full transcript

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 drink alone in the least cool corner of the room. And that's because when it comes to this subject - all the nubemrs, formulas, somlybs, and calculations - the vast majority of us are oiudtress, 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 miknag maths my craeer, 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, edide. What would you know? You're a maths teacher. You went to a slvciteee school. You wear glasses, and you're Asian." (leaguhtr) Firstly, that's racist. (Laughter) Secondly, that's wrong. When I was in school, my favorite subjects were English and hiortsy. And this caused a lot of agsnt 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 raotlyy. Each year, our school entered the prestigious Australian Mathematics Competition and would print out a list of everyone in the shcool in oredr of our scores. setutnds 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 egnlsih and history teacher. But a chance encounter at Sydney University altered my life forever. I was in line to enroll at the faculty of eoduiactn when I started the conversation with one of its professors. He noticed that while my academic life had been dnaoteimd by humanities, I had actually attempted some high-level mhats 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 fthasnrid at school what a lasting and positive difference a great teacher can make. I wnaetd to do that for someone, and it didn't matter to me what seujcbt 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 magirnt chlid, I dutifully learned to play the piano when I was young. (Laughter) My weekends were filled with elndelssy repeating scales and memorizing every note in the piece, siprng and winter. I lasted two years before my career was abruptly enedd when my teacher told my parents, "His fingers are too sorht. I will not teach him anymore." (Laughter) At seven years old, I thought of music like tourrte. It was a dry, solitary, joyless exercise that I only engaged with because someone else feorcd 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 ccrhuh, and there was also a girl I was fairly keen on impressing. So I cvcnnoeid my brother to teach me a few chords. And slowly, but surely, my mind changed. I was eeggnad in a creative psocers. I was making muisc, and I was hokeod. 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 miacsul 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 asernws 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 comnbinig 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 sesne of rhythm. Mathematics is our sense for patterns, rpnhioieltass, and loagcil connections. It's a whole new way to see the world. Now, I want to show you a mathematical reality that I gartunaee you've seen before but perhaps never really perceived. It's been hidden in plain sgiht your entire life. This is a river delta. It's a beautiful piece of gertoemy. Now, when we hear the word geometry, most of us think of tlerniags and circles. But geometry is the mathematics of all shapes, and this meeting of land and sea has craeted shapes with an undeniable pattern. It has a mathematically recursive structure. Every part of the river delta, with its twists and truns, 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 revir 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 sheaps look so remarkably alike? Why should they have anything in cmoomn? 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 dspaepiar so quickly that we soeldm have the opportunity to ponder their geometry. But their shape is so unmistakable and so smiliar 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 cibuc centimeter of your body is pceakd with blood vessels that trace out this same pattern. There's a mathematical reality woven into the fabirc of the universe that you share with winding rivers, towering trees, and raging storms. These shapes are epxlemas of what we call "fractals," as mathematicians. Fractals get their name from the same place as fractions and fractures - it's a rnceefere to the broekn 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 ptaiccre. It's just like developing perfect pitch or a taste for wines. You can lraen to perceive the mamhtectias 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 wrtlesed 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 haumn 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 peatetrnd - 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, pertanis, 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 mtaateaimchl patterns are in Islamic art and desgin. 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 sobmyl of beauty. Every culture around the pleant and throughout history has regaredd them as objects of wonder. And one aepsct of their beauty is that they exhibit a special kind of sytremmy. Flowers grow organically from a center that expands outwards in the shape of a spiral, and this ceaters what we call "rotational symmetry." You can spin a flower around and around, and it still looks blcalsiay the same. But not all spirals are created eaqul. It all depends on the angle of rotation that goes into cniraetg the sapril. For instance, if we build a spiral from an angle of 90 degrees, we get a cross that is neither bufiaetul nor efficient. Huge parts of the flowers area are wasted and don't produce sedes. Using an angle of 62 degrees is better and produces a nice circular spahe, like what we usually associate with flowers. But it's still not great. There's still lgrae prtas 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 ptertan. It's astonishing, and it is exactly the kind of pattern used by that most majestic of flowers - the sfewunlor. Now, 137.5 degrees might seem pretty rnoadm, 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 finegrs to the pillars of the Parthenon. That's why even at a party of 5000 pleope, I'm proud to drelace, "I love mathematics!" (Cheers) (apaslupe)

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 drink alone in the least cool corner of the room. And that's because when it comes to this subject - all the _______, formulas, _______, and calculations - the vast majority of us are _________, 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 ______ maths my ______, 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, _____. What would you know? You're a maths teacher. You went to a _________ school. You wear glasses, and you're Asian." (________) Firstly, that's racist. (Laughter) Secondly, that's wrong. When I was in school, my favorite subjects were English and _______. And this caused a lot of _____ 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 _______. Each year, our school entered the prestigious Australian Mathematics Competition and would print out a list of everyone in the ______ in _____ of our scores. ________ 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 _______ and history teacher. But a chance encounter at Sydney University altered my life forever. I was in line to enroll at the faculty of _________ when I started the conversation with one of its professors. He noticed that while my academic life had been _________ by humanities, I had actually attempted some high-level _____ 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 _________ at school what a lasting and positive difference a great teacher can make. I ______ to do that for someone, and it didn't matter to me what _______ 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 _______ _____, I dutifully learned to play the piano when I was young. (Laughter) My weekends were filled with _________ repeating scales and memorizing every note in the piece, ______ and winter. I lasted two years before my career was abruptly _____ when my teacher told my parents, "His fingers are too _____. I will not teach him anymore." (Laughter) At seven years old, I thought of music like _______. It was a dry, solitary, joyless exercise that I only engaged with because someone else ______ 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 ______, and there was also a girl I was fairly keen on impressing. So I _________ my brother to teach me a few chords. And slowly, but surely, my mind changed. I was _______ in a creative _______. I was making _____, and I was ______. 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 _______ 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 _______ 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 _________ 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 _____ of rhythm. Mathematics is our sense for patterns, _____________, and _______ connections. It's a whole new way to see the world. Now, I want to show you a mathematical reality that I _________ you've seen before but perhaps never really perceived. It's been hidden in plain _____ your entire life. This is a river 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 shapes, and this meeting of land and sea has _______ shapes with an undeniable pattern. It has a mathematically recursive structure. Every part of the river delta, with its twists and _____, 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 _____ 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 ______ look so remarkably alike? Why should they have anything in ______? 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 _________ so quickly that we ______ have the opportunity to ponder their geometry. But their shape is so unmistakable and so _______ 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 _____ centimeter of your body is ______ with blood vessels that trace out this same pattern. There's a mathematical reality woven into the ______ of the universe that you share with winding rivers, towering trees, and raging storms. These shapes are ________ of what we call "fractals," as mathematicians. Fractals get their name from the same place as fractions and fractures - it's a _________ 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 clouds in the sky. Like the other senses, our mathematical sense can be refined with ________. It's just like developing perfect pitch or a taste for wines. You can _____ to perceive the ___________ 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 ________ 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 _____ 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 _________ - 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, ________, 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 ____________ patterns are in Islamic art and ______. 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 ______ of beauty. Every culture around the ______ and throughout history has ________ them as objects of wonder. And one ______ of their beauty is that they exhibit a special kind of ________. Flowers grow organically from a center that expands outwards in the shape of a spiral, and this _______ what we call "rotational symmetry." You can spin a flower around and around, and it still looks _________ the same. But not all spirals are created _____. It all depends on the angle of rotation that goes into ________ the ______. For instance, if we build a spiral from an angle of 90 degrees, we get a cross that is neither _________ nor efficient. Huge parts of the flowers area are wasted and don't produce _____. Using an angle of 62 degrees is better and produces a nice circular _____, like what we usually associate with flowers. But it's still not great. There's still _____ _____ 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 _______. It's astonishing, and it is exactly the kind of pattern used by that most majestic of flowers - the _________. Now, 137.5 degrees might seem pretty ______, 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 _______ to the pillars of the Parthenon. That's why even at a party of 5000 ______, I'm proud to _______, "I love mathematics!" (Cheers) (________)

Solution

  1. logical
  2. mathematics
  3. spring
  4. students
  5. mathematical
  6. equal
  7. eddie
  8. combining
  9. applause
  10. process
  11. packed
  12. creating
  13. sense
  14. practice
  15. disappear
  16. answers
  17. history
  18. design
  19. parts
  20. wanted
  21. symbols
  22. laughter
  23. numbers
  24. wrestled
  25. english
  26. music
  27. seeds
  28. seldom
  29. spiral
  30. creates
  31. patterned
  32. royalty
  33. endlessly
  34. common
  35. turns
  36. outsiders
  37. musical
  38. ended
  39. subject
  40. reference
  41. convinced
  42. shapes
  43. basically
  44. forced
  45. short
  46. regarded
  47. sight
  48. angst
  49. painters
  50. broken
  51. guarantee
  52. shape
  53. planet
  54. firsthand
  55. education
  56. similar
  57. human
  58. cubic
  59. fingers
  60. learn
  61. aspect
  62. fabric
  63. pattern
  64. migrant
  65. maths
  66. random
  67. river
  68. relationships
  69. beautiful
  70. order
  71. geometry
  72. symbol
  73. selective
  74. school
  75. engaged
  76. triangles
  77. examples
  78. hooked
  79. torture
  80. people
  81. symmetry
  82. large
  83. church
  84. career
  85. sunflower
  86. child
  87. dominated
  88. declare
  89. created
  90. making

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

  1. abruptly
  2. abstract
  3. academic
  4. acoustic
  5. acute
  6. aesthetic
  7. alike
  8. altered
  9. amazing
  10. angle
  11. angst
  12. animals
  13. answers
  14. anymore
  15. appeared
  16. applause
  17. area
  18. arrangements
  19. art
  20. artists
  21. asian
  22. aspect
  23. associate
  24. astonishing
  25. attempted
  26. australian
  27. average
  28. aversion
  29. band
  30. basically
  31. beautiful
  32. beauty
  33. began
  34. beings
  35. blood
  36. blur
  37. body
  38. bolts
  39. born
  40. brings
  41. broccoli
  42. broken
  43. brother
  44. brought
  45. build
  46. calculations
  47. call
  48. career
  49. caused
  50. center
  51. centimeter
  52. chance
  53. change
  54. changed
  55. chaos
  56. characters
  57. cheers
  58. child
  59. chords
  60. church
  61. cinematographers
  62. circles
  63. circular
  64. class
  65. classes
  66. close
  67. clouds
  68. combining
  69. common
  70. compare
  71. competition
  72. connections
  73. conversation
  74. convinced
  75. cool
  76. corner
  77. correlated
  78. corridor
  79. cosmos
  80. counting
  81. couple
  82. created
  83. creates
  84. creating
  85. creative
  86. creators
  87. critical
  88. cross
  89. crunching
  90. cubic
  91. culture
  92. dark
  93. dawned
  94. day
  95. declare
  96. deep
  97. degree
  98. degrees
  99. delight
  100. delta
  101. depends
  102. depicting
  103. design
  104. developing
  105. difference
  106. disappear
  107. discovered
  108. disorderly
  109. distinctions
  110. dominated
  111. dream
  112. drew
  113. drink
  114. dry
  115. dutifully
  116. earlier
  117. earth
  118. eddie
  119. education
  120. educators
  121. efficient
  122. emerge
  123. emerges
  124. encounter
  125. encouraged
  126. ended
  127. endlessly
  128. engaged
  129. english
  130. enroll
  131. entered
  132. entire
  133. equal
  134. examples
  135. exercise
  136. exhibit
  137. expands
  138. experience
  139. eyes
  140. eyesight
  141. fabric
  142. fact
  143. faculty
  144. favorite
  145. feel
  146. felt
  147. fern
  148. filled
  149. find
  150. finding
  151. fingers
  152. firsthand
  153. firstly
  154. flower
  155. flowers
  156. focus
  157. forced
  158. forming
  159. forms
  160. formulas
  161. fractals
  162. fractions
  163. fractures
  164. genetic
  165. geometric
  166. geometry
  167. girl
  168. glasses
  169. golden
  170. good
  171. gradually
  172. great
  173. greater
  174. grow
  175. guarantee
  176. guess
  177. guidance
  178. guitar
  179. head
  180. hear
  181. hidden
  182. high
  183. history
  184. home
  185. honored
  186. hooked
  187. hours
  188. huge
  189. human
  190. humanities
  191. humans
  192. humor
  193. identical
  194. imagination
  195. immensely
  196. impact
  197. impairment
  198. impressing
  199. includes
  200. inscrutable
  201. insight
  202. instance
  203. intangible
  204. intricate
  205. islamic
  206. joy
  207. joyless
  208. keen
  209. kind
  210. knew
  211. land
  212. large
  213. lasted
  214. lasting
  215. laughter
  216. learn
  217. learned
  218. learning
  219. leaves
  220. led
  221. life
  222. lightning
  223. line
  224. list
  225. live
  226. lives
  227. logical
  228. long
  229. lot
  230. lottery
  231. love
  232. majestic
  233. majority
  234. making
  235. mathematical
  236. mathematically
  237. mathematicians
  238. mathematics
  239. maths
  240. matter
  241. means
  242. meet
  243. meeting
  244. memorizing
  245. microversion
  246. migrant
  247. mind
  248. mindlessly
  249. mistake
  250. music
  251. musical
  252. musicians
  253. narratives
  254. natural
  255. naturally
  256. nature
  257. nice
  258. note
  259. noticed
  260. number
  261. numbers
  262. objects
  263. ocean
  264. open
  265. opportunity
  266. opposed
  267. order
  268. orderly
  269. organically
  270. originally
  271. outsider
  272. outsiders
  273. outwards
  274. packed
  275. painters
  276. parents
  277. part
  278. parthenon
  279. parts
  280. party
  281. pattern
  282. patterned
  283. patterns
  284. people
  285. perceive
  286. perceived
  287. perfect
  288. perplexing
  289. persevered
  290. person
  291. personal
  292. perspective
  293. phalanges
  294. piano
  295. picked
  296. piece
  297. pillars
  298. pinned
  299. pitch
  300. place
  301. plain
  302. planet
  303. plants
  304. play
  305. playing
  306. ponder
  307. poor
  308. positive
  309. practical
  310. practice
  311. prestigious
  312. pretty
  313. print
  314. prizes
  315. problem
  316. problems
  317. process
  318. produce
  319. produces
  320. professors
  321. proud
  322. pulsing
  323. questions
  324. quickly
  325. racist
  326. raging
  327. raised
  328. random
  329. ranked
  330. ratio
  331. realities
  332. reality
  333. realize
  334. realized
  335. received
  336. recursive
  337. reference
  338. refined
  339. regarded
  340. relationships
  341. remains
  342. remarkably
  343. repeating
  344. resources
  345. rest
  346. rhythm
  347. rich
  348. river
  349. rivers
  350. room
  351. rotation
  352. rote
  353. royalty
  354. sad
  355. sails
  356. scales
  357. school
  358. schools
  359. scores
  360. sculptors
  361. sea
  362. seeds
  363. seldom
  364. selective
  365. sense
  366. senses
  367. set
  368. shameful
  369. shape
  370. shapes
  371. share
  372. sharper
  373. shattered
  374. short
  375. shortage
  376. show
  377. side
  378. sight
  379. similar
  380. similarities
  381. single
  382. sipping
  383. sky
  384. slowly
  385. solitary
  386. solve
  387. sounds
  388. special
  389. spend
  390. spin
  391. spiral
  392. spirals
  393. spring
  394. start
  395. started
  396. status
  397. steel
  398. stories
  399. storms
  400. straw
  401. striking
  402. string
  403. structure
  404. struggle
  405. struggled
  406. students
  407. studied
  408. subject
  409. subjects
  410. sunflower
  411. surely
  412. surprisingly
  413. surrounded
  414. suspicious
  415. swim
  416. sydney
  417. symbol
  418. symbols
  419. symmetry
  420. systems
  421. talk
  422. taste
  423. teach
  424. teacher
  425. teaching
  426. teenager
  427. thought
  428. tile
  429. time
  430. today
  431. told
  432. torture
  433. towering
  434. trace
  435. tree
  436. trees
  437. triangles
  438. turns
  439. twists
  440. undeniable
  441. understand
  442. universal
  443. universe
  444. university
  445. unmistakable
  446. vast
  447. vessels
  448. wanted
  449. wasted
  450. water
  451. ways
  452. wear
  453. weekends
  454. winding
  455. wines
  456. winter
  457. wired
  458. word
  459. world
  460. woven
  461. wrestled
  462. wrong
  463. year
  464. years
  465. young