Placeholder Image

字幕表 動画を再生する

  • Voiceover:Say [unintelligible], you're in math class

  • and your teacher's talking about ...

  • Well, who knows what your teacher's talking about.

  • Probably a good time to start doodling.

  • And you're feeling spirally today, so yeah.

  • Oh, and because of overcrowding in your school,

  • your math class is taking place in

  • greenhouse number three.

  • Plants.

  • Anyway.

  • You've decided there are three basic

  • types of spirals.

  • There's the kind where, as you spiral out,

  • you keep the same distance.

  • Or you could start big but make it tighter

  • and tighter as you go around, in which case

  • the spiral ends.

  • Or you could start tight but make the spiral

  • bigger as you go out.

  • The first kind is good if you really want

  • to fill up a page with lines.

  • Or if you want to draw curled up snakes.

  • You can start with a wonky shape to spiral around

  • but you've noticed that, as you spiral out,

  • it gets rounder and rounder.

  • Probably something to do with how the ratio

  • between two different numbers approaches one

  • as you repeatedly add the same number to both.

  • But you can bring the wonk back by

  • exaggerating the bumps and it gets all

  • optical illusiony.

  • Anyway, you're not sure what the second

  • kind of spiral is good for, but I guess it's

  • a good way to draw snuggled up slug cats,

  • which are a species you've invented just

  • to keep this kind of spiral from feeling useless.

  • This third spiral, however, is good for

  • all sorts of things.

  • You could draw a snail or a nautilus shell.

  • And elephant with a curled up trunk,

  • the horns of a sheep, a fern frond, a cochlea

  • in an inner ear diagram, an ear itself.

  • Those other spirals can't help but be jealous

  • of this clearly superior kind of spiral.

  • But I draw more slug cats.

  • Here's one way to draw a really perfect spiral.

  • Start with one square and draw another

  • next to it that is the same height.

  • Make the next square fit next to both together,

  • that is each side is length two.

  • The next square has length three.

  • The entire outside shape will always

  • be a rectangle.

  • Keep spiraling around, adding bigger

  • and bigger squares.

  • This one has side length one, two, three,

  • four, five, six, seven, eight, nine,

  • 10, 11, 12, 13.

  • And now 21.

  • Once you do that you can add a curve going

  • through each square, arcing from one corner

  • to the opposite corner.

  • Resist the urge to zip quickly across

  • the diagonal, if you want a nice smooth spiral.

  • Have you ever looked at the spirally pattern

  • on a pine cone and thought, "Hey, sure are

  • "spirals on this pine cone?"

  • I don't know why there's pine cones

  • in your greenhouse.

  • Maybe the greenhouse is in a forest.

  • Anyway, there's spirals and there's not

  • just one either.

  • There's one, two, three, four, five, six,

  • seven, eight going this way.

  • Or you could look at the spirals going

  • the other way and there's one, two, three,

  • four, five, six, seven, eight, nine, 10, 11, 12, 13.

  • Look familiar?

  • Eight and 13 are both numbers in the

  • Fibonacci series.

  • That's the one where you start by adding

  • one and one to get two, then one and two

  • to get three, two and three to get five.

  • Three plus five is eight, five plus eight

  • is 13, and so on.

  • Some people think that instead of starting

  • with one plus one you should start with

  • zero and one.

  • Zero plus one is one, one plus one is two,

  • one plus two and three, and it continues

  • on the same way as starting with one and one.

  • Or, I guess you could start with one plus zero

  • and that would work too.

  • Or why not go back one more to negative one

  • and so on?

  • Anyway, if you're into the Fibonacci series,

  • you probably have a bunch memorized.

  • I mean, you've got to know one, one, two,

  • three, five.

  • Finish off the single digits with eight

  • and, ooh with 13, how spooky.

  • And once you're memorizing double digits,

  • you might as well know 21, 34, 55, 89 so that

  • whenever someone turns a Fibonacci number

  • you can say, "Happy Fib Birthday."

  • And then, isn't it interesting that 144, 233, 377?

  • But 610 breaks that pattern, so you'd better

  • know that one too.

  • And oh my goodness, 987 is a neat number

  • and, well, you see how these things get

  • out of hand.

  • Anyway, 'tis the season for decorative

  • scented pine cones and if you're putting

  • glitter glue spirals on your pine cones

  • during math class, you might notice that

  • the number of spirals are five and eight

  • or three and five or three and five again.

  • Five and eight.

  • This one was eight and thirteen and one

  • Fibonacci pine cone is one thing, but all of them?

  • What is up with that?

  • This pine cone has this wumpy weird part.

  • Maybe that messes it up.

  • Let's count the top.

  • Five and eight.

  • Now let's check out the bottom.

  • Eight and 13.

  • If you wanted to draw a mathematically

  • realistic pine cone, you might start

  • by drawing five spirals one way and eight

  • going the other.

  • I'm going to mark out starting and ending

  • points for my spirals first as a guide

  • and then draw the arms.

  • Eight one way and five the other.

  • Now I can fill in the little pine coney things.

  • So there's Fibonacci numbers in pine cones

  • but are there Fibonacci numbers in other things

  • that start with pine?

  • Let's count the spirals on this thing.

  • One, two, three, four, five, six, seven, eight.

  • And one, two, three, four, five, six, seven,

  • eight, nine, 10, 11, 12, 13.

  • The leaves are hard to keep track of,

  • but they're in spirals too.

  • Of Fibonacci numbers.

  • What if we looked at these really tight spirals

  • going almost straight up?

  • One, two, three, four, five, six, seven,

  • eight, nine, 10, 11, 12, 13, 14, 15, 16, 17,

  • 18, 19, 20, 21.

  • A Fibonacci number.

  • Can we find a third spiral on this pine cone?

  • Sure, go down like this.

  • And one, two, three, four, five, six, seven,

  • eight, nine, 10, 11, 12, 13 (muttering) 19, 20, 21.

  • But that's only a couple examples.

  • How about this thing I found

  • on the side of the road?

  • I don't know what it is.

  • It probably starts with pine, though.

  • Five and eight.

  • Let's see how far the conspiracy goes.

  • What else has spirals in it?

  • This artichoke has five and eight.

  • So does this artichoke looking flower thing.

  • And this cactus fruit does too.

  • Here's an orange cauliflower with five and eight

  • and a green one with five and eight.

  • I mean, five and eight.

  • Oh, it's actually five and eight.

  • Maybe plants just like these numbers though.

  • Doesn't mean it has anything to do

  • with Fibonacci, does it?

  • So let's go for some higher numbers.

  • We're going to need some flowers.

  • I think this is a flower.

  • It's got 13 and 21.

  • These daisies are hard to count, but they have

  • 21 and 34.

  • Now let's bring in the big guns.

  • One, two, three, four, five, six, seven,

  • eight, nine, 10, 11, 12, 13, 14, 15, 16, 17,

  • 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,

  • 31, 32, 33, 34.

  • And one, two, three, four, five, six, seven,

  • eight, nine, 10, 11, (muttering) 17, 24,

  • (muttering) 42, 53, 54, 55.

  • I promise, this is a random flower and I didn't

  • pick it out specially to trick you into thinking

  • there's Fibonacci numbers in things, but you should

  • really count for yourself next time you see

  • something spirally.

  • There's even Fibonacci numbers in how

  • the leaves are arranged on this stalk,

  • or this one, or the Brussels sprouts on this stalk

  • are a beautiful delicious three and five.

  • Fibonacci is even in the arrangement of

  • the petals on this rose, and sunflowers have

  • shown Fibonacci numbers as high as 144.

  • It seems pretty cosmic and wondrous, but the cool

  • thing about the Fibonacci series and spiral

  • is not that it's this big complicated

  • mystical magical super math thing beyond

  • the comprehension of our puny human minds

  • that shows up mysteriously everywhere.

  • We'll find that these numbers aren't weird at all.

  • In fact, it would be weird if they weren't there.

  • The cool thing about it is that these

  • incredibly intricate patterns can result

  • from utterly simple beginnings.

Voiceover:Say [unintelligible], you're in math class

字幕と単語

動画の操作 ここで「動画」の調整と「字幕」の表示を設定することができます

B1 中級

数学の落書き。螺旋、フィボナッチ、植物であること [1/3] (Doodling in Math: Spirals, Fibonacci, and Being a Plant [1 of 3])

  • 0 0
    林宜悉 に公開 2021 年 01 月 14 日
動画の中の単語