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 :)