The mathematical secrets of Pascal’s triangle - Wajdi Mohamed Ratemi

翻译人员: Winnie Ling 校对人员: Di SUN

This may look like a neatly arranged stack of numbers,

这些看上去 可能只是一堆排列整齐的数字，

but it's actually a mathematical treasure trove.

实际上，它可是一个数学的宝藏。

Indian mathematicians called it the Staircase of Mount Meru.

印度数学家称它为"须弥山之梯"。

In Iran, it's the Khayyam Triangle.

在伊朗，它是"海亚姆三角"。

And in China, it's Yang Hui's Triangle.

而在中国，它被称为"杨辉三角"。

To much of the Western world, it's known as Pascal's Triangle

在大部分西方国家， 它叫”帕斯卡三角“。

得名于法国数学家, 布莱斯 ·帕斯卡。

after French mathematician Blaise Pascal,

这似乎有点不太公平。

which seems a bit unfair since he was clearly late to the party,

因为帕斯卡的发现比其他人更晚，

but he still had a lot to contribute.

但帕斯卡也对此做出了许多贡献。

So what is it about this that has so intrigued mathematicians the world over?

那么，是什么让世界各地的 数学家们对它如此感兴趣？

In short, it's full of patterns and secrets.

简单地说，它充满了各种形式和秘密。

First and foremost, there's the pattern that generates it.

首先，这是构造三角的形式。

Start with one and imagine invisible zeros on either side of it.

从 1 开始， 并假设两边各有一个看不见的 0，

Add them together in pairs, and you'll generate the next row.

把相邻的数字加起来， 你就会得到下一行。

Now, do that again and again.

现在，重复这样的操作，

Keep going and you'll wind up with something like this,

反复进行， 你最终会得到这样一个图形。

though really Pascal's Triangle goes on infinitely.

实际上，帕斯卡三角是无限大的。

Now, each row corresponds to what's called the coefficients of a binomial expansion

它每一行的数字都对应 (x+y)^n 二项式展开的系数，

of the form (x+y)^n,

where n is the number of the row,

其中 n 是行的序号，

and we start counting from zero.

从 0 开始算。

So if you make n=2 and expand it,

当 n=2时， 二项式展开你会得到

you get (x^2) + 2xy + (y^2).

x^2 + 2xy + y^2。

The coefficients, or numbers in front of the variables,

那些系数，就是每一项变量前的数字，

are the same as the numbers in that row of Pascal's Triangle.

和帕斯卡三角对应行的数字相同。

You'll see the same thing with n=3, which expands to this.

n=3 也是一样，展开得到这个。

So the triangle is a quick and easy way to look up all of these coefficients.

所以，这个三角能让我们 快速得到二项式的系数。

But there's much more.

然而，奥秘远远不止这些。

For example, add up the numbers in each row,

比如说，把每一行的数字加起来，

and you'll get successive powers of two.

你会得到连续的2的次方。

Or in a given row, treat each number as part of a decimal expansion.

或者在某一行，把每一个数字 当成十进制的一部分。

In other words, row two is (1x1) + (2x10) + (1x100).

换句话说，第二行是 (1x1) + (2x10) + (1x100)，

You get 121, which is 11^2.

你会得到 121，也就是 11^2。

And take a look at what happens when you do the same thing to row six.

那么，同理到第六行，看看会发生什么。

It adds up to 1,771,561, which is 11^6, and so on.

总和是 1,771,561， 也就是 11^6，其他也一样。

There are also geometric applications.

除此之外，也有一些几何的应用。

Look at the diagonals.

看看那些对角线，

The first two aren't very interesting: all ones, and then the positive integers,

开头两条并不是很有趣，全都是 1。

接下来是正整数，也被称为自然数。

also known as natural numbers.

But the numbers in the next diagonal are called the triangular numbers

而下一条对角线的数字，则被称为三角数。

because if you take that many dots,

因为如果你用那些数量的点，

you can stack them into equilateral triangles.

可以把它们堆成等边三角形。

The next diagonal has the tetrahedral numbers

下一条对角线是四面体数。

because similarly, you can stack that many spheres into tetrahedra.

同理，你可以把那些球堆成四面体。

Or how about this: shade in all of the odd numbers.

或者这样︰ 把所有的奇数画上阴影，

It doesn't look like much when the triangle's small,

当三角形还小，你还看不出什么。

but if you add thousands of rows,

不过如果你加上成千上万行，

you get a fractal known as Sierpinski's Triangle.

你会得到一个分形， 也就是谢尔宾斯基三角形。

This triangle isn't just a mathematical work of art.

这个三角形不仅是一个数学的艺术品，

It's also quite useful,

它还很有用，

especially when it comes to probability and calculations

尤其是在组合学中的概率计算中。

in the domain of combinatorics.

Say you want to have five children,

假设，你想要五个小孩,

and would like to know the probability

你想要知道

拥有三个女孩和两个男孩 这样理想家庭的概率是多少。

of having your dream family of three girls and two boys.

In the binomial expansion,

在二项展开式中，

that corresponds to girl plus boy to the fifth power.

它对应的就是女孩加男孩的五次方。

So we look at the row five,

所以我们看第五行，

where the first number corresponds to five girls,

第一个数字代表五个女孩的可能性，

and the last corresponds to five boys.

最后一个数字代表五个男孩的可能性。

The third number is what we're looking for.

第三个数字就是我们要找的。

Ten out of the sum of all the possibilities in the row.

这一行所有可能性的总和分之10，

so 10/32, or 31.25%.

那就得到 10/32，或者31.25%。

Or, if you're randomly picking a five-player basketball team

再者，如果你从十二个朋友中

out of a group of twelve friends,

随机选出5人组成一个篮球队，

how many possible groups of five are there?

一共可能有多少种五人组合呢？

In combinatoric terms, this problem would be phrased as twelve choose five,

从组合学上看， 这个问题可以看成是从12中挑5，

and could be calculated with this formula,

并可以用这个公式计算，

or you could just look at the sixth element of row twelve on the triangle

或者你可以找到这个三角形的 第十二行第六项，

and get your answer.

就是你要的答案。

The patterns in Pascal's Triangle

帕斯卡三角的诸多形式，

are a testament to the elegantly interwoven fabric of mathematics.

是数学元素优美交织的证明。

And it's still revealing fresh secrets to this day.

到现在，它仍然揭示着新秘密。

For example, mathematicians recently discovered a way to expand it

例如，数学家最近发现了

一个展开这种多项式的方法。

to these kinds of polynomials.

What might we find next?

接下来我们还可能发现什么？

Well, that's up to you.

这就看你了。