TED演讲：用数学感知这个世界！
这个世界充满了基于数学的循环模式。
在这场精彩的演讲中，全澳洲最红数学老师Eddie Woo分享了他对数学的热情。他认为数学存在于冥冥万物之中，可以被每个人接受甚至享受，因为我们天生就对它具有感知。演讲中，他以真实世界的几何图形为例，鼓励我们每个人都去寻找它们周围的模式，以“一种全新的方式来观察世界”，感受数学的魅力。
演讲者：吴宇森 (Eddie Woo)
被誉为“澳洲最出色的数学老师”，获得国家级奖项，因为教学方式幽默风趣，圈粉无数，有不少学生坦言，因他而迷上数学。
「我爱数学！」如果你们想在接下来的几个小时里在宴会最不显眼的角落里独啜饮料，那正是该在聚会上要说的话。
And that's because when it comes to this subject - all the numbers, formulas, symbols, and calculations - the vast majority of us are outsiders, and that includes me.
那正是因为一旦涉及到这个主题——所有数字、公式、符号和计算的主题——我们当中的绝大多数人是局外人，包括我在内。
That's why today I want to share with you an outsider's perspective of mathematics - what I understand of it, from someone who's always struggled with the subject. And what I've discovered, as someone who went from being an outsider to making math my career, is that, surprisingly, we are all deep down born to be mathematicians.
这就是为什么今天我想与大家分享我这个一直挣扎着努力学习数学的局外人，我的观点和我对数学的理解。身为从局外人转向以教数学为职志的人，我惊讶地发现我们都天生就是数学家。
But back to me being an outsider. I know what you're thinking: "Wait a second, Eddie. What would you know? You're a math teacher. You went to a selective school. You wear glasses, and you're Asian." Firstly, that's racist. Secondly, that's wrong.
但是先回到我身为局外人。我知道你们在想什么：「艾迪，且慢，你哪会知道，你是数学老师，你上过好学校。你戴眼镜，你还是亚洲人。」首先，那是种族歧视。其次，那是错的。
When I was in school, my favorite subjects were English and history. And this caused a lot of angst for me as a teenager because my high school truly honored mathematics.
上学的时候，我最喜欢的科目是英语和历史。十几岁时这就使我非常焦虑，因为我的高中真的很重视数学。
Your status in the school pretty much correlated with which mathematics class you ranked in. There were eight classes. So if you were in math 4, that made you just about average. If you were in math 1, you were like royalty.
学生在学校的地位与所修的数学课密切相关。数学课程有八级。因此，如果修的是数学四，那就是中段的水平。如果修数学一，就像是皇室等级。
Each year, our school entered the prestigious Australian Mathematics Competition and would print out a list of everyone in the school in order of our scores. Students who received prizes and high distinctions were pinned up at the start of a long corridor, far, far away from the dark and shameful place where my name appeared.
我们学校每年都参加著名的澳大利亚数学竞赛，还按照分数的高低列出每一个人的排行榜。在长长的走廊的尽头钉着的是获得奖项和杰出成就学生的名字，远在走廊的另一头是我名字出现的黑暗和羞惭之地。
Math was not really my thing. Stories, characters, narratives - this is where I was at home. And that's why I raised my sails and set course to become an English and history teacher. But a chance encounter at Sydney University altered my life forever.
数学不是我的强项。故事、人物、叙述——我才在行。这就是为什么我扬帆立志要成为英语和历史老师的原因。但在悉尼大学的某个偶遇永远改变了我的生命。
I was in line to enroll at the faculty of education when I started the conversation with one of its professors. He noticed that while my academic life had been dominated by humanities, I had actually attempted some high-level math at school. What he saw was not that I had a problem with math, but that I had persevered with math.
当时我正准备进入教育学院，与一位该学院的教授面谈。他注意到，虽然我高中时主要选修的是人文学科，实际上却在校里修过一些高等数学。他不是看到我数学学得不怎么样，而是看到我坚持到底学了数学。
And he knew something I didn't - that there was a critical shortage of mathematics educators in Australian schools, a shortage that remains to this day. So he encouraged me to change my teaching area to mathematics.
他还知道我所不知——澳大利亚严重短缺数学教师，直到今天仍然持续短缺。因此他鼓励我将教学领域改为数学。
Now, for me, becoming a teacher wasn't about my love for a particular subject. It was about having a personal impact on the lives of young people.
对我来说，当老师的原因并不是我对特定学科的热爱，而是想影响年轻人的生命。
I'd seen firsthand at school what a lasting and positive difference a great teacher can make. I wanted to do that for someone, and it didn't matter to me what subject I did it in. If there was an acute need in mathematics, then it made sense for me to go there.
我在学生时代亲身体验过出色的老师能够带来多么持久而积极的改变。我也想那样。对我来说，这与教哪个学科无关。如果迫切需要数学教师，那我去当数学老师就合情合理。
As I studied my degree, though, I discovered that mathematics was a very different subject to what I'd originally thought. I'd made the same mistake about mathematics that I'd made earlier in my life about music.
然而在修学位时，我发现数学与我最初想的完全不同。我在数学上犯的错误与我早年在音乐上犯的错误相同。
Like a good migrant child, I dutifully learned to play the piano when I was young. My weekends were filled with endlessly repeating scales and memorizing every note in the piece, spring and winter.
像每个听话的移民小孩一样，我从小就乖乖地学钢琴。我的周末充满了无止境的重复音阶练习，强记春季和冬季乐章中的每个音符。
I lasted two years before my career was abruptly ended when my teacher told my parents, "His fingers are too short. I will not teach him anymore." At seven years old, I thought of music like torture. It was a dry, solitary, joyless exercise that I only engaged with because someone else forced me to. It took me 11 years to emerge from that sad place.
我两年的音乐生涯在老师对我的父母说了这话后嘎然而止：「他的手指太短。我不再教他了。」七岁的我一想到音乐就像受酷刑一样。那是一种枯燥、孤独、无聊的锻炼，我只因被逼而学。我花 11 年的时间才从那悲惨之地走了出来。
In year 12, I picked up a steel string acoustic guitar for the first time. I wanted to play it for church, and there was also a girl I was fairly keen on impressing. So I convinced my brother to teach me a few chords. And slowly, but surely, my mind changed. I was engaged in a creative process.
在第 12 年，我第一次拿起钢弦木吉他。我想为教堂演奏，还很想给某个女孩留下深刻的印象。所以我说服哥哥教我一些和弦。可以肯定的是我的想法慢慢变了。我进入创作的过程。
I was making music, and I was hooked. I started playing in a band, and I felt the delight of rhythm pulsing through my body as we brought our sounds together. I'd been surrounded by a musical ocean my entire life, and for the first time, I realized I could swim in it. I went through an almost identical experience when it came to mathematics.
我迷上了音乐，开始在乐队里演奏。每当我们将乐声汇聚在一起时，我感到节奏在我体内跳动。我一生都被音乐之洋包围着，那却是我首次意识到自己能够悠游其中。我在数学方面也经历了几乎相同的经历。
I used to believe that math was about rote learning inscrutable formulas to solve abstract problems that didn't mean anything to me. But at university, I began to see that mathematics is immensely practical and even beautiful,
我曾经认为数学是死记硬背深奥难懂的公式来解决抽象的问题，对我来说没有任何意义。但在大学里，我初见数学非常实用，甚至很美，
that it's not just about finding answers but also about learning to ask the right questions, and that mathematics isn't about mindlessly crunching numbers but rather about forming new ways to see problems so we can solve them by combining insight with imagination.
数学不仅在于寻找答案，还在于学会正确提问。数学不是无意识地处理数字，而是以新的方式看问题，结合洞察力和想象力来解决问题。
It gradually dawned on me that mathematics is a sense. Mathematics is a sense just like sight and touch; it's a sense that allows us to perceive realities which would be otherwise intangible to us. You know, we talk about a sense of humor and a sense of rhythm. Mathematics is our sense for patterns, relationships, and logical connections. It's a whole new way to see the world.
我渐渐意识到数学是一种知觉。数学就像视觉和触觉一样，是种使我们能够感知现实的知觉，要不然我们就看不见这些现实了。我们谈幽默感和节奏感。数学是我们对模式、关系和逻辑链接的感知。这是一种看待世界的全新方式。
Now, I want to show you a mathematical reality that I guarantee you've seen before but perhaps never really perceived. It's been hidden in plain sight your entire life.
现在，我想向你们展示一个数学上的现实，我保证你们以前见过，但也许未曾真正察觉，一直以来视而不见。
This is a river delta. It's a beautiful piece of geometry. Now, when we hear the word geometry, most of us think of triangles and circles. But geometry is the mathematics of all shapes, and this meeting of land and sea has created shapes with an undeniable pattern.
这是河流的三角洲。它是块美丽的几何图形。在听到「几何」一词时，我们大多数人会想到三角形和圆形。但是几何是所有形状的数学，这陆地和海洋相会之处创造出具有不可否认模式的形状。
It has a mathematically recursive structure. Every part of the river delta, with its twists and turns, is a micro version of the greater whole. So I want you to see the mathematics in this.
它具有数学递归结构。三角洲的每个部分有其曲折，是整体的微观版本。我希望你们能看到其中的数学。
But that's not all. I want you to compare this river delta with this amazing tree. It's a wonder in itself. But focus with me on the similarities between this and the river. What I want to know is why on earth should these shapes look so remarkably alike? Why should they have anything in common?
但这还不是全部。我希望你们将此河的三角洲与这棵令人惊奇的树相比较。它本身就是一个奇迹。但是，请把重点放在树与河的相似之处。我想知道为什么这些形状看起来如此相似？它们为什么有共同点？
Things get even more perplexing when you realize it's not just water systems and plants that do this. If you keep your eyes open, you'll see these same shapes are everywhere. Lightning bolts disappear so quickly that we seldom have the opportunity to ponder their geometry. But their shape is so unmistakable and so similar to what we've just seen that one can't help but be suspicious.
意识到不仅水流系统和植物这样更加令人困惑。睁开眼睛，到处都看得到这些相同的形状。闪电消失得如此之快，以至于我们很少有机会思考它们的几何形状。但是它们的形状如此明显，与我们刚刚看到的如此相似，让人不禁疑惑。
And then there's the fact that every single person in this room is filled with these shapes too. Every cubic centimeter of your body is packed with blood vessels that trace out this same pattern. There's a mathematical reality woven into the fabric of the universe that you share with winding rivers, towering trees, and raging storms.
还有另一事实就是这里的每个人也都充满了这些形状。身体的每立方厘米都充满了血管，这些血管可以勾画出相同的模式。在与蜿蜒的河流、参天大树和汹涌的暴风雨共享的宇宙中，编织着一种数学现实。
These shapes are examples of what we call "fractals," as mathematicians. Fractals get their name from the same place as fractions and fractures - it's a reference to the broken and shattered shapes we find around us in nature.
这些形状是数学家称为「碎形」的例子。碎形与分数和断裂的得名相同——是我们给予自然界中所发现的破碎形状的名字。
Now, once you have a sense for fractals, you really do start to see them everywhere: a head of broccoli, the leaves of a fern, even clouds in the sky. Like the other senses, our mathematical sense can be refined with practice. It's just like developing perfect pitch or a taste for wines. You can learn to perceive the mathematics around you with time and the right guidance.
一旦对碎形有感，就处处看得到它们：青花菜、蕨类植物的叶子，甚至天上的云彩。与其他知觉一样，我们的数学感可以经由练习来改善。就像发展完美的音感或品味葡萄酒一样。可以通过时间和正确的指导学习感知周围的数学。
Naturally, some people are born with sharper senses than the rest of us, others are born with impairment. As you can see, I drew a short straw in the genetic lottery when it came to my eyesight. Without my glasses, everything is a blur.
当然，有些人天生比我们其他人敏锐，其他人天生就有些障碍。如大家所见，我的视力属于基因乐透中的劣势。不戴眼镜就一片模糊。
I've wrestled with this sense my entire life, but I would never dream of saying, "Well, seeing has always been a struggle for me. I guess I'm just not a seeing kind of person."
我一生都为视力挣扎，然而我永远不会说：「好吧，既然一直以来我的视力差，我大概不是个【善视者】吧。」
Yet I meet people every day who feel it quite natural to say exactly that about mathematics. Now, I'm convinced we close ourselves off from a huge part of the human experience if we do this. Because all human beings are wired to see patterns.
然而我每天都会遇到一些人坦然宣称自己没有数学细胞。我深信这样子会让我们错失许多人类的经验，因为全人类与生俱来会看到模式。
We live in a patterned universe, a cosmos. That's what cosmos means - orderly and patterned - as opposed to chaos, which means disorderly and random. It isn't just seeing patterns that humans are so good at. We love making patterns too. And the people who do this well have a special name.
我们生活在有序的宇宙体系里。宇宙的含义就是——有秩序、有模式——与混乱恰恰相反，混乱意味着杂乱和随机。人类不仅善于看到模式，也喜欢制作模式。做得很好的人有特殊的名字。
We call them artists, musicians, sculptors, painters, cinematographers - they're all pattern creators. Music was once described as the joy that people feel when they are counting but don't know it.
我们称他们为艺术家、音乐家、雕塑家、画家、摄影师——他们都是模式创作者。音乐曾经被描述为人们在不自觉地计数时所感受到的快乐。
Some of the most striking examples of mathematical patterns are in Islamic art and design. An aversion to depicting humans and animals led to a rich history of intricate tile arrangements and geometric forms.
数学模式最引人注目的一些例子在伊斯兰艺术和设计里见得到，其对描绘人类和动物的禁忌引领出错综复杂的瓷砖排列和几何形状的丰富历史。
The aesthetic side of mathematical patterns like these brings us back to nature itself. For instance, flowers are a universal symbol of beauty. Every culture around the planet and throughout history has regarded them as objects of wonder. And one aspect of their beauty is that they exhibit a special kind of symmetry.
这些数学模式的美学将我们带回了自然界。例如，花朵是普遍的美丽象征。地球上、整个历史上的每种文化都将花朵视为奇观。花朵美丽的特点之一是它们表现出特殊的对称性。
Flowers grow organically from a center that expands outwards in the shape of a spiral, and this creates what we call "rotational symmetry." You can spin a flower around and around, and it still looks basically the same. But not all spirals are created equal. It all depends on the angle of rotation that goes into creating the spiral.
花以螺旋状由内向外有机地扩展生长，这产生了我们所谓的「旋转对称」。任意旋转一朵花，它看起来基本上还是一样。但并非所有螺旋都相同。这完全取决于产生螺旋的旋转角度。
For instance, if we build a spiral from an angle of 90 degrees, we get a cross that is neither beautiful nor efficient. Huge parts of the flowers area are wasted and don't produce seeds. Using an angle of 62 degrees is better and produces a nice circular shape, like what we usually associate with flowers. But it's still not great. There's still large parts of the area that are a poor use of resources for the flower.
例如，如果以 90 度角构建螺旋，就会得到既不美观也无效率的十字。大部分花的面积被浪费掉，不产生种子。使用 62 度角好一些，会产生一个很好的圆形，就像通常与花朵关联的形状那样。但这仍然不够好，仍有很大一部分未充分利用花卉的面积资源。
However, if we use 137.5 degrees, we get this beautiful pattern. It's astonishing, and it is exactly the kind of pattern used by that most majestic of flowers - the sunflower. Now, 137.5 degrees might seem pretty random, but it actually emerges out of a special number that we call the "golden ratio."
然而，如果用 137.5 度，就得到这个美丽的图案。令人惊讶，这正是最壮观的花朵——向日葵所用的那种图案。137.5 度看似随机，但它实际上出现在一个特殊的数字中，我们称之为「黄金比例」。
The golden ratio is a mathematical reality that, like fractals, you can find everywhere - from the phalanges of your fingers to the pillars of the Parthenon. That's why even at a party of 5000 people, I'm proud to declare, "I love mathematics!"
黄金比例是一个数学上的现实，就像碎形一样，处处找得到它——从手指的指骨到帕台农神庙的柱子。这就是为什么即使身处五千人的聚会之中，我仍然自豪地宣称：「我爱数学！」
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