中英文稿

大家好。?我想談?wù)劺斫夂屠斫獾谋举|(zhì)，?理解到底是什么，?因?yàn)槲覀兌荚谧非罄斫狻?我們想理解世間萬(wàn)物。?我認(rèn)為理解是一種能力，?轉(zhuǎn)變（固有）觀點(diǎn)的能力。?如果我們?nèi)狈λ?就說明我們?nèi)狈斫饬Α?這是我的結(jié)論。

我想重點(diǎn)講講數(shù)學(xué)。?很多人認(rèn)為，數(shù)學(xué)就是?加，減，乘，除，?分?jǐn)?shù)，百分?jǐn)?shù)，幾何，代數(shù)等等。?但今天，我也想講講數(shù)學(xué)的本質(zhì)，?我的觀點(diǎn)是，數(shù)學(xué)跟模式有關(guān)。

在我身后，是一個(gè)美麗的圖案，?而這個(gè)圖案，實(shí)際上是通過特定方式?不斷畫圓組成的。?所以我對(duì)數(shù)學(xué)有一個(gè)的定義?非常直白：?首先，數(shù)學(xué)的關(guān)鍵是尋找模式。?這里的模式指的是 某種聯(lián)系、結(jié)構(gòu)，或者規(guī)律、規(guī)則，?這些東西控制了我們所見的事物。?其次，我認(rèn)為數(shù)學(xué)是一種語(yǔ)言， 用來描述各種模式。?如果沒有現(xiàn)成的語(yǔ)言， 就需要?jiǎng)?chuàng)造一種。?在數(shù)學(xué)中，這點(diǎn)尤為重要。?同時(shí)，數(shù)學(xué)也需要進(jìn)行假設(shè)，?對(duì)假設(shè)進(jìn)行多方驗(yàn)證，看看結(jié)果如何。?我們一會(huì)兒就會(huì)這么做。?最后，數(shù)學(xué)可以用來做很酷的事情。?能幫我們完成很多事。

下面我們來看一些模式。?如果你想系領(lǐng)帶，?會(huì)有很多種樣式。?每一種都有名字。?因此領(lǐng)帶結(jié)也包含數(shù)學(xué)。?這是從左側(cè)繞出，右側(cè)繞入， 中間抽出然后系緊的東方結(jié)。?這是從左側(cè)繞入，右側(cè)繞出，再左側(cè)繞入， 中間抽出，最后系緊的四手結(jié)。?這就是我們專門 為領(lǐng)帶結(jié)創(chuàng)造的語(yǔ)言，?最后還有半溫莎結(jié)。?這是一本關(guān)于系鞋帶的數(shù)學(xué)書，?大學(xué)級(jí)別的，?因?yàn)橄敌瑤б灿泻芏喾N模式。?你可以用成千上萬(wàn)種方式來系鞋帶。?我們可以進(jìn)行分析。?然后為系鞋帶也創(chuàng)造一種語(yǔ)言。

這些都可以用數(shù)學(xué)方法來表達(dá)。?這是萊布尼茨 在1675年使用的符號(hào)。?他創(chuàng)造了一種語(yǔ)言， 來描述自然界的模式。?當(dāng)我們把物品拋向空中，?它會(huì)掉下來。?為什么？?我們并不確定，但我們可以 用數(shù)學(xué)把其歸結(jié)成一種模式。

這也是一種模式。?是一種被發(fā)明的語(yǔ)言。?你能猜到這是什么嗎？?這是一套表示 舞蹈動(dòng)作的符號(hào)，踢踏舞。?這能讓舞蹈編排者， 編一些炫酷的，新的動(dòng)作，?因?yàn)樗苡梅?hào)來描述動(dòng)作。

請(qǐng)大家想一想， 表達(dá)是多么神奇的東西。?這里寫的是“數(shù)學(xué)”這個(gè)詞。?實(shí)際上就是一些點(diǎn)，對(duì)吧？?一些點(diǎn)怎么能表示單詞呢？?確實(shí)可以。?他們代表了單詞“數(shù)學(xué)”，?這些符號(hào)也一樣，?這次我們可以用聽的。?聽起來就像這樣。

可以說，這些聲音 也代表了這個(gè)詞和它的含義。?這是怎么做到的呢？?表達(dá)是一種很神奇的過程。

所以我想跟你們 討論一下在表達(dá)過程中?發(fā)生的神奇的事情。?現(xiàn)在你們看到的 只是不同寬度的線條。?這些線條代表了一本書。?強(qiáng)烈推薦這本書，非常不錯(cuò)。

真的，不騙你們。

好吧，讓我們來做一個(gè)實(shí)驗(yàn)。?來玩一下直線。?這是一條直線。?再畫另外一條。?每一次我們都往下、往右移動(dòng)一格，?畫出一條新的直線。?如此反復(fù)，?從中尋找一種模式。?我們得到了這個(gè)圖案，?是一個(gè)非常好看的圖案。?它看起來就像一道弧，對(duì)吧？?我們僅僅畫了些簡(jiǎn)單的直線。

現(xiàn)在，稍微改變一下角度，旋轉(zhuǎn)一下。?再看這段弧。?像什么？?是不是像圓的一部分？?其實(shí)它不是圓的一部分。?所以我繼續(xù)探尋，找出真正的模式。?也許我可以復(fù)制它，畫一幅畫？?好像不行。?也許我應(yīng)該延長(zhǎng)這些線條，?再來尋找模式。?再多畫一些線條。?然后這樣。?把它縮小，再變換角度。?然后我們就會(huì)發(fā)現(xiàn)，開始的直線?變成了拋物線。?這可以用一個(gè)簡(jiǎn)單的等式表達(dá)，?很美的圖案。

這就是我們所做的。?找到某種模式，然后表達(dá)出來。?這是一種很直白的定義。?但是今天，我想討論得更深入一些，?思考它們的本質(zhì)是什么。?是什么造就了這一切？?要看得更深入一些，?就要求我們有轉(zhuǎn)換角度的能力。?當(dāng)你換一種角度來看問題，?當(dāng)你接受另一種觀點(diǎn)，?你就能在所見所聞中，?學(xué)到新的東西。?我認(rèn)為這一點(diǎn)非常重要。

讓我們看看這個(gè)簡(jiǎn)單的方程，?x+x=2x?這是一個(gè)很好的模式，也是正確的。?因?yàn)?+5=2x5。?這個(gè)等式我們司空見慣了。?但是仔細(xì)想一想：這是一個(gè)等式。?它代表一個(gè)事物 與另一個(gè)事物相等，?這么表述有兩種角度。?一種是總和。?是相加的過程。?另一種是相乘。?這是兩種不同的角度。?我會(huì)進(jìn)一步說， 每個(gè)等式都像這樣，?每一個(gè)使用等號(hào)連接的數(shù)學(xué)方程?實(shí)際上都是隱喻。?是兩種事物間的類比。?你觀察一件事情，產(chǎn)生兩種觀點(diǎn)，?然后用一種語(yǔ)言來表達(dá)。

看這個(gè)方程。?它是最美的等式之一。?簡(jiǎn)單表明了，?等式兩邊都是-1。?左手邊的是-1，右邊的也是。?我認(rèn)為這是數(shù)學(xué)中很重要的部分?——采取不同的觀點(diǎn)。

我們繼續(xù)。?選一個(gè)數(shù)字好了。?我們知道4/3，知道它的含義。?就是1.333……，但是 一定要加上后面的省略號(hào)，?否則就不是準(zhǔn)確的4/3了。?但只有在使用十進(jìn)制時(shí)才如此。?我們的數(shù)字系統(tǒng)用的是10位計(jì)數(shù)。?如果我們改成2位計(jì)數(shù)，?也就是二進(jìn)制。?就變成了這樣。?我們現(xiàn)在在討論數(shù)字。?討論4/3這個(gè)數(shù)字。?我們也可以這樣表示，?我們改變進(jìn)制，改變數(shù)位，?就可以用不同的方式書寫。

所有這些都代表同一個(gè)數(shù)。?我們甚至可以把它 簡(jiǎn)單寫作1.3或1.6。?取決于我們選用哪種進(jìn)制。?或者我們還可以簡(jiǎn)單寫成這樣。?我喜歡這種，因?yàn)樗硎?被3除。?表現(xiàn)了兩個(gè)數(shù)字間的關(guān)系。?上邊是4，下邊是3。?你可以用許多方式 來把這個(gè)數(shù)字可視化。?從不同的角度來看這個(gè)數(shù)字。?我在不斷嘗試。?改變觀察事物的角度。?我是故意這么做的。?讓我們畫一個(gè)網(wǎng)格。?假如為4行3列， 那么這條線就始終代表5。?肯定如此。這是一個(gè)美麗的圖案。?4和3和5。?這個(gè)長(zhǎng)方形，長(zhǎng)寬比為4:3，?你們見過很多次的。?就是你們的屏幕大小的平均值。?800 x 600 或是 1600 x 1200?分別是電腦和電視的屏幕。

這都是很好的表達(dá)方式，?但是我還想再深入一點(diǎn)點(diǎn)， 再玩一下這些數(shù)字。?現(xiàn)在，你能看到兩個(gè)圓。 我要像這樣旋轉(zhuǎn)它們。?看一下左上角的那個(gè)，?它轉(zhuǎn)得更快一點(diǎn)兒，對(duì)吧？?你們都能看到。?準(zhǔn)確來說，它的旋轉(zhuǎn)速度 是慢速的4/3倍。?也就是說，它每轉(zhuǎn)4圈，?另一個(gè)圓就會(huì)轉(zhuǎn)3圈。?現(xiàn)在，畫兩條線， 并標(biāo)明相交處的點(diǎn)。?我們就能得到一個(gè)跳舞的點(diǎn)。

這個(gè)點(diǎn)就來源于4/3這個(gè)數(shù)字。?是吧？現(xiàn)在，讓我來看看它的軌跡。?把軌跡畫出來，看看是什么樣子。?這就是數(shù)學(xué)。?就是不斷探索會(huì)發(fā)生什么。?而這來自于4/3這個(gè)數(shù)字。?我覺得，這就是4/3的肖像。?比數(shù)字好看多了——（歡呼）

（掌聲）?其實(shí)這不算新鮮事了。?很早以前就被發(fā)現(xiàn)了， 但是——

但是這僅僅是4/3。

讓我們?cè)僮鲆粋€(gè)實(shí)驗(yàn)。?讓我們選一個(gè)聲音， 是這樣的：（嘟）

這是一個(gè)完美的A，440Hz。?把它翻倍。?就得到了這個(gè)聲音。（嘟）

同時(shí)播放這兩種聲音， 聽起來是這個(gè)效果。?這是一個(gè)八度音，對(duì)吧？?我們來玩一個(gè)游戲。 我們?cè)俜乓淮蜛。?然后我們把它翻為1.5倍。

我們稱之為純五度音。

把它們一起播放，聽起來很不錯(cuò)。?讓我們把這個(gè)聲音翻4/3倍。

會(huì)怎么樣？?你們會(huì)得到這個(gè)聲音。

純四度音。?如果第一個(gè)音是A， 那么這就是一個(gè)D。?一起播放，是這樣的聲音。

這就是4/3的聲音。?這就是改變角度。?我是在從另一個(gè)角度看一個(gè)數(shù)字。

也可以用節(jié)奏來表示。?我可以選一個(gè)節(jié)奏， 在一段時(shí)間內(nèi)敲3下（鼓點(diǎn)聲）

一段固定的時(shí)間，?然后在同樣的時(shí)間內(nèi)敲4下。

單獨(dú)聽很枯燥， 但如果放在一起。

嘿！好多了。

我還可以加點(diǎn)兒踩镲聲。

聽到了嗎？?所以，這就是4/3的聲音。?4/3的節(jié)律。

我還可以繼續(xù)玩， 用這個(gè)數(shù)字做游戲。?4/3是一個(gè)超棒的數(shù)字。 我愛死4/3了！

說真的——4/3的價(jià)值被低估了。?如果你拿一個(gè)球體，看看它的體積，?會(huì)發(fā)現(xiàn)其實(shí)球體體積就是 某個(gè)圓柱體積的4/3倍。?所以4/3出現(xiàn)在了球體里， 是球的體積。

好，我為什么玩這些？?是想跟你們談?wù)?理解一件事物的意義，?談?wù)勎覀兯f的理解是什么。?這就是我的目的。?我認(rèn)為，只有當(dāng)我們 從多個(gè)角度去審視同一事物時(shí)，?才能說我們理解了它。?讓我們看看這個(gè)字母。 這是一個(gè)漂亮的R，對(duì)吧？?你們?cè)趺磁袛噙@是個(gè)R？?因?yàn)槟銈兛催^各種各樣的R，?然后進(jìn)行歸納，?提取它們的共性， 找到了一種模式。?然后你們確認(rèn)這是一個(gè)R。

所以，我要說的是?理解事物和變換角度?是有關(guān)的。?我是一名教師和演講者，?我可以利用這一點(diǎn)去教課，?因?yàn)槲矣秒[喻和類比的方法， 給學(xué)生們換一種方式講故事，?從不同的角度去講述一件事，?我就能讓他們真正理解。?我讓理解變?yōu)榱丝赡埽?因?yàn)槟銈儽仨氁?歸納自己的所見所聞，?如果我給你們另一個(gè)角度， 你們做起來就會(huì)更容易。

讓我們?cè)倥e一個(gè)例子。?這是4和3。 這是4個(gè)三角形。?這也是某種4/3。?讓我們把它們連起來。?現(xiàn)在我們?cè)偻嬉粋€(gè)游戲， 把它們折疊起來，?形成一個(gè)三維結(jié)構(gòu)。?我喜歡這個(gè)。?這是一個(gè)金字塔形。?讓我們?cè)僮鲆粋€(gè)，把它們放在一起。?就形成了一個(gè)八面體。?這是5種正多面體 （又叫柏拉圖立體）之一。?現(xiàn)在我們可以真的來改變角度，?繞各種軸旋轉(zhuǎn)它，?從其它角度來觀察。?我可以改變旋轉(zhuǎn)軸，?改變觀察角度，?還是同一個(gè)物體， 只是看起來有一些不同。?我可以再做一次。

我每調(diào)整一次， 就會(huì)有新東西出現(xiàn)，?所以通過改變角度，?我能更加了解這個(gè)物體。?我可以把它作為創(chuàng)造理解的工具。?我可以把兩個(gè)正四面體， 像這樣穿起來，?看看會(huì)發(fā)生什么。?有點(diǎn)兒像正八面體。?把它旋轉(zhuǎn)起來再看。?發(fā)生了什么？?如果你把這兩個(gè)物體 拼在一起，旋轉(zhuǎn)它，?你就又得到了一個(gè)正八面體，?一個(gè)漂亮的結(jié)構(gòu)。?如果你把它平攤在地上，?這就是一個(gè)正八面體。?正八面體的平面結(jié)構(gòu)圖。?我還可以繼續(xù)玩。?在正八面體周圍畫三個(gè)大圈，?轉(zhuǎn)動(dòng)看看，?三個(gè)大圈實(shí)際上 是與正八面體相連的。?如果我拿一個(gè)自行車泵， 把它充滿氣，?你會(huì)發(fā)現(xiàn)，它看起來還是 有點(diǎn)兒像正八面體的。?看出來我在做什么了嗎？?我在不停改變角度。

讓我們退后一步?——這其實(shí)是一個(gè)隱喻，退后一步——?看看我們?cè)谧龅氖虑椤?我在使用隱喻，?在變換角度，進(jìn)行類比。?變換不同的角度，?來講同一個(gè)故事。?我在敘述， 而且做了好幾種敘述。?我認(rèn)為這一切使得理解變成可能。?我認(rèn)為這是理解事物的關(guān)鍵。?我深信這點(diǎn)。

所以，關(guān)于改變你們的角度——?對(duì)人類來說十分重要。?讓我們來看看地球。?讓我們放大到海洋，看看海洋。?我們可以放大任何事物。?我們以海洋為例，仔細(xì)的看看它。?我們能觀察海浪。?或是沙灘。?我們也可以從另一個(gè)角度看海洋。?每變一次角度，我們就能 對(duì)海洋了解得多一些。?如果我們走到海邊， 就能聞到海水的味道，對(duì)吧？?能聽到海浪的聲音。?能嘗到風(fēng)中咸咸的味道。?所有這些，都是不同的角度。?而這個(gè)（角度）是最棒的。?我們進(jìn)入水中。?從內(nèi)部來觀察。?你們知道嗎？?這對(duì)數(shù)學(xué)和計(jì)算機(jī)科學(xué) 來說都絕對(duì)重要。?如果你能從一個(gè) 結(jié)構(gòu)的內(nèi)部去進(jìn)行觀察，?那你就能夠真正認(rèn)識(shí)它。?認(rèn)識(shí)到它的本質(zhì)。

所以，當(dāng)我們一路前行，?進(jìn)入海洋，?我們發(fā)揮了想象力。?我認(rèn)為這又更深入了一層，?是改變角度的必然要求。?我們可以做個(gè)游戲。?想象一下你正坐在那兒。?然后你同時(shí)又在上面。?你就可以從外部審視你自己了。?這聽起來很奇怪。?你在改變你的角度。?你在使用你的想象力，?你在從外部審視你自己。?這需要有想象力。

數(shù)學(xué)和計(jì)算機(jī)科學(xué)是 最具想象力的藝術(shù)形式。?還有一種改變角度的方式，?可能更被你們熟知，?因?yàn)槲覀兠刻於荚谧觯?叫做共情。?當(dāng)我從你的角度看世界的時(shí)候，?我就與你產(chǎn)生了共情。?如果我能夠真正的?理解你們眼中的世界，?那我就與你產(chǎn)生了共情。?這需要想象力。?這就是我們獲得理解的方式。?而這種方式充斥了 數(shù)學(xué)和計(jì)算機(jī)科學(xué)領(lǐng)域。?共情和這些學(xué)科間有著深刻的聯(lián)系。

所以，我的結(jié)論是：?深入的理解一件事?與轉(zhuǎn)換角度的能力密切相關(guān)。?所以我的建議是： 嘗試轉(zhuǎn)換你的角度。?你可以學(xué)習(xí)數(shù)學(xué)。?這是鍛煉大腦的好方法。?變換你們的角度， 讓思維變得更靈活。?它能夠讓你們易于接受新事物，?能夠理解事物。?請(qǐng)?jiān)试S我再使用一次隱喻：?讓思維像水一樣吧。?會(huì)很不錯(cuò)的。謝謝大家。

?Hi.?I want to talk about understanding, and the nature of understanding,?and what the essence of understanding is,?because understanding is something we aim for, everyone.?We want to understand things.?My claim is that understanding has to do?with the ability to change your perspective.?If you don't have that, you don't have understanding.?So that is my claim.

And I want to focus on mathematics.?Many of us think of mathematics as addition, subtraction,?multiplication, division,?fractions, percent, geometry, algebra -- all that stuff.?But actually, I want to talk about the essence of mathematics as well.?And my claim is that mathematics has to do with patterns.

Behind me, you see a beautiful pattern,?and this pattern actually emerges just from drawing circles?in a very particular way.?So my day-to-day definition of mathematics that I use every day?is the following:?First of all, it's about finding patterns.?And by "pattern," I mean a connection, a structure, some regularity,?some rules that govern what we see.?Second of all,?I think it is about representing these patterns with a language.?We make up language if we don't have it,?and in mathematics, this is essential.?It's also about making assumptions?and playing around with these assumptions and just seeing what happens.?We're going to do that very soon.?And finally, it's about doing cool stuff.?Mathematics enables us to do so many things.

So let's have a look at these patterns.?If you want to tie a tie knot,?there are patterns.?Tie knots have names.?And you can also do the mathematics of tie knots.?This is a left-out, right-in, center-out and tie.?This is a left-in, right-out, left-in, center-out and tie.?This is a language we made up for the patterns of tie knots,?and a half-Windsor is all that.?This is a mathematics book about tying shoelaces?at the university level,?because there are patterns in shoelaces.?You can do it in so many different ways.?We can analyze it.?We can make up languages for it.

And representations are all over mathematics.?This is Leibniz's notation from 1675.?He invented a language for patterns in nature.?When we throw something up in the air,?it falls down.?Why??We're not sure, but we can represent this with mathematics in a pattern.

This is also a pattern.?This is also an invented language.?Can you guess for what??It is actually a notation system for dancing, for tap dancing.?That enables him as a choreographer to do cool stuff, to do new things,?because he has represented it.

I want you to think about how amazing representing something actually is.?Here it says the word "mathematics."?But actually, they're just dots, right??So how in the world can these dots represent the word??Well, they do.?They represent the word "mathematics,"?and these symbols also represent that word?and this we can listen to.?It sounds like this.

Somehow these sounds represent the word and the concept.?How does this happen??There's something amazing going on about representing stuff.

So I want to talk about that magic that happens?when we actually represent something.?Here you see just lines with different widths.?They stand for numbers for a particular book.?And I can actually recommend this book, it's a very nice book.

Just trust me.

OK, so let's just do an experiment,?just to play around with some straight lines.?This is a straight line.?Let's make another one.?So every time we move, we move one down and one across,?and we draw a new straight line, right??We do this over and over and over,?and we look for patterns.?So this pattern emerges,?and it's a rather nice pattern.?It looks like a curve, right??Just from drawing simple, straight lines.

Now I can change my perspective a little bit. I can rotate it.?Have a look at the curve.?What does it look like??Is it a part of a circle??It's actually not a part of a circle.?So I have to continue my investigation and look for the true pattern.?Perhaps if I copy it and make some art??Well, no.?Perhaps I should extend the lines like this,?and look for the pattern there.?Let's make more lines.?We do this.?And then let's zoom out and change our perspective again.?Then we can actually see that what started out as just straight lines?is actually a curve called a parabola.?This is represented by a simple equation,?and it's a beautiful pattern.

So this is the stuff that we do.?We find patterns, and we represent them.?And I think this is a nice day-to-day definition.?But today I want to go a little bit deeper,?and think about what the nature of this is.?What makes it possible??There's one thing that's a little bit deeper,?and that has to do with the ability to change your perspective.?And I claim that when you change your perspective,?and if you take another point of view,?you learn something new about what you are watching?or looking at or hearing.?And I think this is a really important thing that we do all the time.

So let's just look at this simple equation,?x + x = 2 ? x.?This is a very nice pattern, and it's true,?because 5 + 5 = 2 ? 5, etc.?We've seen this over and over, and we represent it like this.?But think about it: this is an equation.?It says that something is equal to something else,?and that's two different perspectives.?One perspective is, it's a sum.?It's something you plus together.?On the other hand, it's a multiplication,?and those are two different perspectives.?And I would go as far as to say that every equation is like this,?every mathematical equation where you use that equality sign?is actually a metaphor.?It's an analogy between two things.?You're just viewing something and taking two different points of view,?and you're expressing that in a language.

Have a look at this equation.?This is one of the most beautiful equations.?It simply says that, well,?two things, they're both -1.?This thing on the left-hand side is -1, and the other one is.?And that, I think, is one of the essential parts?of mathematics -- you take different points of view.

So let's just play around.?Let's take a number.?We know four-thirds. We know what four-thirds is.?It's 1.333, but we have to have those three dots,?otherwise it's not exactly four-thirds.?But this is only in base 10.?You know, the number system, we use 10 digits.?If we change that around and only use two digits,?that's called the binary system.?It's written like this.?So we're now talking about the number.?The number is four-thirds.?We can write it like this,?and we can change the base, change the number of digits,?and we can write it differently.

So these are all representations of the same number.?We can even write it simply, like 1.3 or 1.6.?It all depends on how many digits you have.?Or perhaps we just simplify and write it like this.?I like this one, because this says four divided by three.?And this number expresses a relation between two numbers.?You have four on the one hand and three on the other.?And you can visualize this in many ways.?What I'm doing now is viewing that number from different perspectives.?I'm playing around.?I'm playing around with how we view something,?and I'm doing it very deliberately.?We can take a grid.?If it's four across and three up, this line equals five, always.?It has to be like this. This is a beautiful pattern.?Four and three and five.?And this rectangle, which is 4 x 3,?you've seen a lot of times.?This is your average computer screen.?800 x 600 or 1,600 x 1,200?is a television or a computer screen.

So these are all nice representations,?but I want to go a little bit further and just play more with this number.?Here you see two circles. I'm going to rotate them like this.?Observe the upper-left one.?It goes a little bit faster, right??You can see this.?It actually goes exactly four-thirds as fast.?That means that when it goes around four times,?the other one goes around three times.?Now let's make two lines, and draw this dot where the lines meet.?We get this dot dancing around.

And this dot comes from that number.?Right? Now we should trace it.?Let's trace it and see what happens.?This is what mathematics is all about.?It's about seeing what happens.?And this emerges from four-thirds.?I like to say that this is the image of four-thirds.?It's much nicer -- (Cheers)

Thank you!

(Applause)?This is not new.?This has been known for a long time, but --

But this is four-thirds.

Let's do another experiment.?Let's now take a sound, this sound: (Beep)

This is a perfect A, 440Hz.?Let's multiply it by two.?We get this sound. (Beep)

When we play them together, it sounds like this.?This is an octave, right??We can do this game. We can play a sound, play the same A.?We can multiply it by three-halves.

This is what we call a perfect fifth.

They sound really nice together.?Let's multiply this sound by four-thirds. (Beep)

What happens??You get this sound. (Beep)

This is the perfect fourth.?If the first one is an A, this is a D.?They sound like this together. (Beeps)

This is the sound of four-thirds.?What I'm doing now, I'm changing my perspective.?I'm just viewing a number from another perspective.

I can even do this with rhythms, right??I can take a rhythm and play three beats at one time (Drumbeats)

in a period of time,?and I can play another sound four times in that same space.

Sounds kind of boring, but listen to them together.

I can even make a little hi-hat.

(Drumbeats and cymbals)

Can you hear this??So, this is the sound of four-thirds.?Again, this is as a rhythm.

(Drumbeats and cowbell)

And I can keep doing this and play games with this number.?Four-thirds is a really great number. I love four-thirds!

Truly -- it's an undervalued number.?So if you take a sphere and look at the volume of the sphere,?it's actually four-thirds of some particular cylinder.?So four-thirds is in the sphere. It's the volume of the sphere.

OK, so why am I doing all this??Well, I want to talk about what it means to understand something?and what we mean by understanding something.?That's my aim here.?And my claim is that you understand something?if you have the ability to view it from different perspectives.?Let's look at this letter. It's a beautiful R, right??How do you know that??Well, as a matter of fact, you've seen a bunch of R's,?and you've generalized?and abstracted all of these and found a pattern.?So you know that this is an R.

So what I'm aiming for here is saying something?about how understanding and changing your perspective?are linked.?And I'm a teacher and a lecturer,?and I can actually use this to teach something,?because when I give someone else another story, a metaphor, an analogy,?if I tell a story from a different point of view,?I enable understanding.?I make understanding possible,?because you have to generalize over everything you see and hear,?and if I give you another perspective, that will become easier for you.

Let's do a simple example again.?This is four and three. This is four triangles.?So this is also four-thirds, in a way.?Let's just join them together.?Now we're going to play a game; we're going to fold it up?into a three-dimensional structure.?I love this.?This is a square pyramid.?And let's just take two of them and put them together.?So this is what is called an octahedron.?It's one of the five platonic solids.?Now we can quite literally change our perspective,?because we can rotate it around all of the axes?and view it from different perspectives.?And I can change the axis,?and then I can view it from another point of view,?but it's the same thing, but it looks a little different.?I can do it even one more time.