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What is Pascal's Triangle? (Part 2)

Learn about the history of Pascal’s famous triangle and discover some of the amazing number patterns—including the Fibonacci sequence—hidden within it.

By
Jason Marshall, PhD
4-minute read
Episode #124

What is Pascal's Triangle? (Part 2)

 

Now that we’ve learned how to draw Pascal’s famous triangle and use the numbers in its rows to easily calculate probabilities when tossing coins, it’s time to dig a bit deeper and investigate the properties of the triangle itself. Why is that an interesting thing to do? Because it turns out that Pascal’s triangle is not a one trick pony—it’s useful for a surprising number of things.

And not only is it useful, if you look closely enough, you’ll also discover that Pascal’s triangle contains a bunch of amazing patterns—including, kind of strangely, the famous Fibonacci sequence. Where is it? And what other patterns are hidden in the triangle? Stay tuned because that’s exactly what we’re talking about today.

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Did Pascal Discover Pascal’s Triangle?

Before looking for patterns in Pascal’s triangle, let’s take a minute to talk about what it is and how it came to be. As you’ll recall, this triangle of numbers has a 1 in the top row and 1s along both edges, and each subsequent row is built by adding pairs of numbers from the previous. So the first row is just 1; the second row is 1, 1; the third row is 1, 2, 1; the fourth row is 1, 3, 3, 1; then 1, 4, 6, 4, 1; and so on.

We keep calling this pattern “Pascal’s triangle,” but who is that? And was he actually the first person to study this pattern? Well, Pascal was a French mathematician who lived in the 17th century. And, no, he was not the first person to study this triangle…not by a long shot. It turns out that people around the world had been looking into this pattern for centuries. So why is it named after him? It’s probably partly due to cultural biases, and partly because his investigations were the most extensive and well organized. Which meant that soon after publishing his 1653 book on the subject, “Pascal’s triangle” was born!

Patterns in Pascal’s Triangle

With that bit of history out of the way, let’s turn our attention to patterns. As we’ve seen before, looking for patterns in numbers is a big (and very fun) part of math. And, in this sense, Pascal’s triangle provides quite a feast! For example, one immediately obvious—although extremely simple—pattern is that the top-most diagonal of numbers in Pascal’s triangle contains only 1s.

 

That might not be very interesting, but what about the other diagonals? Does anything more interesting show up? Indeed it does in the very next one. The second diagonal contains all the counting numbers: 1, 2, 3, 4, 5, and so on. Can you see any other patterns? Perhaps in the next diagonal? Or elsewhere? Go ahead and take a few minutes to investigate. Don’t worry though if you can’t find anything—getting good at looking for patterns in numbers takes practice.

Triangular and Tetrahedral Numbers

So, did you find any interesting patterns? In truth, besides the fact that Pascal’s triangle is symmetric—meaning that its left side is a mirror image of its right side—the remaining patterns aren’t obvious. But there is one sitting right beneath the diagonal of counting numbers we looked at earlier. I’m talking about the diagonal that contains the numbers 1, 3, 6, 10, 15, 21, and so on. Why are those numbers special? Because they form a sequence known as the triangular numbers.

What exactly are these triangular numbers? It’s easiest to see with a drawing—so grab yourself a sheet of paper and a pen. The name of the game with our drawing is to build triangles using an evenly spaced grid of dots. The first triangle has 1 dot on each side, which—kind of strangely—means that this “triangle” contains a total of 1 dot. The second triangle has 2 dots on each side. When you draw this triangle using an evenly spaced grid of dots, you’ll find that the entire triangle requires 3 total dots to draw. The next triangle with 3 dots per side requires 6 total dots to draw, the triangle with 4 dots per side requires 10 total dots to draw, the next will have 15, then 21, and so on.

 

Does this pattern of numbers—1, 3, 6, 10, 15, 21, and so on—look familiar? That’s right—it’s precisely the sequence of triangular numbers we found along the third diagonal of Pascal’s triangle. And now you know why these numbers are called “triangular”! What about the fourth diagonal of the triangle? Is that some special sequence too? Yep, they’re called the tetrahedral numbers…and the fifth diagonal contains the pentatope numbers. Both of these are perfectly interesting in their own right, but I want to move on to talking about something even more interesting: the Fibonacci sequence.

The Fibonacci Sequence

As I mentioned earlier, the Fibonacci sequence is hidden in Pascal’s triangle—but it’s kind of hard to find. The easiest way to discover it is to start by drawing Pascal’s triangle left-justified so that each of its diagonals from before are lined up in a column. If you add up the numbers along these new diagonals, you’ll obtain the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, and so on. Pretty amazing, right?

Although we’re going to stop our search here, this is by no means the last of the patterns to be found in Pascal’s triangle. For example, the pattern of odd and even numbers in the triangle is related to another very cool object called the Sierpinski triangle, each row of the triangle is curiously related to the powers of 11, and—as I encourage you to investigate—the sum of the numbers in each row of the triangle forms a rather surprising pattern. As I said, with Pascal’s triangle, the surprises just keep coming!

Wrap Up

Okay, that’s all the math we have time for today. Remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too. Finally, please send your math questions my way via Facebook, Twitter, or email at mathdude@quickanddirtytips.com.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!

About the Author

Jason Marshall, PhD

Jason Marshall is the author of The Math Dude's Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.