Pythagoras theorem is one of the most fundamental theorems in
mathematics. It says that if you have a right angle triangle like
this. Then the square of the length of the hypotenuse is
equal to the sum of the squares of the lengths of these other
two sides. And that's something that has important in
trigonometry as well. For example, if I have an angle here
theater, and if I choose the unit so that the length of the
hypotenuse is one.
Then This distance is just the cosine of the angle Theta Cause
Theta. And this distance is the sign of the angle theater.
And so Pythagoras Theorem says that the square of cosine, Cos
squared theater, plus the square of the sign.
Sine squared theater
Equals 1.
Now this has applications in the real world. If we want to
measure positions and lengths using systems of coordinates.
So let's suppose that I have a pencil.
And I imagine that my pencil is
positioned. Just here.
So it's exactly on the hypotenuse of a triangle.
And it supposed that I've got some
coordinate axes. The usual X axis there and
AY axis here.
So now with my pencil back here
this point. Is the
point? XY.
And so. Pythagoras theorem tells me that X squared plus Y
squared must equal 1.
But of course, in the real world we don't have a given set of
coordinate axes. I might choose one set. You might choose a
different set. Let's suppose that you chose a set UV with the
same origin, but pointing in
different directions. So if I draw a new set of axes.
I'll have my U access going this way.
And my
viaccess Going this
way.
And then with respect to these
new axes. The pencil tip will be at the point.
UV.
So if I now drop perpendiculars.
Then this distances you
along here. This
distance. His V and this new angle.
Is 5.
But the length of the pencil still the same, it's still
one that hasn't changed.
And so we must have caused square 5 plus sign
squared Phi equals 1.
Be cause you squared plus B squared equals 1.
So called squared Theta plus sign squared theater equals 1
Costco at 5 plus sign. Squared 5 equals 1. It doesn't matter what
these angles, Theta or Phi are. This equation must always hold,
so it's not just an equation, it's actually what we call an
identity. Something that's always true for any angle
theater or any angle fire.
Well, that's two dimensions, but of course we live in a 3
dimensional world. So how would pythagoras theorem work there?
Well our pencil.
Would now be sticking up out of the paper.
And we'd have to have not just an X coordinate and AY
coordinate, but also as Ed coordinate going vertically
upwards.
And then it would no longer be true that the X squared plus Y
squared value will be the square of the length of the pencil.
It would be a bit shorter because it would be the distance
to where the shadow of the pencil falls on the paper.
Instead, we have to add the square of the vertical distance
as well to get us to the tip of
the pencil. So in three dimensions we would have not X
squared plus Y squared equals the length of the pencil squared
plucked, but instead X squared plus Y squared.
Plus, Z squared equals the length of the pencil squared.
And if the pencil happened to be in three dimensions, but lying
flat in the plane.
That would mean that zed was zero, and so in our
formula, if said is zero, we just get the X squared
plus Y squared back again.
Well, that's three dimensions, and what mathematicians like to
do, of course, is to generalize. We've had two dimensions, 3
dimensions. What about four
dimensions? Well, that might seem a bit physically unreal,
but in fact in the middle of the 19th century physicists did
start to use 4 dimensions to describe what was going on in
the world. They have XY and Z as the three dimensions of space.
They also have an extra fourth
dimension. T for time and in fact this was very important in
Einstein's theory of Relativity, which he published in 19105.
So how does that work?
Well, if we look at what a four
dimensional length might be.
Well, it's Square would be X squared plus Y squared plus set
squared. And then you might think it would be plus T
squared, but actually there's a
little twist. Because we don't have plus T squared.
In fact we have minus T squared because time is physically
different from the spatial
coordinates. And really, in there there's a hidden factor
of C squared. Really, it's X squared plus Y squared +6
squared minus C squared T squared. To get the
dimensions right and see is the speed of light.
But I suppose that we've chosen our units so that the speed of
light is actually equal to 1, and so X squared plus Y squared
plus Z squared minus T squared is the formula for the square of
the four dimensional length.
And now, as in the three dimensional case, we could ask
what happens if we. Just look at length line one particular
plane. Well, let's take the XY plane then. That would mean that
zed and T were both 0.
And so we would get for our formula X squared plus
Y squared.
But what if we look instead not at the XY plane, but at the XT
plane. And then.
Y and said will be 0 and we would have.
X squared not plus T squared.
But minus T squared.
And now that gives us a problem.
Give us a problem if we try to express the angle between
our 4 dimensional length and a fixed direction.
Because if we say that the angle
was theater. Then we would be looking at Cos squared
Theta minus.
Sine squared Theta.
And the problem is that this is not a constant for different
values of Theta. If there were a plus there, it would be constant
with a minus. It's not a constant, so that doesn't work.
We can't use trigonometric functions to talk about these
angles. Instead, we have to use new kinds of functions called
hyperbolic functions. And these are similar to the trigonometric
functions, calls them sign, but we write them with the letter H
on the end. So this one.
Is caution and this one is a bit harder to pronounce, so I'll
pronounce it as shine.
And for the angle I'll put you rather than theater, because
this isn't a real angle. It's a hyperbolic angle, and it works
in a different sort of way.
And now.
We can use these to obtain a formula like this one that
didn't work for the trigonometric functions, but one
which does work with the
hyperbolic functions. The reason
it works? Is to do with where these functions come from. The
trick functions come from.
A circle. Where is the
hyperbolic functions? Come
from. A hyperbola.
And if you know about the equations of these curves, the
equation of a circle is X squared plus Y squared equals
the radius squared.
But for I hyperbola like this.
The equation is similar, but without a plus sign. Instead it
has a minus sign.
Uncaught a minus sign is just what we want.
And we find that rather than having this formula, which
doesn't work for trig functions, the corresponding one for
hyperbolic functions, cosh squared U minus.
Shine squared you.
Does work and that this is
actually equal? To one, no matter what the hyperbolic angle
you might be.
So the way that scientists think about the four dimensions of
space time is very similar to the way in which they think
about 3 dimensions of ordinary
space. But there is this little twist that in the formula for
the square of the length, there's a minus sign in front of
tea, and that means that when they look at rotations involving
the time coordinate, they're not ordinary rotations using
trigonometric functions, they are hyperbolic rotations using
these new hyperbolic functions.