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The Fibonacci sequence floats over the Atlantic coastline under our home spiral galaxy, the Milky Way.

斐波那契数列漂浮在我们的螺旋星系——银河系下面的大西洋海岸线上。

Is there a magic equation to the universe? Probably not, but there are some pretty common ones that we find over and over in the natural world. Take, for instance, the Fibonacci numbers — a sequence of numbers and a corresponding ratio that reflect various patterns found in nature, from the swirl of a pine cone's seeds to the curve of a nautilus shell to the twist of a hurricane.

宇宙中是否存在一个神奇的方程式?也许不是，但有一些我们在自然界中反复发现的非常常见的。以斐波那契数列为例，斐波那契数列是一系列数字和相应的比率，反映了自然界中发现的各种模式，从松果种子的漩涡到鹦鹉螺壳的曲线，再到飓风的扭曲。

Humans have probably known about this numerical sequence for millennia — it can be found in ancient Sanskrit texts — but in modern times we have associated it with one medieval man's obsession with rabbits.

人类可能在几千年前就知道这个数字序列了——它可以在古梵文文本中找到——但在现代，我们把它与一个中世纪男人对兔子的痴迷联系在一起。

In 1202, Italian mathematician Leonardo Pisano (also known as Leonardo Fibonacci, meaning "son of Bonacci") wondered how many rabbits could come from a single set of parents. More specifically, he posed the question: Given optimal conditions, how many pairs of rabbits can be produced from a single pair of rabbits in one year? This thought experiment dictates that the female rabbits always give birth to pairs, and each pair consists of one male and one female.

1202年，意大利数学家莱昂纳多·皮萨诺(也被称为莱昂纳多·斐波那契，意思是“波那契的儿子”)想知道一组父母能生多少只兔子。更具体地说，他提出了一个问题:在最佳条件下，一对兔子在一年内可以繁殖出多少对兔子?这个思想实验表明，雌性兔子总是生一对，每一对由一只雄性和一只雌性组成。

Think about it: Two newborn rabbits are placed in a fenced-in yard and left to, well, breed like rabbits. Rabbits can't bear young until they are at least 1 month old, so for the first month, only one pair remains. At the end of the second month, the female gives birth to a new pair, leaving two pairs total. When month three rolls around, the original pair of rabbits produces yet another pair of newborns while their earlier offspring grow to adulthood. This leaves three pairs of rabbit, two of which will give birth to two more pairs the following month for a total of five pairs of rabbits.

想想看:两只刚出生的兔子被放在一个有围栏的院子里，让它们像兔子一样繁殖。兔子至少要到1个月大的时候才能生育后代，所以在第一个月里，只剩下一对。在第二个月的月底，雌性生下一对新的，总共留下两对。当第三个月到来时，原来的一对兔子又生了一对新生儿，而它们早期的后代则长大成人。这样就剩下三对兔子，其中两对将在下个月再生两对，总共是五对兔子。

The first Fibonacci numbers go as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 and on to infinity. The equation that describes it looks like this: Xn+2= Xn+1 + Xn. Basically, each integer is the sum of the preceding two numbers. This set of infinite sums is known as the Fibonacci series or the Fibonacci sequence. The ratio between the numbers in the Fibonacci sequence (1.6180339887498948482...) is frequently called the golden ratio or golden number. The ratios of successive Fibonacci numbers approach the golden ratio as the numbers approach infinity.

第一个斐波那契数是这样的:0、1、1、2、3、5、8、13、21、34、55、89、144，一直到无穷大。描述它的方程是这样的:Xn+2= Xn+1 + Xn。基本上，每个整数都是前两个数字的和。这个无穷和的集合被称为斐波那契级数或斐波那契数列。斐波那契数列中数字之间的比率(1.6180339887498948482…)通常被称为黄金比例或黄金数。连续斐波那契数的比率接近黄金比例，因为数字接近无穷大。

Want to see how these fascinating numbers are expressed in nature? No need to visit your local pet store; all you have to do is look around you.

想看看这些迷人的数字在自然界中是如何表达的吗?不需要去当地的宠物店;你所要做的就是看看你周围。

Contents 内容

1. The Golden Ratio in Nature 自然界的黄金比例
2. Misconceptions About the Fibonacci Sequence 关于斐波那契数列的误解

The Golden Ratio in Nature 自然界的黄金比例

Take a good look at this Roman cauliflower. Its spiral follows the Fibonacci series.

好好看看这个罗马花椰菜。它的螺旋遵循斐波那契数列。

While some plant seeds, petals and branches, etc., follow the Fibonacci sequence, it certainly doesn't reflect how all things grow in the natural world. And just because a series of numbers can be applied to an astonishing variety of objects, that doesn't necessarily imply there's any correlation between figures and reality. As with numerological superstitions such as famous people dying in sets of three, sometimes a coincidence is just a coincidence.

虽然一些植物的种子、花瓣和树枝等遵循斐波那契数列，但它肯定不能反映自然界中所有事物的生长方式。仅仅因为一系列数字可以应用于各种各样的物体，这并不一定意味着数字和现实之间有任何关联。就像数字命理迷信一样，比如名人是三次死亡，有时巧合只是巧合。

But while some would argue that the prevalence of the Fibonacci numbers in nature are exaggerated, they appear often enough to prove that they reflect some naturally occurring patterns. You can commonly spot these by studying the manner in which various plants grow. Here are a few examples:

但是，尽管有些人会争辩说，斐波那契数列在自然界中的流行程度被夸大了，但它们经常出现，足以证明它们反映了一些自然发生的模式。你可以通过研究各种植物的生长方式来发现它们。以下是一些例子:

Seed heads, pinecones, fruits and vegetables: Look at the array of seeds in the center of a sunflower and you'll notice what looks like spiral patterns curving left and right. Amazingly, if you count these spirals, your total will be a Fibonacci number. Divide the spirals into those pointed left and right and you'll get two consecutive Fibonacci numbers. You can decipher spiral patterns in pine cones, pineapples and cauliflower that also reflect the Fibonacci sequence in this manner.

瓜子头、松果、水果和蔬菜:看看向日葵中央排列的种子，你会发现它们看起来像是左右弯曲的螺旋图案。令人惊讶的是，如果你计算这些螺旋，你的总数将是一个斐波那契数。将螺旋分为那些指向左边和右边，你会得到两个连续的斐波那契数。你可以破译松果、菠萝和花椰菜的螺旋图案，它们也以这种方式反映了斐波那契数列。

Flowers and branches: Some plants express the Fibonacci sequence in their growth points, the places where tree branches form or split. One trunk grows until it produces a branch, resulting in two growth points. The main trunk then produces another branch, resulting in three growth points. Then the trunk and the first branch produce two more growth points, bringing the total to five. This pattern continues, following the Fibonacci numbers. Additionally, if you count the number of petals on a flower, you'll often find the total to be one of the numbers in the Fibonacci sequence. For example, lilies and irises have three petals, buttercups and wild roses have five, delphiniums have eight petals and so on.

花和树枝:一些植物在它们的生长点(树枝形成或分叉的地方)表达斐波那契序列。一根树干会长出分支，产生两个生长点。主干然后产生另一个分支，从而产生三个生长点。然后树干和第一个分支又产生两个生长点，使总数达到五个。这种模式继续，遵循斐波那契数。此外，如果你数一朵花上的花瓣数量，你经常会发现总数是斐波那契数列中的一个数字。例如，百合和鸢尾花有三个花瓣，毛茛和野玫瑰有五个花瓣，飞燕草有八个花瓣，等等。

Honeybees: A honeybee colony consists of a queen, a few drones and lots of workers. The female bees (queens and workers) have two parents: a drone and a queen. Drones, on the other hand, hatch from unfertilized eggs. This means they have only one parent. Therefore, Fibonacci numbers express a drone's family tree in that he has one parent, two grandparents, three great-grandparents and so forth.

蜜蜂:一个蜂群由一只蜂王、几只雄蜂和许多工蜂组成。雌蜂(蜂王和工蜂)有两个父母:雄蜂和蜂王。另一方面，雄蜂是从未受精卵中孵化出来的。这意味着它们只有一个父母。因此，斐波那契数列表达了一个无人机的家谱，他有一个父母，两个祖父母，三个曾祖父母等等。

The Fibonacci spiral, also known as a golden spiral, is a visual expression of the golden ratio. In the illustration above, areas of the shell's growth are mapped out in an interesting pattern of squares that use only Fibonacci numbers. If the two smallest squares have a width and height of 1, then the box below has a measurement of 2. The other boxes represent squared numbers in the Fibonacci series.

斐波那契螺旋，也被称为黄金螺旋，是黄金比例的视觉表达。在上面的插图中，壳的生长区域以一种有趣的正方形模式绘制出来，只使用斐波那契数。如果最小的两个正方形的宽和高为1，那么下面的方框的尺寸为2。其他方框表示斐波那契数列中的平方数。

Storms: Storm systems like hurricanes and tornados often follow the Fibonacci sequence. Next time you see a hurricane spiraling on the weather radar, check out the unmistakable Fibonacci proportions of the spiral of clouds on the screen.

风暴:像飓风和龙卷风这样的风暴系统通常遵循斐波那契数列。下次你在天气雷达上看到飓风盘旋时，看看屏幕上云螺旋的斐波那契比例。

The human body: Take a good look at yourself in the mirror. You'll notice that most of your body parts follow the numbers one, two, three and five. You have one nose, two eyes, three segments to each limb and five fingers on each hand. The proportions and measurements of the human body can also be divided up in terms of the golden ratio. DNA molecules follow this sequence, measuring 34 angstroms long and 21 angstroms wide for each full cycle of the double helix.

人体:对着镜子好好看看自己。你会注意到你身体的大部分部位都遵循数字1、2、3和5。你有一个鼻子，两只眼睛，四肢各有三节，每只手上有五个手指。人体的比例和尺寸也可以根据黄金比例进行划分。DNA分子遵循这个序列，每一个完整的双螺旋周期长度为34埃，宽度为21埃。

Why do so many natural patterns reflect the Fibonacci sequence? Scientists have pondered the question for centuries. In some cases, the correlation may just be coincidence. In other situations, the ratio exists because that particular growth pattern evolved as the most effective. In plants, this may mean maximum exposure for light-hungry leaves or maximized seed arrangement.

为什么如此多的自然模式反映斐波那契数列?科学家们已经思考这个问题几个世纪了。在某些情况下，这种相关性可能只是巧合。在其他情况下，这个比例之所以存在，是因为这种特定的增长模式是最有效的。在植物中，这可能意味着最大限度地暴露在阳光下的叶片或最大限度地安排种子。

Misconceptions About the Fibonacci Sequence 关于斐波那契数列的误解

While experts agree that the Fibonacci sequence is common in nature, there is less agreement is whether the Fibonacci sequence is expressed in art and architecture. Although some books say that the Great Pyramid and the Parthenon (as well as some of Leonardo da Vinci's paintings) were designed using the golden ratio, when this is tested, it's found to not be true.

虽然专家们一致认为斐波那契数列在自然界中很常见，但对于斐波那契数列是否在艺术和建筑中表达却没有那么一致的意见。虽然有些书说大金字塔和帕台农神庙(以及一些列奥纳多·达·芬奇的画作)是使用黄金比例设计的，但经过测试后，发现事实并非如此。

Mathematician George Markowsky pointed out that both the Parthenon and the Great Pyramid have parts that don't fit inside a golden rectangle or conform to the golden ratio, something left out by people determined to prove that the golden ratio exists in everything. The term "the golden mean" was used in ancient times to denote something that avoided access in either direction, and some people have conflated the golden mean with the golden ratio, which is a more recent term that came into existence in the 19th century.

数学家乔治·马可夫斯基指出，帕台农神庙和大金字塔都有不符合黄金分割的部分，这是那些决心证明黄金分割存在于万物的人所遗漏的。“中庸之道”一词在古代被用来指避免朝任何方向接近的东西，有些人把中庸之道与黄金分割率混为一谈，黄金分割率是一个出现在19世纪的较新的术语。