In Search of Symmetry
I watched a TED talk called Symmetry: Reality’s Riddle given by Marcus du Sautoy, an Oxford mathematics professor:
http://www.ted.com/talks/marcus_du_sautoy_symmetry_reality_s_riddle.html
He studies the concept of symmetry by means of mathematical structures such as groups and the building blocks of prime numbers, which are the “atoms” of number theory. This mathematical concept of symmetry underlies a lot of physical phenomena.
It was a fascinating talk and it brought me back to the days when mathematics was the polestar around which my academic world revolved back in junior high school…
1. When Logic Ruled my World
My eighth-grade math teacher was especially encouraging, and she allowed me the creative outlet of doing special reports on various subjects like symbolic logic, complex numbers, or abstract algebra. These reports kept getting longer and longer, until the last one I did covered two full spiral notebooks. It was a comparison between the symbolic logic notation our algebra books used and the so-called Polish notation that was popularized with the logic game called Wff ‘n Proof.
My interest in the syntax of mathematical logic was connected with my growing interest in the syntax of natural languages. In the same way that there are natural languages that have different syntax in the form of differing word order, logic notations can have a different order of the logic variables (x, y) and the logic operators (AND, OR, NOT).
So the logic notation used in our algebra books was an infix notation, where the logic operator comes in between the two variables, such as A OR B. The Polish notation was a prefix notation, so the operator comes before the variables, such as OR A B. Something called Reverse Polish Notation or RPN was yet another syntax where the operator came after the variables, such as A B OR.
Why is this significant? When you have an infix notation, such as the one used in our algebra books, you need brackets to make sure you do the operations in the correct order if you have a more complex expression such as A AND B OR C. This can be interpreted as (A AND B) OR C or A AND (B OR C). But if you say A B AND C OR in postfix notation or RPN you get the same as the first result unambiguously without parentheses, and the second you would get with B C OR A AND. The elimination of clumsy parentheses made RPN useful for handheld calculators, and many engineers got used to using it.
As my education progressed, my passion for mathematics started to be rivaled by other passions: foreign languages and music in high school, physics and philosophy in college, art and literature between my undergraduate and graduate school years, and finally history in graduate school.
2. Adventures in Mathematics
However, Marcus du Sautoy’s talk brought me back to the primacy of mathematics as my main intellectual interest back in junior high school. As I mentioned above, I did special reports in the areas of abstract algebra, which I learned these through a series of advanced mathematics textbooks that my father got me called Adventures in Mathematics published by Science Research Associates. It tried to introduce various concepts of higher mathematics in a form that would be palatable to those in the upper school grades. For example, it taught abstract algebra by introducing “clock mathematics.” If it’s 9:00 AM, and you have a four-hour meeting, it ends at 1:00 PM, which you get through the following logic: 9 + 4 = 13, but any time your answer is over 12, you subtract 12 from it until you get an answer that it between 1 and 12, in this case 1. You can perform arithmetical operations on a clock that has a different number of hours than 12, such as 5, and you would get the following if you constructed an addition table of the hours for a 5-hour clock, with the 5^{th} hour being designated by a 0:
Figure 1: Addition on a five-hour clock (modulo 5)
+ |
0 |
1 |
2 |
3 |
4 |
0 |
0 |
1 |
2 |
3 |
4 |
1 |
1 |
2 |
3 |
4 |
0 |
2 |
2 |
3 |
4 |
0 |
1 |
3 |
3 |
4 |
0 |
1 |
2 |
4 |
4 |
0 |
1 |
2 |
3 |
You notice how each of the hours is represented in each column and each row. This is a property that shows that there is some underlying symmetry in this operation (addition) on this particular set of numbers (those on a 5-hour clock).
This “occurs once in each column and row” symmetry is precisely the symmetry which underlies the popular math game called Sudoku. You figure out the puzzle by using the underlying symmetry to deduce what the various numbers should be in the blank spaces.
Just in the same way that you can do an addition table using clock or modular arithmetic as it is known, you can also make a multiplication table
Figure 2. Multiplication (modulo 5)
* |
0 |
1 |
2 |
3 |
4 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
2 |
3 |
4 |
2 |
0 |
2 |
4 |
1 |
3 |
3 |
0 |
3 |
1 |
4 |
2 |
4 |
0 |
4 |
3 |
2 |
1 |
Except for the 0 row and column, which give you 0’s all the way down or across, the other rows and columns also have the property of having every number appear from 0 to 4. However, if you were to do a multiplication table for a 4-hour clock or a 6-hour clock, you would not have this same property of symmetry. I was trying to figure out why, and then it suddenly dawned on me: an n-hour clock would have this Sudoku-like property for addition AND multiplication if and only if n was a prime number. I still remember the thrill I got in having my mind pierce to the heart of a mathematical truth.
This was the first mathematical insight I remember having with respect to abstract algebra. Although this is several orders of magnitude less significant than British mathematician Andrew Wiles’ proof of Fermat’s last theorem (namely that a^{n}+b^{n}=c^{n }has no solutions among positive integers a, b, and c if n > 2), I still feel some sort of spiritual kinship with him, because I know what that thrill of discovery and challenge of proof feel like. You don’t have to be able to climb Mt. Everest to experience the thrill of mountain climbing.
I thank Marcus du Sautoy of reminding me of one of my first academic interests and bringing all of those years of “mountain climbing” back to me.
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