Welcome to another Mathologer video. Have a look at this. What do you see? Yes, some

cool Mathologer toys in the background but ignore those. Yes, I know, it's hard but

ignore the toys :) Focus on the black UFO at the bottom. On top of the UFO is a

circular mirror and on top of the mirror is a coin featuring pretty Princess Leia

and her trusty robot r2d2. Now let me pick up the coin. What!?

My fingers are passing right through the coin. Is this some kind of Jedi mind

trick?

What I want to do today is to explain our ghostly Princess Leia as well as a

closely related way to conjure up ghostly voices. It's really cool to be

able to create these ghosts but today's maths is also super applicable. In fact, if

this maths ceased to exist from one day to the next you wouldn't recognize the

world you live in anymore. Well let's get to it. It's all got to do with our high

school friend good old x-squared. Sticking with movie

references, x squared is a little bit like Clark Kent.

Most of Clark's friend think they know all as to know about him but only a

select few are aware that he's actually Superman. Same with x squared which also

has some hidden superpowers that hardly anybody knows about. Okay, here we go.

Did you know that the point (0,1/4) and the horizontal at y= -1/4

are super special for the parabola y = x^2. The

point is called the focus and the line is called directrix of the parabola.

What's special is that every point on the parabola is exactly the same

distance from the focus and the directrix. So these two distances there

are always the same. That looks tricky but to show it we just

need help from our other school friend Mr. Pythagoras. Here we go. If our

parabola point has coordinates x and x squared, then Pythagoras tells us the

square of the green distance. And going straight down to the directrix the

square of the yellow distance is this. Now it's just a matter of going on

algebra autopilot to check that these two expressions are equal. And that shows

the two distances are the same. Easy-peasy. The directrix is the secret

ingredient for lots of parabolic magic. For the first magic trick, let's position

the parabola on a piece of paper so that the red directrix coincides with the

bottom edge of the paper. Now look at any point on the

bottom edge, that one there. Fold the paper so that the black point ends up on the

focus. So there fold, fold, fold. Right on top. And unfold again. Okay it looks as

if the paper crease is a tangent of the parabola and that the touching point is

right above the black point. And looks are not deceiving. Starting with any

point at the bottom, folding results in a crease that is tangent to the parabola.

If you do this for all the points of the directrix, you get all the tangents of the

parabola. So why does this work? All those tangents suggests calculus but

you really don't need it. All you need is a little middle school OWL-gebra :)

Anyway I'll leave it as a challenge for you to give a proof in the comments and

if you're desperate I'll give one possible proof at the very

end of this video. Anyway for the record let's note that proving the second super

property of the parabola is also easy peasy. The second property gives a really

pretty way to create a parabola from scratch without having to calculate

anything. Start with a piece of paper, mark a point close to the middle of one

of the sides and perform the folding action for a bunch of points on the side.

Then the parabola materializes as if by magic. Super super nice :) Okay after this

little piece of paper magic we're almost ready to conjure our ghosts. We just need

one more super property of the parabola and actually you probably all know this

one, although I'm guessing that only a few of you will have seen a proper

explanation. Looking again at our paper folding

notice that this green triangle there gets folded smack on top of this

identical pink triangle and that means that the green angle and the pink angle

are the same. Right? Then the angle opposite the green is

also of the same size. Now for what we are after we just need that these two

angles here are the same. We also don't need the directrix anymore, so let's get

rid of that too. There's not much left of our picture but it tells us something

super interesting. Imagine the parabola is a mirror and the

vertical line is a ray of light hitting the mirror. Then this ray of light will

be reflected like this and the reflected ray will pass through the focus. But of

course the same is true for any vertical ray of light and so all the vertical

rays get focused on the, well, focus :) Lucky that that's what we chose to call it.

Of course, this also works in the opposite direction: any ray emanating from the

focus will be reflected into a vertical ray. I'm sure that many of you are aware

of this focusing property of the parabola and it's myriad applications

in the guise of parabolic reflectors and mirrors. Now finally we are ready to

conjure some ghosts. To begin let's add another parabola to this picture like so.

Then it's clear that a whole sector of rays emanating from the red focus will

end up passing through the green focus. Chances are you've seen the setup before

in the guise of the mysterious whispering dishes at science museums.

A whispering dish is exactly a circular paraboloid, the shape you get when you

spin a parabola around its axis of symmetry. As such the paraboloid inherits

all the nice reflective properties of the parabola. The whispering dish in the

picture is located at Scienceworks the Science Museum in my hometown of

Melbourne. The focus of the dish is located inside the ring I'm pointing at.

Okay, now take two of these dishes and place them 50 meters apart. Then if the

junior mathologeress Lara whispers at the green focus

Mathologer junior Karl will hear her disembodied voice at the

read focus. Really quite a stunning effect.

It's a great experiment but what's not so great is the explanation on the

whispering dish. What it says there is: "The other person hears you clearly

because the curved shape of the dish focuses the sound into the ring at their

end." Pretty damn nothingy, isn't it? Well most science museums try a little

harder and at least feature this suggestive drawing here but there's one

very obvious question about this effect and it's a question that is seldom asked:

Why isn't a sound muffled? Specifically, why doesn't a sound wave leaving the

green focus in different directions, then arrive at the red focus at different

times. Well that amounts to asking whether all the yellow parts in this

diagram are the same length. They don't look it but surprisingly they

are. And there's an easy explanation just using our focal-directrix super property.

Don't believe me? Just watch! Let's bring back the two directrices of the two

paraboli and let's take a careful look at one of the yellow paths. What can we

say about the length of this path? Well, let's see. Because of the focus-directrix

super property the red distance from the focus to the reflecting point on the

parabola is the same as the distance from the reflecting point to the

directrix. And, of course, the same is true up on top. But this means that the length

of the path from the red focus to the green focus is exactly the distance

between the two directrix lines. And since this is the case for all paths all

paths have the same lengths. How easy and how pretty a proof is that. This equal

length property is also important for many other really significant

applications of parabolic reflectors, but strangely unlike the focusing property

the equal lengths property is rarely mentioned by anybody. Okay, now what about

ghostly Princess Leia? How do we conjure her?

For this we use proper parabolic mirrors and instead of moving them

apart, we move them close together. We then place Leia and her robot friend in

the middle of the bottom mirror, at the green focus. Then we cut a hole in the

middle of the upper mirror just above the red focus. Then a hologram of Leia

materializes at the second focus.

Now, this is real mathematical magic :) !! And on that happy note I will declare I am

NOT happy. It's time for a Mathologer sermon. These days here in Melbourne my

junior Mathologers Karl and Lara seem to spend half their time in maths class

torturing quadratics but they never get to see any of the beautiful maths I've

shown you today. Much less figure out why it works. If they didn't happen to have

an annoying Mathologer for a father they'd never find out about any of this.

Well except for the science museum's explanation which turns out to be a

masterpiece of explaining nothing. This is especially puzzling and especially

annoying because the simple maths that you need to explain all this super

important and super applicable stuff properly is exactly the school maths

that it's done ad nauseam. At the same time

Victoria's maths textbooks are chock-a-block with pseudo applications

like parabolic bike paths, quadratic types of cheese, and so on (I'm not making

this up :) And this is just the tip of a parabolic iceberg. As my colleague Marty

likes to say: our educational authorities never miss an opportunity to miss an

opportunity. I'd be very interested in finding out from you guys what's the

state of educational affairs where you are. Do kids learn about the things I

talked about today in maths class. Properly? At all? Let us know in the

comments. And that's it from me for today. Okay, let's end on a satisfying

mathematical note and tie up a loose end. Here's a simple explanation for why our

paper crease from before just touches the parabola. If you want to figure this

out for yourself now is the time to avert your eyes and ears. Well, it's easy

to read the equation of the yellow creased line off this diagram. For this we

just need its slope and its y-intercept. The slope of the creased line is 2a, the

negative reciprocal of the slope of the thin segment and it's green y-intercept

is minus a squared (may take you a second or two to convince yourself of this.)

And so we get this for the equation of the line. And then where does this crease

line hit the parabola? Well we just equate the line with x squared and solve

for x. Okay, so there's exactly one solution at x=a and that

means that the crease touches the parabola at exactly the point we predicted.

All is good, no loose ends, I'm happy again and we'll all be able to

sleep tonight. And that's really it for today :)