TedTest

From the Ted Talk by Eddie Woo: How math is our real sixth sense

Unscramble the Blue Letters

"I love mathematics" (Laughter) is exactly what to say at a party if you want to spend the next couple of hours sipping your dnirk alone in the least cool corner of the room. And that's because when it comes to this subject - all the numbers, formulas, sblymos, and calculations - the vast majority of us are outsiders, and that idnuelcs me. 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 mikang mthas my career, is that, surprisingly, we are all deep down born to be mathematicians. (Laughter) But back to me being an outsider. I know what you're thinking: "Wait a second, Eddie. What would you know? You're a maths teacher. You went to a selective school. You wear glasses, and you're Asian." (Laughter) Firstly, that's risact. (Laughter) Secondly, that's wrong. When I was in school, my fiavtore subjects were English and hsroity. And this ceausd a lot of angst for me as a teenager because my high school truly honored mathematics. Your status in the school pretty much correlated with which mathematics class you ranked in. There were eight classes. So if you were in maths 4, that made you just about avgaere. If you were in maths 1, you were like royalty. 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 safhmuel place where my name appeared. Maths 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. I was in line to elronl at the faculty of eiactoudn when I started the citvenaorosn with one of its professors. He noticed that while my academic life had been dominated by htnuaieims, I had actually attempted some high-level maths at school. What he saw was not that I had a prloebm with maths, but that I had persevered with maths. And he knew something I didn't - that there was a critical shortage of mathematics educators in Australian schools, a shtgaroe that ranemis to this day. So he encouraged me to change my teaching area to mathematics. Now, for me, becoming a teacher wasn't about my love for a particular scbujet. It was about having a personal impact on the lives of young people. I'd seen fsairhtnd 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. As I sdeutid 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 ealrier in my life about music. Like a good migrant child, I dutifully learned to play the piano when I was young. (Laughter) My weekends were filled with endlessly repeating scales and memorizing every note in the peice, sinrpg and winter. I lasted two years before my career was abruptly ended when my teacher told my prnetas, "His fingers are too short. I will not teach him anymore." (Laughter) At seven years old, I thought of misuc like totrure. It was a dry, solitary, joyless exercise that I only eaneggd with because someone else forced me to. It took me 11 years to emerge from that sad place. 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 crodhs. And slowly, but surely, my mind changed. I was engaged in a cievrate process. I was making music, and I was hkoeod. I started playing in a band, and I felt the delight of rhythm pulsing through my body as we brhogut our sounds together. I'd been sendouurrd by a musical ocean my eitrne life, and for the first time, I realized I could swim in it. I went through an almost idnatiecl experience when it came to mathematics. I used to believe that maths was about rote learning inscrutable formulas to solve atcsrbat problems that didn't mean anything to me. But at university, I baegn to see that mathematics is immensely practical and even beautiful, that it's not just about finding answers but also about learning to ask the right questions, and that mathematics isn't about mindlessly cnrnihucg numbers but rather about forming new ways to see poblerms so we can solve them by combining insight with imagination. It gradually dawned on me that mathematics is a snese. Mathematics is a sense just like sight and touch; it's a sense that allows us to perceive reiaitles 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. 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. This is a revir delta. It's a beautiful piece of geemotry. Now, when we hear the word geometry, most of us think of tlinegars and circles. But geometry is the mathematics of all sehpas, and this meeting of land and sea has created shapes with an unbienldae pattern. It has a maiehcamaltlty recursive strurutce. Every part of the river delta, with its twists and turns, is a microversion of the greater whole. So I want you to see the mathematics in this. 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 ertah should these shapes look so remarkably alike? Why should they have anything in common? Things get even more perplexing when you realize it's not just water systems and ptlans that do this. If you keep your eyes open, you'll see these same shapes are everywhere. lihigtnng blots disappear so quickly that we seldom have the onotutrppiy to ponder their geometry. But their sphae is so unmistakable and so similar to what we've just seen that one can't help but be spicouusis. And then there's the fact that every single preosn in this room is filled with these shapes too. Every cubic centimeter of your body is pkeacd with blood vseless that trace out this same praettn. There's a mathematical reality woven into the fabric of the usnirvee that you share with wniindg rivers, tonreiwg trees, and raging sotmrs. 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 bekron and shattered shapes we find around us in nature. 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 coduls 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 wnies. You can learn to perceive the mathematics around you with time and the right guidance. Naturally, some people are born with sherpar senses than the rest of us, others are born with impairment. As you can see, I drew a short straw in the genetic ltretoy when it came to my eyheigst. Without my glasses, everything is a blur. 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." (Laughter) Yet I meet people every day who feel it quite nautral to say exactly that about mihcteaatms. 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 wried to see patterns. We live in a patterned universe, a cosmos. That's what cosmos means - oreldry and patterned - as opposed to chaos, which means disorderly and ranodm. 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 sipeacl name. 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. (Laughter) Some of the most sikrting epmlxaes of mthcmiateaal patterns are in iialsmc art and design. An aoeivrsn to depicting humans and ailmnas led to a rich history of intricate tile arrangements and geometric forms. 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 pnalet and throughout history has regarded them as objects of wonder. And one aepsct of their beauty is that they exhibit a special kind of setrmmyy. Flowers grow oliarlacngy 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 spilras are created euaql. It all depends on the angle of rotation that goes into creating the spiral. For instance, if we build a spiral from an angle of 90 degrees, we get a csros that is neither beautiful nor enfifeict. 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 gaert. There's still large parts of the area that are a poor use of resources for the flower. However, if we use 137.5 dgerees, (lhaeutgr) we get this beautiful pattern. It's astonishing, and it is exactly the kind of pattern used by that most majestic of frlewos - the sunflower. Now, 137.5 degrees might seem pretty random, but it actually egmrees out of a special number that we call the "golden ratio." The golden ratio is a mathematical reality that, like fractals, you can find everywhere - from the phalanges of your fingers to the prialls of the Parthenon. That's why even at a party of 5000 people, I'm proud to declare, "I love mathematics!" (Cheers) (Applause)

Open Cloze

"I love mathematics" (Laughter) is exactly what to say at a party if you want to spend the next couple of hours sipping your _____ alone in the least cool corner of the room. And that's because when it comes to this subject - all the numbers, formulas, _______, and calculations - the vast majority of us are outsiders, and that ________ me. 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 ______ _____ my career, is that, surprisingly, we are all deep down born to be mathematicians. (Laughter) But back to me being an outsider. I know what you're thinking: "Wait a second, Eddie. What would you know? You're a maths teacher. You went to a selective school. You wear glasses, and you're Asian." (Laughter) Firstly, that's ______. (Laughter) Secondly, that's wrong. When I was in school, my ________ subjects were English and _______. And this ______ a lot of angst for me as a teenager because my high school truly honored mathematics. Your status in the school pretty much correlated with which mathematics class you ranked in. There were eight classes. So if you were in maths 4, that made you just about _______. If you were in maths 1, you were like royalty. 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 ________ place where my name appeared. Maths 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. I was in line to ______ at the faculty of _________ when I started the ____________ with one of its professors. He noticed that while my academic life had been dominated by __________, I had actually attempted some high-level maths at school. What he saw was not that I had a _______ with maths, but that I had persevered with maths. And he knew something I didn't - that there was a critical shortage of mathematics educators in Australian schools, a ________ that _______ to this day. So he encouraged me to change my teaching area to mathematics. Now, for me, becoming a teacher wasn't about my love for a particular _______. It was about having a personal impact on the lives of young people. I'd seen _________ 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. As I _______ 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 _______ in my life about music. Like a good migrant child, I dutifully learned to play the piano when I was young. (Laughter) My weekends were filled with endlessly repeating scales and memorizing every note in the _____, ______ and winter. I lasted two years before my career was abruptly ended when my teacher told my _______, "His fingers are too short. I will not teach him anymore." (Laughter) At seven years old, I thought of _____ like _______. It was a dry, solitary, joyless exercise that I only _______ with because someone else forced me to. It took me 11 years to emerge from that sad place. 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 ______. And slowly, but surely, my mind changed. I was engaged in a ________ process. I was making music, and I was ______. I started playing in a band, and I felt the delight of rhythm pulsing through my body as we _______ our sounds together. I'd been __________ by a musical ocean my ______ life, and for the first time, I realized I could swim in it. I went through an almost _________ experience when it came to mathematics. I used to believe that maths was about rote learning inscrutable formulas to solve ________ problems that didn't mean anything to me. But at university, I _____ to see that mathematics is immensely practical and even beautiful, that it's not just about finding answers but also about learning to ask the right questions, and that mathematics isn't about mindlessly _________ numbers but rather about forming new ways to see ________ so we can solve them by combining insight with imagination. It gradually dawned on me that mathematics is a _____. Mathematics is a sense just like sight and touch; it's a sense that allows us to perceive _________ 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. 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. This is a _____ delta. It's a beautiful piece of ________. Now, when we hear the word geometry, most of us think of _________ and circles. But geometry is the mathematics of all ______, and this meeting of land and sea has created shapes with an __________ pattern. It has a ______________ recursive _________. Every part of the river delta, with its twists and turns, is a microversion of the greater whole. So I want you to see the mathematics in this. 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 _____ should these shapes look so remarkably alike? Why should they have anything in common? Things get even more perplexing when you realize it's not just water systems and ______ that do this. If you keep your eyes open, you'll see these same shapes are everywhere. _________ _____ disappear so quickly that we seldom have the ___________ to ponder their geometry. But their _____ is so unmistakable and so similar to what we've just seen that one can't help but be __________. And then there's the fact that every single ______ in this room is filled with these shapes too. Every cubic centimeter of your body is ______ with blood _______ that trace out this same _______. There's a mathematical reality woven into the fabric of the ________ that you share with _______ rivers, ________ trees, and raging ______. 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 ______ and shattered shapes we find around us in nature. 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 ______ 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 _____. You can learn to perceive the mathematics around you with time and the right guidance. Naturally, some people are born with _______ senses than the rest of us, others are born with impairment. As you can see, I drew a short straw in the genetic _______ when it came to my ________. Without my glasses, everything is a blur. 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." (Laughter) Yet I meet people every day who feel it quite _______ to say exactly that about ___________. 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 _____ to see patterns. We live in a patterned universe, a cosmos. That's what cosmos means - _______ and patterned - as opposed to chaos, which means disorderly and ______. 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 _______ name. 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. (Laughter) Some of the most ________ ________ of ____________ patterns are in _______ art and design. An ________ to depicting humans and _______ led to a rich history of intricate tile arrangements and geometric forms. 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 ______ and throughout history has regarded them as objects of wonder. And one ______ of their beauty is that they exhibit a special kind of ________. Flowers grow ___________ 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 _______ are created _____. It all depends on the angle of rotation that goes into creating the spiral. For instance, if we build a spiral from an angle of 90 degrees, we get a _____ that is neither beautiful nor _________. 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 _____. There's still large parts of the area that are a poor use of resources for the flower. However, if we use 137.5 _______, (________) we get this beautiful pattern. It's astonishing, and it is exactly the kind of pattern used by that most majestic of _______ - the sunflower. Now, 137.5 degrees might seem pretty random, but it actually _______ out of a special number that we call the "golden ratio." The golden ratio is a mathematical reality that, like fractals, you can find everywhere - from the phalanges of your fingers to the _______ of the Parthenon. That's why even at a party of 5000 people, I'm proud to declare, "I love mathematics!" (Cheers) (Applause)

Solution

includes
hooked
aversion
cross
shape
striking
islamic
laughter
firsthand
creative
shameful
organically
abstract
crunching
aspect
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favorite
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chords
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planet
maths
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spirals
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person
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opportunity
random
enroll
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bolts
racist
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conversation
began
humanities
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sense
studied
realities
lightning
pattern
broken
engaged
eyesight
identical
spring
piece
plants
lottery
making
winding
suspicious
shortage
subject

Original Text

"I love mathematics" (Laughter) is exactly what to say at a party if you want to spend the next couple of hours sipping your drink alone in the least cool corner of the room. 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. 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 maths my career, is that, surprisingly, we are all deep down born to be mathematicians. (Laughter) But back to me being an outsider. I know what you're thinking: "Wait a second, Eddie. What would you know? You're a maths teacher. You went to a selective school. You wear glasses, and you're Asian." (Laughter) Firstly, that's racist. (Laughter) Secondly, that's wrong. 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. Your status in the school pretty much correlated with which mathematics class you ranked in. There were eight classes. So if you were in maths 4, that made you just about average. If you were in maths 1, you were like royalty. 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. Maths 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. 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 maths at school. What he saw was not that I had a problem with maths, but that I had persevered with maths. 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. 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. 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. 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. Like a good migrant child, I dutifully learned to play the piano when I was young. (Laughter) My weekends were filled with endlessly repeating scales and memorizing every note in the piece, spring and winter. 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." (Laughter) 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. 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. 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. I used to believe that maths 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, 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. 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. 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. 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. It has a mathematically recursive structure. Every part of the river delta, with its twists and turns, is a microversion of the greater whole. So I want you to see the mathematics in this. 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? 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. 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. 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. 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. 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. 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." (Laughter) 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. 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. 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. (Laughter) 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. 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. 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. 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. However, if we use 137.5 degrees, (Laughter) 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." 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!" (Cheers) (Applause)

Frequently Occurring Word Combinations

ngrams of length 2

collocation	frequency
mathematical reality	3
river delta	2
mathematical patterns	2

Important Words

abruptly
abstract
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acute
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guidance
guitar
head
hear
hidden
high
history
home
honored
hooked
hours
huge
human
humanities
humans
humor
identical
imagination
immensely
impact
impairment
impressing
includes
inscrutable
insight
instance
intangible
intricate
islamic
joy
joyless
keen
kind
knew
land
large
lasted
lasting
laughter
learn
learned
learning
leaves
led
life
lightning
line
list
live
lives
logical
long
lot
lottery
love
majestic
majority
making
mathematical
mathematically
mathematicians
mathematics
maths
matter
means
meet
meeting
memorizing
microversion
migrant
mind
mindlessly
mistake
music
musical
musicians
narratives
natural
naturally
nature
nice
note
noticed
number
numbers
objects
ocean
open
opportunity
opposed
order
orderly
organically
originally
outsider
outsiders
outwards
packed
painters
parents
part
parthenon
parts
party
pattern
patterned
patterns
people
perceive
perceived
perfect
perplexing
persevered
person
personal
perspective
phalanges
piano
picked
piece
pillars
pinned
pitch
place
plain
planet
plants
play
playing
ponder
poor
positive
practical
practice
prestigious
pretty
print
prizes
problem
problems
process
produce
produces
professors
proud
pulsing
questions
quickly
racist
raging
raised
random
ranked
ratio
realities
reality
realize
realized
received
recursive
reference
refined
regarded
relationships
remains
remarkably
repeating
resources
rest
rhythm
rich
river
rivers
room
rotation
rote
royalty
sad
sails
scales
school
schools
scores
sculptors
sea
seeds
seldom
selective
sense
senses
set
shameful
shape
shapes
share
sharper
shattered
short
shortage
show
side
sight
similar
similarities
single
sipping
sky
slowly
solitary
solve
sounds
special
spend
spin
spiral
spirals
spring
start
started
status
steel
stories
storms
straw
striking
string
structure
struggle
struggled
students
studied
subject
subjects
sunflower
surely
surprisingly
surrounded
suspicious
swim
sydney
symbol
symbols
symmetry
systems
talk
taste
teach
teacher
teaching
teenager
thought
tile
time
today
told
torture
towering
trace
tree
trees
triangles
turns
twists
undeniable
understand
universal
universe
university
unmistakable
vast
vessels
wanted
wasted
water
ways
wear
weekends
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wines
winter
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word
world
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year
years
young