Meme that’s making the rounds this week: This New Year, 2024, is tetrahedral.
What does that mean? No, it has nothing to do with those containers various juices and milk and the like come in. Then again, maybe it does. Tetra Paks were originally crafted as tetrahedrons—thus their name—but at some point they morphed into today’s ubiquitous cuboids.
Before your eyes glaze over, let me explain. Cuboids are box-shaped objects—Tetra Paks, sweet boxes, cartons, and others like that. Tetrahedrons are pyramids on a triangular base. This means they have four sides—the triangle at the bottom and, rising from its sides, three more triangles that meet at a point on top. The earliest Tetra Paks had this shape. I’m not sure why or when they changed, but you can still occasionally buy things in a tetrahedral Tetra Pak.
With that pyramid in mind, imagine a small experiment to build a tetrahedron. Start with one small marble, that you place on a table. Not much of a tetrahedron there, but indulge me as I call it that: looks like a sphere, but imagine it as a tetrahedron. Now pick up that marble and lay down three more, touching each other to form a triangle. That’s the triangular base. Place the first marble on those three, forming a two-level pyramid using four marbles. There’s a tetrahedron again, and this time you can certainly discern the shape. Four identical equilateral triangles form the four sides.
Next, a one-step bigger pyramid—three levels. Lay down six marbles to form an equilateral triangle. On top of it, place three more as above, forming a smaller equilateral triangle. Finally, a single marble on top. Once more a tetrahedron. Once more, four identical equilateral triangles—bigger ones, of course—form the four sides. You’ve used 10 marbles this time.
There’s a series forming here: one marble, then four, then 10. What’s the next number? That’s the number of marbles you need for a four-level pyramid. You will start with a base that uses 10 marbles, for a tetrahedral total of 20.
That is, 20 is the fourth tetrahedral number. Keep going like this, building ever taller pyramids, and eventually you’ll have crafted a pyramid that uses 253 marbles at the bottom and rises 22 marble levels from there. That gives us the 22nd tetrahedral, which is. . .yes ma’am, 2024.
Savour that. It’s probably been at least 150 years since the last human who was alive in the last tetrahedral year, 1771, died; and it will probably be at least 170 years until the first human is born who will see the next, 2300. So, you and I, and the rest of humanity today, are rather lucky to live through a tetrahedral year: this one, 2024.
Still, even with my suggestion to savour it, I imagine all this doesn’t do much for you. These are just more numbers, after all. Except, mathematicians know that when it comes to numbers, there are endless patterns to discover—which is what fuels their research efforts. So with tetrahedral numbers.
For example, take the series of numbers I’ve also touched on above, that make up the successive triangular bases of the pyramids. That series goes like this: 1, 3, 6, 10. . .They are called the triangular numbers. Of course, the 22nd entry there is 253, the base of the tetrahedron that has 2024 marbles.
Look at that series like this. You have crafted a given tetrahedron, say, the two-level pyramid with four marbles. To now produce the three-level pyramid, you need your existing one to sit on top of a triangular base made up of six marbles. That is, the third tetrahedral number is the sum of its triangular base and the second tetrahedral number. More generally, any given tetrahedral is the sum of its triangular base and the previous tetrahedral.
That will persuade you that the triangular numbers are the successive differences between the tetrahedral numbers. So, a little thought—visualize the pyramid if you like—takes us here: a given tetrahedral number, then, is the sum of its triangular base and every smaller triangular base. Or, as a mathematician might put it: “The nth tetrahedral number is the sum of the first n triangular numbers."
And remember, you can form those triangles by successively laying down one marble, then two just below, then three below those, etc. The process shows that adding the natural numbers, or integers, gives us triangular numbers.
Triangular numbers have their own charms that have fascinated mathematicians for centuries. There’s a famous story, possibly apocryphal, of how a boy named Carl Friedrich Gauss, who grew up to be one of the world’s most influential mathematicians, found the formula that produces any given triangular number.
But the triangulars also turn up in the just-as-famous “handshake problem". Suppose you are at a party at which everyone shakes hands exactly once with everyone else. Question that I’m sure you have puzzled over at many parties: How many handshakes?
The answer is in the triangular numbers. Let’s say there are six people there: you, me, Priya, Rahul, Alain, and Dhana. You start, and shake all five hands. Since I’ve already shaken hands with you, I shake four other hands. Priya has shaken hands with you and me, so she has to shake three more. On we go down the line to Dhana, who has no hands left unshaken. You can see: the number of handshakes is 5 4 3 2 1 = 15, the 5th triangular number.
The point of all this? Again, the patterns. We like to find something special or unique about the numbers around us, like 2024. That it is tetrahedral is certainly unique to this particular moment in human history, as spelled out above. You may or may not find meaning in that, sure. But there are those who do, and who even go digging for more. Fuel, remember, for ever more research efforts. Truly, there’s no end to the patterns we can discover in numbers.
All of this brings me to another meme that’s making the rounds this week: 2024 = 2³ 3³ 4³ 5³ 6³ 7³ 8³ 9³ (The sum of the cubes from 2 to 9.)
Which is pretty enough, you’ll agree. But as the mathematician John Allen Paulos pointed out on Twitter, we might want to wait till next year, when this pattern will be “even a bit more remarkable".
Small puzzle for you: Why does Paulos say so?
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun.
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