When you hear the word symmetry, maybe you picture a simple geometric shape like a square or a triangle,
当你听到“对称”这个词时，你的脑海里大概会出现一个简单的几何图形，比如一个正方形或三角形，
or the complex pattern on a butterfly's wings.
或者是更复杂一点的图案，诸如蝴蝶翅膀。
If you are artistically inclined, you might think of the subtle modulations of a Mozart concerto,
如果你是一个富有艺术感的人，你可能会想到莫扎特协奏曲中微小的转调，
or the effortless poise of a prima ballerina.
或者芭蕾舞首席女演员毫不费力的舞姿。
When used in every day life, the word symmetry represents vague notions of beauty, harmony and balance.
在日常生活中，对称这个词含糊地代表了唯美、和谐以及平衡的观念。
In math and science, symmetry has a different, and very specific, meaning.
在数学和科学中，对称有着不同而且非常特殊的定义。
In this technical sense, a symmetry is the property of an object.
从技术角度而言，对称是物体的一种特性。
Pretty much any type of object can have symmetry, from tangible things like butterflies, to abstract entities like geometric shapes.
基本上所有物体都有对称性，从有形的物体，比如蝴蝶，到抽象物体，比如几何图形。
So, what does it mean for an object to be symmetric?
那么，一个物体有“对称性”是什么意思呢？
Here's the definition: a symmetry is a transformation that leaves that object unchanged.
定义如下：对称是使物体维持原样的一种变换。
Okay, that sounds a bit abstract, so let's unpack it.
嗯，听起来有点抽象，让我们来具体解释一下。
It will help to look at a particular example, like this equilateral triangle.
举个等边三角形的例子会有助于我们理解。
If we rotate our triangle through 120 degrees, around an access through its center,
如果我们把这个三角形以中心为轴旋转120度，
we end up with a triangle that's identical to the original.
我们会得到一个完全一样的三角形。
In this case, the object is the triangle,
在本例中，物体是三角形，
and the transformation that leaves the object unchanged is rotation through 120 degrees.
而使物体维持原样的变换是旋转120度。
So we can say an equilateral triangle is symmetric with respect to rotations of 120 degrees around its center.
所以我们可以说，等边三角形以中心为轴的120度旋转是有对称性的。
If we rotated the triangle by, say, 90 degrees instead, the rotated triangle would look different to the original.
如果我们把这个三角形旋转90度，这个三角形和原来那个就不同了。
In other words, an equilateral triangle is not symmetric with respect to rotations of 90 degrees around its center.
换句话来说，等边三角形以中心为轴的90度旋转是没有对称性的。
But why do mathematicians and scientists care about symmetries?
但是数学家和科学家们为什么要在意对称呢？
Turns out, they're essential in many fields of math and science.
答案是，对称在数学及科学的很多领域中至关重要。
Let's take a close look at one example: symmetry in biology.
让我们举一个例子来说明：生物学中的对称。
You might have noticed that there's a very familiar kind of symmetry we haven't mentioned yet:
你可能会注意到，我们还没有提及一类很常见的对称：
the symmetry of the right and left sides of the human body.
人体左右两侧的对称。
The transformation that gives this symmetry is reflection by an imaginary mirror that slices vertically through the body.
这个对称所对应的变换是反射，我们可以想象一面镜子把人体从中间竖直分开。
Biologists call this bilateral symmetry.
生物学家们把这叫做“两侧对称”。
As with all symmetries found in living things, it's only approximate, but still a striking feature of the human body.
如同所有可以在生物体上找到的对称，这只是一种近似的对称，但这依旧是人体的一个显著特征。
We humans aren't the only bilaterally symmetric organisms.
我们人类并不是唯一一种两侧对称的生物。
Many other animals, foxes, sharks, beetles, that butterfly we mentioned earlier,
还有很多动物也是，比如狐狸、鲨鱼、甲壳虫，还有我们之前提到的蝴蝶等，
have this kind of symmetry, as do some plants like orchid flowers.
都有这种对称性，一些植物比如兰花，也是如此。

Other organisms have different symmetries, ones that only become apparent when you rotate the organism around its center point.
其他生物体有不同的对称性，有一些对称只有你在把它绕中心点旋转时才看得出来。
It's a lot like the rotational symmetry of the triangle we watched earlier.
这和我们之前看到的三角形的旋转对称非常相似。
But when it occurs in animals, this kind of symmetry is known as radial symmetry.
不过当这种对称出现在动物身上时，我们称它为“辐射对称”。
For instance, some sea urchins and starfish have pentaradial or five-fold symmetry,
例如，某些海胆还有海星，呈五辐射对称，或五次对称。
that is, symmetry with respect to rotations of 72 degrees around their center.
意思是，这些生物以中心为轴进行72度旋转是有对称性的。
This symmetry also appears in plants, as you can see for yourself by slicing through an apple horizontally.
植物中也会出现这种对称，你去把一个苹果从水平方向切开就会发现了。
Some jellyfish are symmetric with respect to rotations of 90 degrees,
有些水母是关于90度旋转对称的，
while sea anemones are symmetric when you rotate them at any angle.
而海葵则是对于任意角度都是旋转对称的。
Some corals, on the other hand, have no symmetry at all. They are completely asymmetric.
另一方面，某些珊瑚不具有任何对称性。它们完全不对称。
But why do organisms exhibit these different symmetries?
但是为什么生物体会呈现这些不同的对称呢？
Does body symmetry tell us anything about an animal's lifestyle?
物种身体的对称性是否向我们揭示了动物的生活习性？
Let's look at one particular group: bilaterally symmetric animals.
让我们来重点观察一个群体：两侧对称的动物。
In this camp, we have foxes, beetles, sharks, butterflies, and, of course, humans.
在这个分组里，我们有狐狸、甲壳虫、鲨鱼、蝴蝶，当然，还有人类。
The thing that unites bilaterally symmetric animals is that their bodies are designed around movement.
这类两侧对称的生物的特点是它们的身体构造是建立在运动的基础上的。
If you want to pick one direction and move that way,
如果你想朝某个方向移动，
it helps to have a front end where you can group your sensory organs -- your eyes, ears and nose.
有一个集中了各种感官的前端是很有帮助的--包括了你的眼睛，耳朵和鼻子。
It helps to have your mouth there too since you're more likely to run into food or enemies from this end.
把嘴放在前端也很有用，因为这个前端既会的得到食物，也有可能遭遇敌人。
You're probably familiar with a name for a group of organs,
你应该对一个带有包括嘴在内的一堆器官，
plus a mouth, mounted on the front of an animal's body. It's called a head.
伫立在动物身体最前端的东西很熟悉。这个东西叫做头。
Having a head leads naturally to the development of bilateral symmetry.
有一个头的生物会很自然地进化成两侧对称体。
And it also helps you build streamlined fins if you're a fish,
如果你是一条鱼，这也会帮助你长出流线型的鳍，
aerodynamic wings if you're a bird, or well coordinated legs for running if you're a fox.
如果你是一只鸟，那就会长出符合空气动力学的翅膀，或者如果你是狐狸，那你就会有非常协调的腿。
But, what does this all have to do with evolution?
但是这些和物种进化又有什么关系呢？
Turns out, biologists can use these various body symmetries to figure out which animals are related to which.
答案是，生物学家们可以用这些身体对称性来判断物种间的联系。
For instance, we saw that starfish and sea urchins have five-fold symmetry.
比如，我们知道海胆和海星都具有五次对称。
But really what we should have said was adult starfish and sea urchins.
但我们要注意的是只有成年海星和海胆才这样。
In their larval stage, they're bilateral, just like us humans.
在它们幼年时期，它们就像我们人类一样是两侧对称的。
For biologists, this is strong evidence that we're more closely related to starfish than we are, to say,
对于生物学家来说，这是一个有力的证据，表明我们人类与海星的关系要近得多，
corals, or other animals that don't exhibit bilateral symmetry at any stage in their development.
相比于珊瑚或者其他在任何生长阶段都不具备两侧对称的生物而言。
One of the most fascinating and important problems in biology is reconstructing the tree of life,
生物学中有一个迷人且又重要的任务就是重建“生命之树”，
discovering when and how the different branches diverged.
找出物种进化中各分支形成的时间与方式。
Thinking about something as simple as body symmetry
研究身体对称性这种简单的东西，
can help us dig far into our evolutionary past and understand where we, as a species, have come from.
可以帮助我们深入发掘我们的进化历程，从而了解我们作为一个物种是从何而来的。