When you hear the word symmetry, maybe you picture a simple geometric shape like a square or a triangle, or the complex pattern on a butterfly's wings. If you are artistically inclined, you might think of the subtle modulations of a Mozart concerto, or the effortless poise of a prima ballerina. When used in every day life, the word symmetry represents vague notions of beauty, harmony and balance. In math and science, symmetry has a different, and very specific, meaning. In this technical sense, a symmetry is the property of an object. Pretty much any type of object can have symmetry, from tangible things like butterflies, to abstract entities like geometric shapes. So, what does it mean for an object to be symmetric? Here's the definition: a symmetry is a transformation that leaves that object unchanged. Okay, that sounds a bit abstract, so let's unpack it. It will help to look at a particular example, like this equilateral triangle. If we rotate our triangle through 120 degrees, around an access through its center, we end up with a triangle that's identical to the original. In this case, the object is the triangle, and the transformation that leaves the object unchanged is rotation through 120 degrees. So we can say an equilateral triangle is symmetric with respect to rotations of 120 degrees around its center. If we rotated the triangle by, say, 90 degrees instead, the rotated triangle would look different to the original. In other words, an equilateral triangle is not symmetric with respect to rotations of 90 degrees around its center. But why do mathematicians and scientists care about symmetries? Turns out, they're essential in many fields of math and science. Let's take a close look at one example: symmetry in biology. You might have noticed that there's a very familiar kind of symmetry we haven't mentioned yet: the symmetry of the right and left sides of the human body. The transformation that gives this symmetry is reflection by an imaginary mirror that slices vertically through the body. Biologists call this bilateral symmetry. As with all symmetries found in living things, it's only approximate, but still a striking feature of the human body. We humans aren't the only bilaterally symmetric organisms. Many other animals, foxes, sharks, beetles, that butterfly we mentioned earlier, have this kind of symmetry, as do some plants like orchid flowers. Other organisms have different symmetries, ones that only become apparent when you rotate the organism around its center point. It's a lot like the rotational symmetry of the triangle we watched earlier. But when it occurs in animals, this kind of symmetry is known as radial symmetry. For instance, some sea urchins and starfish have pentaradial or five-fold symmetry, that is, symmetry with respect to rotations of 72 degrees around their center. This symmetry also appears in plants, as you can see for yourself by slicing through an apple horizontally. Some jellyfish are symmetric with respect to rotations of 90 degrees, while sea anemones are symmetric when you rotate them at any angle. Some corals, on the other hand, have no symmetry at all. They are completely asymmetric. But why do organisms exhibit these different symmetries? Does body symmetry tell us anything about an animal's lifestyle? Let's look at one particular group: bilaterally symmetric animals. In this camp, we have foxes, beetles, sharks, butterflies, and, of course, humans. The thing that unites bilaterally symmetric animals is that their bodies are designed around movement. If you want to pick one direction and move that way, it helps to have a front end where you can group your sensory organs-- your eyes, ears and nose. It helps to have your mouth there too since you're more likely to run into food or enemies from this end. You're probably familiar with a name for a group of organs, plus a mouth, mounted on the front of an animal's body. It's called a head. Having a head leads naturally to the development of bilateral symmetry. And it also helps you build streamlined fins if you're a fish, aerodynamic wings if you're a bird, or well coordinated legs for running if you're a fox. But, what does this all have to do with evolution? Turns out, biologists can use these various body symmetries to figure out which animals are related to which. For instance, we saw that starfish and sea urchins have five-fold symmetry. But really what we should have said was adult starfish and sea urchins. In their larval stage, they're bilateral, just like us humans. For biologists, this is strong evidence that we're more closely related to starfish than we are, to say, corals, or other animals that don't exhibit bilateral symmetry at any stage in their development. One of the most fascinating and important problems in biology is reconstructing the tree of life, discovering when and how the different branches diverged. Thinking about something as simple as body symmetry can help us dig far into our evolutionary past and understand where we, as a species, have come from.
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, 或者是更复杂一点的图案,诸如蝴蝶翅膀。 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, 或者芭蕾舞首席女演员毫不费力的舞姿。 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。 在数学和科学中,对称有着不同, 而且非常特殊的定义。 从技术角度而言, 对称是物体的一种特性。 基本上所有物体都有对称性, 从有形的物体比如蝴蝶, 到抽象物体如几何图形。 那么,一个物体有“对称性”是什么意思呢? 定义如下: 对称是使物体维持原样的一种变换。 嗯,听起来有点抽象,让我们来具体解释一下。 举个等边三角形的例子 会有助于我们理解。 如果我们把这个三角形以中心为轴 旋转120度, 我们会得到一个完全一样的三角形。 在本例中,物体是三角形, 而使物体维持原样的变换 是旋转120度。 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。 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。 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。 但是数学家和科学家们为什么要在意对称呢? 答案是,对称在数学及科学的很多领域中至关重要。 让我们举一个例子来说明:生物学中的对称。 你可能会注意到,我们还没有提及 一类很常见的对称: 人体左右两侧的对称。 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。 生物学家们把这叫做“两侧对称”。 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。 我们人类并不是唯一一种两侧对称的生物。 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, 都有这种对称性, 一些植物比如兰花,也是如此。 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。 这和我们之前看到的三角形的旋转对称非常相似。 不过当这种对称出现在动物身上时, 我们称它为“辐射对称”。 例如,某些海胆还有海星 呈五辐射对称,或五次对称。 意思是,这些生物以中心为轴进行72度旋转是有对称性的。 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。 有些水母是关于90度旋转对称的, 而海葵则是对于任意角度都是旋转对称的。 另一方面,某些珊瑚不具有任何对称性。 它们完全不对称。 但是为什么生物体会呈现这些不同的对称呢? 物种身体的对称性是否向我们揭示了动物的生活习性? 让我们来重点观察一个群体: 两侧对称的动物。 在这个分组里,我们有狐狸,甲壳虫,鲨鱼,蝴蝶, 当然,还有人类。 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。 如果你想朝某个方向移动, 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。 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。 你应该对一个带有包括嘴在内的一堆器官, 伫立在动物身体最前端的东西很熟悉。 这个东西叫做——头。 有一个头的生物会很自然地进化成两侧对称体。 如果你是一条鱼,这也会帮助你长出流线型的鳍, 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。 但是这些和物种进化又有什么关系呢? 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。 比如,我们知道海胆和海星都具有五次对称。 但我们要注意的是 只有成年海星和海胆才这样。 在它们幼年时期,它们就像我们人类一样是两侧对称的。 对于生物学家来说,这是一个有力的证据, 这表明, 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, 我们人类与海星的关系要近得多。 生物学中有一个迷人且又重要的任务就是 重建“生命之树”, 找出物种进化中各分支形成的时间与方式。 研究身体对称性这种简单的东西 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。