn't a very good way of calculating pi, because it only works if you already know the value of pi.

Reference
D.E. Smith, Y. Makami, “A History of Japanese Mathematics”, Open Court Publishing Company, Chicago, 1914, p111-112

Ersby's Triangle

Pascal's Triangle is a famous and fascinating piece of mathematical architecture. In its sequences are many interesting patterns that also appear in the mathematics of probability, fractals, and geometry.

One day, I decided to take a closer look at it. In particular, I wanted to find a formula to describe the numbers along each diagonal.

So far, so good. But I came a bit unstuck when I reached the fourth diagonal. At first, I thought I found something to do with the differences between the numbers...

 Eventually, after several false starts I guessed that since the first two numbers in the sequence were 1 and 4 perhaps the answer was something to do with square numbers. Before long, I came up with something else...

The fifth diagonal was also difficult, but when I went back to working only with square numbers, the answer came quite quickly.

I noticed that 1 squared was being multiplied by the following numbers: 1, 1, 2, 2, 3, 3...

This time the multiples of 1 squared were 1, 2, 4, 6, 9... But I was also interested in how the multiples seemed to "move along" the sum because now 2 squared was being multiplied by 1, 1, 2, 2, 3... I wondered if something was going on so, for clarity's sake, I wrote these multiples on a copy of Pascal's triangle, beside the number that each sum belonged to.