One of the things I’ve been doing lately is hanging art in my apartment. After four years of living here, it was finally good time. Among the things I’ve hung is this decorative wall quilt made by my maternal grandmother. The quilt is based on the cover illustration of the November 1959 issue of The Scientific American, which represented a then-recent breakthrough concerning a type of mathematical object called a Graeco-Latin square.
A Graeco-Latin square can be understood as a grid, $n$ cells by $n$ cells, in which every cell has two properties (in this case, outer color and inner color). Each property has $n$ possible values and, for each property, each value occurs exactly once in each row and column. Furthermore, each possible pairing of values occurs exactly once in the whole square (so that there is one and only one square with a yellow outer color and a red inner color, for instance).
Another way of illustrating this is to use Greek and Latin letters instead of outer and inner colors. (Hence the name “Graeco-Latin.”) This makes it a bit clearer that the two properties need not be related.
The example above comes from the 1959 SciAm issue. Notice how letters $a$, $b$, $c$, and $d$ each appear exactly once in every row and column, as do letters $\alpha$, $\beta$, $\gamma$, and $\delta$. If you check, you will also find that every possible combination of a Latin letter and a Greek letter appears exactly once.
In the eighteenth century, one of history’s greatest mathematicians, Leonhard Euler, studied Graeco-Latin squares. He found several general methods for constructing them, but was unable to construct a square of order 2 (that is, measuring 2 by 2), 6, or 10, or whose order was any other even number not divisible by 4. (These are called “oddly even” numbers.) He eventually conjectured that it was impossible.
The breakthrough covered by the Scientific American was the discovery by R. C. Bose, S. S. Shrikhande, and E. T. Parker of a method of constructing Graeco-Latin squares of every oddly-even order greater than or equal to 10. This showed that Euler’s conjecture, which had stood for over 150 years, was incorrect. (It had been proven earlier that squares of orders 2 and 6 are indeed impossible, which I find intriguing. Two is small and weird, so it’s not too surprising that it would behave anomalously, but 6 does surprise me.)
As well as mathematicians, this result interested scientists in general. Since Euler’s time, Graeco-Latin squares had become important in the design of scientific experiments.
I don’t have anything very interesting to say about all this. I read the SciAm article, by beloved math communicator and puzzle creator Martin Gardner, which you can check out below. (The scan is mine. I bought the issue on eBay some years ago.)
I also watched two videos on the topic from the Numberphile channel: Euler Squares and Euler Squares (extra). These are very engaging and I recommend them.
Those expositions felt a little lacking in details to me, though, so sometime soon I’m going to have a go at reading Bose and Shrikhande’s original paper, which I found available here.
I’m also looking into ancient Babylonian numerals. I watched a video on trigonometry that discussed the origin of our degrees in their base 60 number system, and it suddenly struck me that they must have had an awful lot of numerals, if, like us, they had a unique one for every natural number smaller than the base. It turns out they did, but the numerals were constructed out of sub-symbols that made them easier to identify. I’ve collected a couple more documents to read on on the topic.
Updates on the project and blog:
I’m having some trouble finding a rhythm in my math explorations. I haven’t been working on them every day, as I intended. I need to think more about what the goal of the project is now and, accordingly, what activities to prioritize.
I also need to think more about how to have math as a consistent part of my life when my life is so constantly disrupted. In college, I had a decoration on my wall (made by myself and printed on a piece of letter paper) that had a photo of James Murray, first editor of the Oxford English Dictionary, along with his slightly famous quotation, “Nothing is better than a most diligent life.” (Being a Victorian scholar, he said it in Latin, but I displayed it in English.) I think I still believe that, but I need to come up with some version of diligence that doesn’t involve pushing myself too hard.
While feeling slightly at sea, I’ve enjoyed the math I’ve done, despite my counselor’s warnings that trying to start this project up again would just make me unhappy.
