Doodling in math: Spirals, Fibonacci, and being a plant [1 of 3] Part 2: http://youtu.be/lOIP_Z_-0Hs Part 3: http://youtu.be/14-NdQwKz9w Re: Pineapple under the Sea: http://youtu.be/gBxeju8dMho
Doodling in math: Spirals, Fibonacci, and being a plant [1 of 3]
- Say you're me and 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 Green House #3.
- 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 1
- 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's 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 –
- an 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.
- Better 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 with it next to both together –
- that is, each side is length 2.
- The next square has length 3.
- The entire outside shape will always be a rectangle.
- Keep spiraling around, adding bigger and bigger squares.
- This one has side length [COUNTS TO] 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! There sure are spirals on this pine cone!"?
- (I don't know why there are pine cones in your greenhouse.
- Maybe the greenhouse is in a forest.)
- Anyway, there are spirals.
- And there's not just one, either.
- There are [COUNTS TO] 8 going this way.
- Or you could look at the spirals going the other way,
- and there's [COUNTS TO] 13.
- Look familiar?
- 8 and 13 are both numbers in the 'Fibonacci series.'
- That's the one where you start by adding 1 and 1 to get 2 –
- then 1 and 2 to get 3.
- 2 and 3 to get 5.
- 3 + 5 is 8.
- 5 + 8 is 13.
- And so on.
- Some people think that instead of starting by adding 1 + 1,
- you should start with 0 and 1.
- 0 + 1 = 1.
- 1 + 1 = 2.
- 2 + 1 = 3.
- And it continues on the same way as starting with 1 + 1.
- Or I guess you could start with 1 + 0
- and that would work, too.
- Or why not go back one more to -1, and so on?
- Anyway, if you're into the Fibonacci series,
- you probably have a bunch of it memorized.
- I mean, you've got to know 1, 1, 2, 3, 5.
- Finish off the single digits with 8.
- And, ooh, 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 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 –
- uh, during math class –
- you might notice that the number of spirals are 5 and 8 –
- or 3 and 5 –
- a or 3 and 5 again –
- 5 and 8.
- This one was 8 and 13.
- 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. 5 and 8.
- Now let's check out the bottom. 8 and 13.
- If you wanted to draw a mathematically realistic pine cone,
- you might start by drawing 5 spirals going one way,
- and 8 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 –
- 8 one way and 5 the other.
- Now I can fill in the little pine coney things.
- So there are 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.
- [COUNTS TO] 8.
- And [COUNTS TO] 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?
- [COUNTS TO] 21 –
- a Fibonacci number.
- Can we find a third spiral on this pine cone?
- Sure. Go down like this and –
- [COUNTS TO] 21.
- But that's only a couple of 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...
- 5 and 8.
- Let's see how far the conspiracy goes.
- What else has spirals in it?
- This artichoke has 5 and 8.
- So does this artichoke-looking flower thing.
- And this cactus fruit does, too.
- Here's an orange cauliflower with 5 and 8.
- And a green one with 5 and 8.
- I mean 5 and 8.
- Oh, it's actually 5 and 8.
- 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.
- [COUNTS TO] 34.
- And [COUNTS TO] 55.
- I promise this is a random flower –
- and I didn't pick it out specially to trick you
- into thinking there are Fibonacci numbers in things.
- But you should really count for yourself
- next time you see something spirally.
- There are 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 3 and 5.
- Fibonacci is even in the arrangement
- of the petals on this rose.
- And some flowers have shown Fibonacci numbers
- as high as 144.
- It's 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.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
Have something that's not a question about this content?
This discussion area is not meant for answering homework questions.
Share a tip
When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831...
Thank the author
This is great, I finally understand quadratic functions!
Have something that's not a tip or thanks about this content?
This discussion area is not meant for answering homework questions.
At 2:33, Sal said "single bonds" but meant "covalent bonds."
For general discussions about Khan Academy, visit our Reddit discussion page.
Here are posts to avoid making. If you do encounter them, flag them for attention from our Guardians.
- disrespectful or offensive
- an advertisement
- low quality
- not about the video topic
- soliciting votes or seeking badges
- a homework question
- a duplicate answer
- repeatedly making the same post
- a tip or thanks in Questions
- a question in Tips & Thanks
- an answer that should be its own question