This may look like a neatly
arranged stack of numbers, but it's actually a mathematical treasure trove.
Indian mathematicians called it the Staircase of Mount Meru. In Iran, it's the
Khayyam Triangle. And in China, it's Yang Hui's Triangle. To much of the
Western world, it's known as Pascal's Triangle after French mathematician
Blaise Pascal, which seems a bit unfair since he was clearly late to the party,
but he still had a lot to contribute. So what is it about this that has so
intrigued mathematicians the world over? In short, it's full of patterns and
secrets.
First and foremost, there's the pattern that generates it. Start with
one and imagine invisible zeros on either side of it. Add them together in
pairs, and you'll generate the next row. Now, do that again and again. Keep
going and you'll wind up with something like this, though really Pascal's
Triangle goes on infinitely. Now, each row corresponds to what's called the
coefficients of a binomial expansion of the form (x+y)^n, where n is the number
of the row, and we start counting from zero. So if you make n=2 and expand it,
you get (x^2) + 2xy + (y^2). The coefficients, or numbers in front of the
variables, are the same as the numbers in that row of Pascal's Triangle. You'll
see the same thing with n=3, which expands to this. So the triangle is a quick
and easy way to look up all of these coefficients. But there's much more. For
example, add up the numbers in each row, and you'll get successive powers of
two. Or in a given row, treat each number as part of a decimal expansion. In
other words, row two is (1x1) + (2x10) + (1x100). You get 121, which is 11^2.
And take a look at what happens when you do the same thing to row six. It adds
up to 1,771,561, which is 11^6, and so on. There are also geometric
applications. Look at the diagonals. The first two aren't very interesting: all
ones, and then the positive integers, also known as natural numbers. But the
numbers in the next diagonal are called the triangular numbers because if you
take that many dots, you can stack them into equilateral triangles. The next
diagonal has the tetrahedral numbers because similarly, you can stack that many
spheres into tetrahedra. Or how about this: shade in all of the odd numbers. It
doesn't look like much when the triangle's small, but if you add thousands of
rows, you get a fractal known as Sierpinski's Triangle. This triangle isn't
just a mathematical work of art. It's also quite useful, especially when it
comes to probability and calculations in the domain of combinatorics.
Say you
want to have five children, and would like to know the probability of having
your dream family of three girls and two boys. In the binomial expansion, that
corresponds to girl plus boy to the fifth power. So we look at the row five,
where the first number corresponds to five girls, and the last corresponds to
five boys. The third number is what we're looking for. Ten out of the sum of
all the possibilities in the row. so 10/32, or 31.25%. Or, if you're randomly
picking a five-player basketball team out of a group of twelve friends, how
many possible groups of five are there?
In combinatoric terms, this problem
would be phrased as twelve choose five, and could be calculated with this
formula, or you could just look at the sixth element of row twelve on the
triangle and get your answer. The patterns in Pascal's Triangle are a testament
to the elegantly interwoven fabric of mathematics. And it's still revealing
fresh secrets to this day. For example, mathematicians recently discovered a
way to expand it to these kinds of polynomials. What might we find next? Well,
that's up to you.
