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(ambient music)
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Mathematics has been described as the science of patterns.
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In mathematics, there are patterns in numbers, in shapes,
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in probability, in motion.
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And nature has visible regularities that I see
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more than I ever use to, because now I'm looking for them.
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They're everywhere, really, and if we have eyes to see them,
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we can really enjoy them as well as try and understand them.
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This is the greatest free show on Earth.
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And I think that if I could just get my students
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to put down their cell phones and concentrate
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on this world around them, they can see wonderful beauty.
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And I hope it will excite them as it does me.
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And what always excites me is that the stunning variety
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of shapes and patterns to be found in the natural world
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are all the result of Mother Nature following
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biological, chemical, physical principles
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and the underlying mathematical structures.
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And in the world of nature, one of the most basic rules
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is always be efficient.
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Nature does appear to seek the most economical,
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the most efficient ways of achieving Her end.
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She seeks to minimize energy.
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It is certainly the case that a lot of mathematical models
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of natural processes involved minimization schemes.
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And one of those natural minimization schemes
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is don't expend energy if you don't have to
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just like some of my students.
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Some of the most primitive creatures,
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such as sponges and corals simply let their food
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waft over them.
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In technical terms, they are asymmetrical.
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For most creatures, some sort of symmetry
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is both important and revealing.
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Look at the sea anemone, it's another creature
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that doesn't move or at least not very much.
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And it displays what is known as radial symmetry.
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You can cut through a sea anemone at basically any angle,
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and the cross-section will always be the same.
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A nice variation on radial symmetry is rotational symmetry.
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The starfish, for example, if you carefully arrange
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the stars on the starfish, then you will have
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pentagonal symmetry,
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which means that there are five axes of symmetry,
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each at an angle of 72 degrees from the next one.
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We see it in flowers, Vincas, I believe,
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are pentagonally symmetric, you've only got to rotate
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through a certain angle to leave the flower
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indistinguishable approximately from where it was before.
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Closer to home, if you carefully cut through an apple,
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you'll find the pits are arranged
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in a pentagonally symmetrical pattern.
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But the commonest type of symmetry by far
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is what's known as bilateral symmetry.
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Nearly all creatures on the planet exhibit
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bilateral symmetry.
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And again, it has to do with mobility.
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Any designer would put an organism's senses and its mouth
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at the head of the organism's motion.
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In bilaterally symmetrical animals
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that immediately presumes an up and a down,
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and also left and right.
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There's so much going on in the natural world,
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so many different causes and effects,
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that one might look for a unifying principle
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to describe that.
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And I would say elegance and efficiency is probably the one.
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Take a look at a beehive for example,
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we're all familiar with the hexagons in a honeycomb,
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but why hexagons?
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Why not circles, which have a minimum perimeter
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for a given area.
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Surely, that's a superb example of efficiency.
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Unfortunately, the problem with circles
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is that you can't pack them together without leaving gaps.
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So the nearest regular polygon that comes to a circle
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is the regular hexagon.
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And so those six sides minimize the perimeter
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for a given area.
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And that may have something to do with the way
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beehives are constructed, because the bees,
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whether they know it or not are superb engineers.
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And it turns out given those space-filling constraints,
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the hexagons use the least wax for the most storage space.
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And the icing on the cake, if you're a bee,
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is that for every six hexagons you make,
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you get another one for free.
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Hexagons are a recurrent theme in nature,
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turning up in geological formations, insect eyes,
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and perhaps most famously, in snowflakes.
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When you hear the statement, and it's often heard,
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that every snowflake is individual, is unique,
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it's true but it depends on the scale at which
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you're looking at it.
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Because if I just look at them on my sleeve
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as they fall on my dark coat, yes some are big,
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and some are smaller, but I don't really get a sense
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of the uniqueness at the level we're talking about.
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And if you go down very, very closely with a microscope,
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if you could imagine the molecules,
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then they're all the same, no one ice molecule
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is different from any other.
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That six-fold symmetry is, in an amazing way,
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perpetuated through the different scales.
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And so we see the six-fold symmetry, the hexagonal symmetry
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in snowflakes because of that.
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And yet, they are different.
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There's little irregularities, perhaps,
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so they're not exactly hexagonally symmetric.
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So in that sense, every one is different.
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But the life history depends on, of course,
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like human beings, where we've been, what we've experienced,
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what we've encountered.
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So each snowflake falls a different path to the Earth,
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and therefore it falls through different regions,
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regions of different humidity, different temperature,
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buffeted by the wind, perhaps, melting a little bit
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and then refreezing.
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So in that sense, yes, they are utterly unique.
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They are an excellent example of one of nature's
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favorite mathematical features, deterministic chaos.
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One of the features of deterministic chaos
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is that there is an in-built sensitivity
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to initial conditions.
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And that applies to the snowflake, in the sense
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that each one has its own unique path
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through its brief lifetime.
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This is one of the features of chaos that,
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a small variation in the initial conditions,
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and this is related to what's known as the butterfly effect
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can induce downstream, so to speak,
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in time, very different circumstances.
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The butterfly effect seems really, really silly.
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A butterfly flapping its wings somewhere in South America
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could give rise to some enormous storm over Japan.
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Now, this sounds almost sort of zen,
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but it's certainly part of what is known
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as deterministic chaos.
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So that's why the weather is actually unpredictable
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beyond a few days because some slight change
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can be replicated en masse downstream, so to speak,
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a few days later.
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And so the structure of the weather
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is inherently unpredictable.
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We can a broad idea over a few days,
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and even perhaps into 10 days,
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but nevertheless, there's an ultimate limitation there.
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Often, the manifestation of deterministic chaos in nature
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produces complex shapes and forms that have
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a fractal-like structure.
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A fractal is essentially, crudely speaking,
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a picture of chaos.
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There are several ways of looking at fractals,
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but essentially, they are things,
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they are geometric entities which have the same geometric
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or statistical properties no matter how small,
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how far down you go, how much you magnify them.
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Let me give you a classic example of the
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Cook or Koch Snowflake Curve
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which is a beloved of many students,
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especially the ones I teach, because it's so fascinating.
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You take an equilateral triangle,
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every side is the same length.
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You then remove the middle third of each side
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and put another little equilateral triangle
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or at least two sides of it on there,
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so you've got a sort of star.
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And you continue that process, this is called iteration.
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You continue it ad infinitum, and what you end up with
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is a crinkly wonderful shape pattern
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which actually has infinite length and finite area,
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because it can be enclosed in a circle,
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its therefore got finite area, and this is the archetype
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or one of them of a fractal.
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Another fractal of note is the Sierpinski Triangle
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or the Sierpinski Gasket.
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If you take an equilateral triangle and mark the midpoint
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of each side and join those points together,
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you've got another triangle pointed downwards as it were,
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paint that black or remove it,
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and then do the same thing with the remaining three
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white triangles.
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If you continue that process, you get a strange shape
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that seems to appear on some shells in which
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has these triangles, these patterns
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to several different scales.
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And that appears to be the result of some chemical process
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that I don't understand that mimics, in some way,
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this Sierpinski Triangle, at least to several levels down.
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And it's quite fascinating, they're beautiful.
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It's wonderful, I try to get across, I don't know,
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the beauty of nature.
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When I translate all this into the classroom,
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it's very easy to get excited and passionate about it,
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and I think it's a wonderful tool
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for younger children as well to try and instill in them
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the curiosity, well they have the curiosity younger children
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it's not been dissipated from them.
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But just to focus them and ask questions.
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What's going on here, what's that?
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It's just wonderful.
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Science is all about asking questions,
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and in that sense, scientists and mathematicians
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are still kids.
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(gentle piano music)
16709
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