Hi I'm Toby and welcome back to another episode of The Joy of Mathematics. I'm so

glad you could join me again today. Are you ready to work on some mathematical

artwork? Because that's what I thought we would do today. No stress, no equations,

just relax and gain an appreciation of our

mathematical world. I'm going to run some ideas across the bottom of the screen

that we will come to understand today. I would also like to thank today's sponsor

brilliant.org. So I'm going to start off my artwork by splitting it in two

I'm going to have a line that separates the sky from the grass. We're going to

get a beautiful blue sky by filling it in like so, just some nice long strokes

there. All right, so in the sky don't forget to add some sunlight and we can

also add some little birds in the corner here. All right, so we're also going to

need to do the grass so grab a green and just like we did the sky, fill in a nice

green healthy grass here. If you can see the backdrop behind me today the grass

isn't so green, it's a little dry, it's also a little smoky from some of the

smoke haze from bush fires we've got going on :(

All right, so we've set up our scene now what I'm going to do is I'm going to

place a little mushroom right here in the middle of the grass. It's the kind of

mushroom that would attract fairies, we may be mathematicians today but

we're not too serious that we can't include a little bit of magic. So what I

want to do next is I want to draw a hill that the mushroom is going to be on but

just so that everything looks nice and aesthetic I don't want the hill to be

too close to the horizon and I don't want it to be too close to the mushroom

either, in fact right in the middle of these two would be perfect, and that's

what we're going to do. We're going to measure out the distance here find

halfway and that is going to be the tip of our hill. I'm just going to switch us

to our little chalk pen here so it's a bit easier to see on the background of

the chalk. All right, so there's the tip of our hill, now I thought about using a

ruler to do this, I even brought a ruler along with me but then I thought at the

last minute that might be a bit of overkill, we can just measure this using

our minds and our brains, so if we have a look here we want the hill to curve down,

we want it to be equally far away from the horizon and the mushroom, so what

would be equidistant? About there and about there, I think that would be the

next point of our hill and the same on the other side if we measure with our

eyes down from here the same distance from the horizon and the mushroom we

might get there. And continuing that little measurement with our eyes all the

way around we start to see what this hill is going to look like

not everything needs to be done perfectly with a ruler, I think maybe

that some of the fun of it is getting it maybe slightly wrong but we know the

idea we're going for. I'm going to go ahead and add some happy little trees to

our beautiful hill in here, not too many we don't want to go overboard but these

are some happy little pine cones. That one didn't quite make it to the hill.

Pine cones are some of my favorite trees to draw. I've actually got a cone sitting up

here on my easel today to celebrate the festive season, I considered wearing it

on my own head but I didn't want to mess up my hair.

You see cones are interesting shapes because they contain other shapes inside

of them, let me show you what I mean. I've got some scissors here and I'm going to

make a few cuts. So I think you can tell that because of the shape of the cone if

I was to cut it straight across I would get a circle but if I was to cut it a

little bit wonky across here it's a little bit less obvious but I would get

an ellipse, a kind of stretched out circle. Let's see if we can get one...

It's a little bit jagged but you can see there I've cut on a wonky angle and I've

got an ellipse out and now what I'm going to do is cut this one more way and

that is going to be parallel to one of the sides. So this is the side that I

want to be parallel to, I'm going to make a cut across here like this and we'll

see what shape we get so here we go...

So I've tried my best there to cut this side parallel and this is the shape that

we've ended up with. It may shock you to know but this shape here is actually

exactly the same shape as our hill that we drew before. That's now two different

places that we've found this shape hiding, let's see if I trace it out.

This is a little precarious but that's the fun of it, so I'm just tracing around the

shape that we cut out from our cone to see what it looks like on our artwork.

All right, moment of truth, there we go yeah that looks about right, looks quite

similar to our hill. Now this shape has a name and it is called a parabola.

Now we've found parabolas already in a couple of places but they actually occur

everywhere in nature as well as in many man-made things here's some footage of a

couple of other parabolas that I found around town. There's a parabola in the

curve of some dishes at the physics department and in the falling water of a

fountain near the library. A water fountain or any object thrown up into

the air at an angle will trace out a parabola, the object retains its

horizontal motion (if we neglect air resistance) and so the only force acting

on it is gravity causing a downward acceleration. Like many of nature's

secrets, humans have taken inspiration from the parabolas. One useful property

that the parabolas have is that they reflect all parallel light rays to a

focus.

A mushroom was the focus of our last parabola in here but if this up here is

a big dish then we could put some electronics here in the focus or in the

case of the one I showed you before from the physics department you could maybe

heat up some water in here as a demonstration of some solar heating.

It works both ways, a torch and headlights on a car would have a parabolic mirror

with the bulb in the middle. Light from the bulb would be reflected off the

parabolic mirror out into a beam of light. So let's just add a little bit

more color to our drawing here, you're basically done we can add in a few more

parabolic hills, don't go overboard just keep it nice and pretty. I hope that this

video has been a little bit of inspiration for you to keep an eye out

for mathematical shapes all around you. If you find a way to work some parabolas

into an artwork of your own then send it to me and I'll share my favorite ones on Twitter.

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lifelong love of learning. I would like to wish you happy studying and I hope

that you have an absolutely mathematical day.

A special thanks to all of my patrons including Helmholtz, our first patron

cat of the day submitted by Rob.