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 sepnd 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, solbmys, and calculations - the vast majority of us are outsiders, and that includes me. That's why today I want to shrae 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) fistlry, that's racist. (Laughter) Secondly, that's wrnog. 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 casls 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 arsuialtan Mathematics Competition and would print out a list of everyone in the socohl 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 apeaeprd. Maths was not really my thing. Stories, cacehartrs, 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 sydeny University aerteld my life forever. I was in line to enroll at the faculty of education when I sreattd the conversation with one of its professors. He noticed that while my academic life had been dominated by heaiiutnms, 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 mthas. And he knew something I didn't - that there was a critical shortage of mathematics ecdaoutrs in Australian schools, a shortage that rieanms to this day. So he encouraged me to change my teaching area to mathematics. Now, for me, becoming a teehacr wasn't about my love for a particular subject. It was about having a personal impact on the lveis of yonug people. I'd seen ftarsinhd at school what a lasting and positive difference a great teacher can make. I wanted to do that for someone, and it didn't meattr 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 drcseivoed 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 craeer was abruptly ended when my teacher told my parents, "His fingers are too short. I will not taech him anymore." (Laughter) At seven yaers 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 pcekid 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 cvnenoicd my brother to teach me a few chords. And slowly, but surely, my mind changed. I was engaged in a cviratee process. I was making music, and I was hoekod. I started playing in a band, and I felt the delight of rhythm pulsing through my body as we borught 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 flmoraus 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 btfuaeiul, that it's not just about finding aersnws but also about leinrang to ask the right questions, and that mathematics isn't about mindlessly crunching numbers but rather about foimnrg new ways to see problems so we can svloe them by combining isinhgt with imagination. It gradually dawned on me that matchaietms is a ssene. Mathematics is a sense just like sight and touch; it's a sense that allows us to pevciree 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 prnttaes, 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 taelrgins and crciles. But germoety is the mathematics of all shapes, and this meeting of land and sea has created shapes with an uniledbane pattern. It has a mathematically recursive structure. Every part of the river delta, with its twists and turns, is a microversion of the gateerr whole. So I want you to see the mathematics in this. But that's not all. I want you to crmoape this river delta with this amazing tree. It's a wonder in itself. But focus with me on the similarities between this and the rievr. What I want to know is why on eatrh should these shapes look so remarkably alike? Why should they have anything in common? Things get even more plpirnexeg 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 qculiky 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 sginle psroen 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 wveon 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. frcatlas 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 srtat 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 weins. 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 eyhisget. Without my glessas, everything is a blur. I've wrsteled 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 nutaral 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 uvrensie, a coosms. That's what cosmos means - oerdrly and patterned - as opposed to chaos, which menas disorderly and random. It isn't just seeing patterns that humans are so good at. We love mnakig patterns too. And the people who do this well have a special name. We call them artists, musicians, spourclts, painters, cinematographers - they're all pattern creators. misuc 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 maethaimctal patterns are in imlsiac art and design. An aversion to depicting humans and aiamnls led to a rich history of intricate tile arrangements and geometric forms. The atsehetic 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 histroy 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 cneetr 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 srplais are created euqal. It all depends on the angle of rtioaton that goes into creating the spiral. For instance, if we build a sairpl from an angle of 90 degrees, we get a cross that is neither beautiful nor efcifneit. Huge parts of the flowers area are wasted and don't produce sdees. Using an angle of 62 degrees is better and produces a nice circular saphe, 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 resuecros 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 fworels - the sflnewuor. Now, 137.5 degrees might seem pertty random, but it actually eegrmes out of a special number that we call the "golden ritao." 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 paroetnhn. That's why even at a patry 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 _____ 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, _______, and calculations - the vast majority of us are outsiders, and that includes me. That's why today I want to _____ 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) _______, that's racist. (Laughter) Secondly, that's _____. 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 _____ 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 __________ Mathematics Competition and would print out a list of everyone in the ______ 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 ________. Maths was not really my thing. Stories, __________, 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 ______ University _______ my life forever. I was in line to enroll at the faculty of education when I _______ the conversation 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 problem with maths, but that I had persevered with _____. And he knew something I didn't - that there was a critical shortage of mathematics _________ in Australian schools, a shortage that _______ to this day. So he encouraged me to change my teaching area to mathematics. Now, for me, becoming a _______ wasn't about my love for a particular subject. It was about having a personal impact on the _____ of _____ 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 ______ 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 __________ 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 ______ was abruptly ended when my teacher told my parents, "His fingers are too short. I will not _____ him anymore." (Laughter) At seven _____ 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 ______ 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 _________ my brother to teach me a few chords. 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 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 ________ 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 _________, that it's not just about finding _______ but also about ________ to ask the right questions, and that mathematics isn't about mindlessly crunching numbers but rather about _______ new ways to see problems so we can _____ them by combining _______ with imagination. It gradually dawned on me that ___________ is a _____. Mathematics is a sense just like sight and touch; it's a sense that allows us to ________ 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 ________, 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 _________ and _______. But ________ is the mathematics of all shapes, and this meeting of land and sea has created shapes with an __________ pattern. It has a mathematically recursive structure. Every part of the river delta, with its twists and turns, is a microversion of the _______ whole. So I want you to see the mathematics in this. But that's not all. I want you to _______ this river delta with this amazing tree. It's a wonder in itself. But focus with me on the similarities between this and the _____. 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 __________ 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 _______ 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 ______ ______ 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 _____ 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. ________ 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 _____ 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 _____. 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 ________. Without my _______, 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 _______ 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 ________, a ______. That's what cosmos means - _______ and patterned - as opposed to chaos, which _____ disorderly and random. It isn't just seeing patterns that humans are so good at. We love ______ patterns too. And the people who do this well have a special name. We call them artists, musicians, _________, painters, cinematographers - they're all pattern creators. _____ 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 _______ art and design. An aversion to depicting humans and _______ led to a rich history of intricate tile arrangements and geometric forms. The _________ 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 _______ 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 ______ 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 ________ that goes into creating the spiral. For instance, if we build a ______ from an angle of 90 degrees, we get a cross that is neither beautiful nor _________. 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 large parts of the area that are a poor use of _________ 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 _______ - the _________. Now, 137.5 degrees might seem ______ random, but it actually _______ out of a special number that we call the "golden _____." 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 _________. That's why even at a _____ of 5000 people, I'm proud to declare, "I love mathematics!" (Cheers) (Applause)

Solution

picked
eyesight
forming
years
young
remains
spirals
spiral
cosmos
wrestled
sculptors
woven
australian
answers
rotation
appeared
undeniable
glasses
quickly
learning
beautiful
teach
flowers
person
sunflower
triangles
humanities
sydney
making
universe
brought
ratio
geometry
symbols
fractals
educators
maths
single
animals
discovered
mathematics
insight
circles
class
greater
started
seeds
creative
solve
share
perplexing
sense
party
career
perceive
wrong
matter
compare
patterns
hooked
aesthetic
river
wines
parthenon
start
convinced
resources
equal
music
islamic
altered
pretty
lives
center
firstly
mathematical
natural
orderly
formulas
emerges
firsthand
teacher
school
characters
earth
efficient
means
spend
shape
history

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
academic
acoustic
acute
aesthetic
alike
altered
amazing
angle
angst
animals
answers
anymore
appeared
applause
area
arrangements
art
artists
asian
aspect
associate
astonishing
attempted
australian
average
aversion
band
basically
beautiful
beauty
began
beings
blood
blur
body
bolts
born
brings
broccoli
broken
brother
brought
build
calculations
call
career
caused
center
centimeter
chance
change
changed
chaos
characters
cheers
child
chords
church
cinematographers
circles
circular
class
classes
close
clouds
combining
common
compare
competition
connections
conversation
convinced
cool
corner
correlated
corridor
cosmos
counting
couple
created
creates
creating
creative
creators
critical
cross
crunching
cubic
culture
dark
dawned
day
declare
deep
degree
degrees
delight
delta
depends
depicting
design
developing
difference
disappear
discovered
disorderly
distinctions
dominated
dream
drew
drink
dry
dutifully
earlier
earth
eddie
education
educators
efficient
emerge
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encounter
encouraged
ended
endlessly
engaged
english
enroll
entered
entire
equal
examples
exercise
exhibit
expands
experience
eyes
eyesight
fabric
fact
faculty
favorite
feel
felt
fern
filled
find
finding
fingers
firsthand
firstly
flower
flowers
focus
forced
forming
forms
formulas
fractals
fractions
fractures
genetic
geometric
geometry
girl
glasses
golden
good
gradually
great
greater
grow
guarantee
guess
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
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persevered
person
personal
perspective
phalanges
piano
picked
piece
pillars
pinned
pitch
place
plain
planet
plants
play
playing
ponder
poor
positive
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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
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selective
sense
senses
set
shameful
shape
shapes
share
sharper
shattered
short
shortage
show
side
sight
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similarities
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sky
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sounds
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spend
spin
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spirals
spring
start
started
status
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stories
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straw
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string
structure
struggle
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students
studied
subject
subjects
sunflower
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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
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water
ways
wear
weekends
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year
years
young