This article was published in Scientific American’s former blog network and reflects the views of the author, not necessarily those of Scientific American
This year, January third falls on a Thursday, meaning we celebrate the semiregular mathematical holiday Thirdsday. Jim Propp, mathematics blogger extraordinaire and creator (or discoverer) of Thirdsday, explains some of the reasons he is so excited about the holiday.
Thirdsday celebrates the number 1/3, arguably the simplest and most natural fraction after 1 and 1/2 and, in my opinion, a tragically underappreciated number. Though it is familiar, and itself quite finite, 1/3 offers one of the first brushes with the infinite for many math students in the form of that eternal riddle: 1/3=0.333333….
Thirdsdays occur every five to eleven years, depending on the placement of leap years. Thirdsday is my mom’s birthday. (By my calculations, she’s ten Thirdsdays old.) She planned ahead and had three children, whom she will talk to on the phone one at a time on Thursday, presuming we all remember to call. If you have not made such arrangements yet, there are other ways to celebrate.
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Propp has some ideas, with an overarching theme: Don’t celebrate Thirdsday halfway. Celebrate it thirdway. Split a pizza or some cake with two friends. Drink a third of a 6-pack of beer. Have a recitation contest of the number of digits of 1/3 you can say in one breath. (He even shares a handy mnemonic to help you remember them.)
My suggestion for Thirdsday fun is to try to relate as many other numbers as you can to a third.
One place to start is music. We perceive ratios of frequencies as musical intervals. An octave—the interval between middle C and the C above or below it on a piano—has the frequency ratio 2:1. If middle C has a frequency of 260 Hertz (vibrations per second, abbreviated Hz), the C below it has a frequency of 130 Hz, and the C above it has a frequency of 520 Hz. A perfect fifth—C to G, the opening jump in “Twinkle, Twinkle, Little Star”—has the frequency ratio 3:2. There are two types of thirds in typical Western music: major and minor. A major third—the interval from C to E, or the first interval of “Kumbaya”—has a ratio of 5:4, and a minor third—C to E♭ or A to C, the first interval of “Greensleeves,” or the “nanny-nanny-boo-boo” interval—has a ratio of 6:5. So already we’ve related 5/4 and 6/5 to a third.
But there’s more! Most instruments aren’t tuned using those precise whole-number intervals. On pianos and other instruments that create a fixed, discrete set of pitches, tuning is a compromise. The octaves couldn’t all be in tune if the fifths were, and vice versa, fifths couldn’t be in tune if thirds were, and vice versa, etc. The properties of prime numbers mean no tuning system can ever yield perfect whole number ratios for all intervals. Equal temperament, one of the most commonly used compromises, squishes all the fifths and minor thirds and stretches out the major thirds. Using equal temperament, a major third is the frequency ratio of 3√2:1, and a minor third is 4√2:1. Hey, the major third already has a third in it! A third root becomes a different type of third. If you go hunting through other tunings and temperaments, you can find a lot of other ways to write a third.
If music isn’t your bag, another place to go hunting for thirds is in the Cantor set. The Cantor set, as I’ve written before, is a classic fractal. It also happens to be very thirdy. To build one, you start with the interval [0,1] on a number line. Remove the middle third (1/3,2/3). You’re left with two intervals: [0,1/3] and [2/3,1]. Now you remove the middle thirds from those two intervals, the middle thirds from the remaining four intervals, and so on. The Cantor set is the collection of stuff left over after iterating the process infinitely many times. Or to put it another way, it’s the set of all the points that are never in the middle third of any interval.
Diagram of several iterations of the creation of the Cantor set. Credit: 127 "rect" W3C Wikimedia
From the way the Cantor set is constructed, it’s clear that some numbers are never removed. 0, 1/3, 2/3, and 1 all remain because the removed middle intervals don’t include the endpoints. More surprising is the fact that there are some other numbers left in there. For example, 1/4 and 1/10 both survive the middle thirds process. A cool trick with base 3 numbers can help you find more numbers, both fractions and irrational numbers, that are in the Cantor set. You can read more about that process in my earlier post on the Cantor set.
If you put your mind to it, I'm sure you can find the third lurking in some other numbers, too.
No matter how you celebrate, be sure to make this Thirdsday a good one! This holiday occurs on average every seven years, but we will have a bit of a dry spell until 2030, when Thirdsday reappears. But don't put too much pressure on yourself: you don’t need to be first on Thirdsday. Aim for bronze!