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.
这些看上去 可能只是一堆排列整齐的数字, 实际上,它可是一个数学的宝藏。 印度数学家称它为"须弥山之梯"。 在伊朗,它是"海亚姆三角"。 而在中国,它被称为"杨辉三角"。 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。 这似乎有点不太公平。 因为帕斯卡的发现比其他人更晚, 但帕斯卡也对此做出了许多贡献。 那么,是什么让世界各地的 数学家们对它如此感兴趣? 简单地说,它充满了各种形式和秘密。 首先,这是构造三角的形式。 从 1 开始, 并假设两边各有一个看不见的 0, 把相邻的数字加起来, 你就会得到下一行。 现在,重复这样的操作, 反复进行, 你最终会得到这样一个图形。 实际上,帕斯卡三角是无限大的。 它每一行的数字都对应 (x+y)^n 二项式展开的系数, 其中 n 是行的序号, 从 0 开始算。 当 n=2时, 二项式展开你会得到 x^2 + 2xy + y^2。 那些系数,就是每一项变量前的数字, 和帕斯卡三角对应行的数字相同。 n=3 也是一样,展开得到这个。 所以,这个三角能让我们 快速得到二项式的系数。 然而,奥秘远远不止这些。 比如说,把每一行的数字加起来, 你会得到连续的2的次方。 或者在某一行,把每一个数字 当成十进制的一部分。 换句话说,第二行是 (1x1) + (2x10) + (1x100), 你会得到 121,也就是 11^2。 那么,同理到第六行,看看会发生什么。 总和是 1,771,561, 也就是 11^6,其他也一样。 除此之外,也有一些几何的应用。 看看那些对角线, 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。 而下一条对角线的数字,则被称为三角数。 因为如果你用那些数量的点, 可以把它们堆成等边三角形。 下一条对角线是四面体数。 同理,你可以把那些球堆成四面体。 或者这样︰ 把所有的奇数画上阴影, 当三角形还小,你还看不出什么。 不过如果你加上成千上万行, 你会得到一个分形, 也就是谢尔宾斯基三角形。 这个三角形不仅是一个数学的艺术品, 它还很有用, 尤其是在组合学中的概率计算中。 假设,你想要五个小孩, 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。 在二项展开式中, 它对应的就是女孩加男孩的五次方。 所以我们看第五行, 第一个数字代表五个女孩的可能性, 最后一个数字代表五个男孩的可能性。 第三个数字就是我们要找的。 这一行所有可能性的总和分之10, 那就得到 10/32,或者31.25%。 再者,如果你从十二个朋友中 随机选出5人组成一个篮球队, 一共可能有多少种五人组合呢? 从组合学上看, 这个问题可以看成是从12中挑5, 并可以用这个公式计算, 或者你可以找到这个三角形的 第十二行第六项, 就是你要的答案。 帕斯卡三角的诸多形式, 是数学元素优美交织的证明。 到现在,它仍然揭示着新秘密。 例如,数学家最近发现了 一个展开这种多项式的方法。 接下来我们还可能发现什么? 这就看你了。