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(mellow music)
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Why does mathematics work?
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Why is it so useful?
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Is mathematics invented or discovered?
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I happen to think we discover it rather than invent it
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but it depends on one's philosophical presuppositions
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what conclusions one will draw.
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One of the most celebrated discoverers
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of the role of mathematics in nature
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was a young Italian merchant's son called Fibonacci.
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Fibonacci was a 12th-century mathematician living in Pisa,
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and he came up with a sequence which is
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prevalent in nature today in one form or another,
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known as the Fibonacci sequence,
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one, one, two, three, five, eight, and so on.
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Each number in the sequence
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is the sum of the two previous numbers.
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So, the next number after one will be one,
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because there was nothing before it.
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The number after that will be two, one plus one.
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After that will be three, two plus one,
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then five, then eight,
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then 13, 21, 34, 55, 89, 144, and so on.
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The whole sequence appears again, and again, and again
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in one form or another, with surprising regularity.
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And, it is amazing, nobody really knows why
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this particular sequence is so important in nature,
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but there it is.
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That's what we see.
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If you look at the distribution
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of seeds on a sunflower head,
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you see these wonderful spirals,
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which are, in a sense optical illusions,
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because they're not representative
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of the order in which the seeds were developed.
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Nevertheless, the number going clockwise,
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and the number going anticlockwise,
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are 95% of the time adjacent terms
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in the Fibonacci sequence, and it's just amazing.
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But there's more to the Fibonacci sequence
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than the arrangement of seeds on a sunflower head.
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The ratio of successive numbers in the sequence
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gets closer and closer to a rather special value,
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known as the golden number.
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This golden number, one plus the square root of five,
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all divided by two, is a natural consequence
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of the geometry of a regular pentagon.
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If you draw a regular Pentagon and you join up the corners,
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these cords, these lines joining the corners,
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intersect one another in a ratio
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that can be shown to be this golden number.
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The ratio of the larger part of that cord
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to the smaller part is the same
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as the whole length of the cord
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divided by the larger part,
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approximately 1.618, the golden number.
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The golden ratio is a linear measure
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in the sense that it's a ratio of two lengths.
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If one translates this to the geometry of a circle,
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one can get something called the golden angle.
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It turns out to be approximately 137.5°
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and that, again, figures quite commonly in nature
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in the arrangement of leaves.
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The most effective arrangement of leaves on a stem
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is when the new ones sprout
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at precisely the golden angle from the one below.
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One consequence appears to be minimal blocking of sunlight
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from leaves below when the sun is high.
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If you look at a cactus from above,
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some types of cactus at least,
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I have a very good photograph of one
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with a picture of my shoe there as well
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which you can determine that this golden angle
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is approximately represented as the leaves grew
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from one to another to another and so,
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the successive leaves fill the space with that angle
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from the previous leaf, and it's quite magnificent.
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I've measured some of these,
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and while I can't measure them accurately,
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it's a pretty darn good approximation, to 137.5°.
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The more you look at natural phenomena
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the more you see the evidence of golden ratios,
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golden angles, and the Fibonacci sequence at work,
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and often, that will result in the emergence of spirals.
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They're everywhere if you care to look.
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Seashells, the nautilus shell, the shell of a snail.
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You can actually relate this spiral
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approximately to the golden ratio.
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If we go back to our golden rectangle
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which has, long side approximately 1.618,
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short side one, and you cut off a square,
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what you have left in that rectangle
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is another rectangle, a smaller one
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which is also a golden rectangle.
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The relationship between the areas of the squares
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is extremely interesting.
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They are in the ratio of successive Fibonacci numbers.
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We might start with one, and then one, and then two,
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and then area three, and then area five,
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and then area eight, and area 13, and so on.
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And if you keep doing that and then join corner to corner
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with an arc of a circle
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you get an approximate equiangular spiral,
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and this is certainly very reminiscent of the spirals,
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particularly the nautilus shell, that we see in nature.
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So, once again, these numbers are almost ubiquitous,
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and one of these days I'm going to ask God why.
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One of the most enduring puzzles of mathematics in nature
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relates to the patterns and markings we see on animals.
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A leopard's spots, a zebra's stripes and so on.
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It fell to one of the greatest mathematicians
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of the 20th century, Alan Turing,
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to shed light on the topic.
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Now, he was a genius.
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He published one paper and one only on the chemical basis
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of morphogenesis, that was his only foray into biology,
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but it was profound, it was seminal.
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Morphogenesis relates in biological terms
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to the various chemical changes that may take place
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in an embryo that will ultimately lead
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to patterns in the adult creature.
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The equations are quite complicated
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but mathematicians have found that by varying the parameters
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in these equations you can get spots,
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you can get stripes, you can get uniform colors.
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These studies have helped answer some age-old questions
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including whether zebras are white horses with black stripes
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of black horses with white stripes.
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People used to think that they were
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black-striped white horses,
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but the prevailing view is now the opposite.
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Zebra embryos are completely black.
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The white stripes appear during the last embryonic stage.
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So, zebras it seems, are black horses with white stripes.
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But not all of the colors in nature are created
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by the interaction of chemical pigments in an animal's skin.
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Some colors are created by microscopic structures
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that split white sunlight into its component colors.
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It's a phenomenon known as known as iridescence.
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Many birds and insects display these beautiful colors.
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Iridescence, I love that word because it is derived
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from a Greek word for rainbow, iris and iridos,
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and over the years I've studied rainbows in great detail.
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Rainbows are all about some basic,
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and yet very subtle geometry, and they will only occur
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when certain very specific conditions are met.
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The sun has to be shining,
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there has to be rain somewhere,
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and if the conditions are right,
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if the sun is not too high in the sky,
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then if you stand with your back to the sun,
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the sunlight is scattered by the raindrops ahead of you,
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and it's scattered in all directions,
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but there's a concentration of the light
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that is refracted inside the raindrop,
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reflected from the backside of the drop,
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and refracted out again.
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The colors you see are from different raindrops.
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There's myriads of raindrops, so it's cumulative effect.
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A good portion of them will scatter right into your eye,
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a good proportion scatter green, orange, whatever.
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In a sense, a rainbow is a highly exotic image of the sun.
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I just love rainbows.
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I think most people probably do.
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If I can go on to another phenomenon
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which is related, the glory.
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If you've ever flown on a plane
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and been on the shadow side of the plane above cloud
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you may well have noticed the shadow of the plane
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surrounded by circular colored rings,
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and that's back scattering of light by cloud droplets.
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The smaller the cloud droplets are
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the larger the radius of the glory.
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This is an amazing phenomenon,
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and the more you look into these
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atmospheric optical phenomena
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the more fascinating they become.
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Interactions between sunlight and water droplets,
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sunlight and ice crystals,
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and always the all-important geometrical configurations
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that link you, the observer, the sun,
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and whatever water droplets or ice crystals
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are creating the effect.
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So, you get sun halos, moon halos,
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fog bows, son dogs,
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circumzenithal arcs,
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circumorizon arcs, and so on.
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It's a mathematical feast.
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As I often say mathematics in nature
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is the greatest show on Earth.
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And, what thrills me is when a student,
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and this happens quite a lot,
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will either come to me at the end of the class
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or sometimes after the course is over,
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and will show me a picture they've taken,
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or sketch out something they've seen,
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and they're excited by it and they
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don't perhaps understand what was going on.
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"What was happening here?", they say, and I say
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"Well I wasn't there, I don't know,
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"but here's a possibility, here's what I suspect."
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They're actually thinking, they're taking this stuff outside
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into the greatest free show on Earth,
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and they're thinking about it and they're thinking
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about the underlying principles
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and whether or not they're correct it doesn't matter
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to that degree, it's the fact that they're thinking
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and they're curiosity has been aroused,
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and so I feel like I've made a difference
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however small, in that students life.
16909
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