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.
當你聽到對稱這個詞時, 浮現在腦海裡的 或許是一個簡單的幾何圖形 像是正方形或是三角形, 又或是像蝴蝶翅膀上較複雜的圖案。 若是站在藝術的觀點來看, 浮現在你腦海裡的可能是 莫札特協奏曲中的微妙的轉調, 或者是首席女舞者不費力的舞姿。 當應用在每天的生活中, 對稱這個字眼代表了對美,和諧及平衡 模糊的詮釋。 而在數學與科學領域,對稱有了個不一樣 且非常具體的意義。 從技術角度來說, 對稱是物體的一種性質。 幾乎任何物體都有對稱性, 從有形的東西例如蝴蝶, 到抽象的東西例如幾何圖形。 那麼,到底物體的對稱性是什麼意思呢? 定義如下: 對稱是使物體維持原樣的一種變換。 是的,這聽起來非常抽象, 讓我們一一解釋它。 讓我們來看個例子吧, 這會幫助我們理解它, 像這個等邊三角形。 如果我們試著轉動這個三角形120度, 保持著中心不變, 我們會得到一個與原本看起來 一模一樣的三角形。 在這個案例中,三角形就是那個物體, 而使物體維持原樣的變換, 就是轉動了120度。 所以我們可以說等邊三角形具有 繞中心點旋轉120度的對稱性。 如果我們旋轉三角形90度, 旋轉後的三角形看起來 跟原本的就不一樣了。 也就是說,一個等邊三角形不具有 繞中心點旋轉90度的對稱性。 但是為何數學家跟科學家 這麼在意對稱呢? 這麼說吧,它們在數學及科學的 很多領域中至關重要。 讓我們舉另一個例子吧: 生物學中的對稱。 你可能已經注意到了 一個看起來很熟悉的對稱 但是我們還沒提到—— 人體左側及右側的對稱性。 這一對稱所對應的變換是反射, 想象有一面鏡子垂直得 將身體分成兩半。 生物學家把這叫做雙邊對稱。 就如同在所有生物身上可以找到的對稱, 這只是一種近似的對稱, 但這依舊是人體的一個顯著特徵。 事實上不是只有我們人類 才是雙邊對稱的有機體。 很多其他的動物例如狐狸,鯊魚,甲蟲, 還有我們先前提到的蝴蝶, 都有這種對稱性, 還有些植物像是蘭花。 其他有機體有不一樣的對稱性, 有些有機體只有在以他的中心點轉動後 才會變得明顯。 就像我們剛剛看到的 旋轉對稱的三角形那樣。 但是,當它在動物身上發生時, 這種對稱性叫做放射對稱。 舉個例子,有些海膽與海星 有五放射對稱或五倍對稱, 也就是說,對中心點轉動72度 會看到一樣的形狀。 這樣的對稱也同時出現在植物身上, 就像你橫切蘋果時看到的那樣。 有些水母的對稱是轉動90度後形狀一樣, 又或是有些海葵的對稱性是 轉動任何角度牠們的形狀都一樣。 而一些珊瑚,可以說是毫無對稱可言的。 牠們完全是不對稱的。 但為何有機體會出現這些不一樣的對稱呢? 身體的對稱性是否也在告訴我們 動物們的生活方式? 讓我們針對特定的群體來看看: 雙邊對稱的動物。 在這個群體裡,我們有 狐狸,甲蟲,鯊魚,蝴蝶, 當然還有人類。 對雙邊對稱的動物來說很重要的一件事 就是牠們身體的設計是因為動作的需要。 如果你選定一個方向前進, 那麼有一個 集中各種感官的前端是有好處的—— 你的眼睛,耳朵跟鼻子。 把嘴巴放到前端也有好處, 因為在這個前端你更有可能撞到食物 或是敵人。 你應該很熟悉這個 帶有一堆器官和一張嘴, 長在動物身體前端的東西。 這個叫做頭。 有頭的生物會自然地發展出雙邊對稱。 它也有助於形成魚類流線型的鰭, 鳥類符合空氣動力學的翅膀, 或者是狐狸奔跑時有協調性的腳。 但是,這些跟演化有什麼關係呢? 有的,生物學家透過各種身體的對稱性 可以知道物種間的關聯。 就像我們看到的海星與海膽有五倍對稱。 但確切來說指的是 成年的海星和海膽。 在他們的幼體階段, 是雙邊對稱的,就跟人類一樣。 對生物學家來說這是一個有力的證據, 也就是說我們跟海星的關係 要比珊瑚 或其他那些在任何生長階段 都不具備雙邊對稱的動物 都更為緊密。 生物學中有一個非常引人入勝且重要的問題 就是重建生命之樹, 尋找不同的分支是在何時以及如何分化的。 思考身體的對稱性這種簡單的問題 可以幫助我們深入發掘過去的演化歷程, 並且了解我們作為一個物種是從哪裡來的。