Topology - Part 3


Uploaded by MyWhyU on 05.07.2011

Transcript:
Professor! What did you do to A.V.? I kind of liked him
... in a weird way.
Uh ... I don’t know.
This is highly irregular.
A.V. Are you okay?
Whoa! This is really weird!
I must have hit warp speed.
I seem to be floating in infinite space.
I hope there are no Klingons around here.
Professor Schmohawk. Can’t you do anything?
Well I, uh ... I guess I’ll have to call Mr. Squeejeeopolis.
He’s the only one around here who actually knows how this thing works.
Hello? Mr. Squeejeeopolis?
Can you come down to the lecture hall for a minute?
We seem to be having a small problem
with the multidimensional conformal space projection configurator.
Yes I know you’re very busy.
I apologize for the inconvenience, but we may have a serious issue with the configurator.
You’ll be right down?
Thank you so much. Goodbye.
Ach! Professor Schmohawk!
What seems to be the problem here?
Oh, Mr. Squeejeeopolis. Thank you for coming down.
As you can see we are having a bit of a problem with ...
Ach! Mine Gott in himmel!
This is not good!
Yes, uh we were hoping that maybe you could help us with ...
He looks terrible!
Yes, uh I know he doesn’t look very well at the moment ...
My poor baby. They have not been feeding you again.
Schmohawk! If I tell you once, I told you a thousand times.
This is a very expensive equipment
and you must feed the fishy twice each day.
Oh ... yes I forgot.
You cannot forget these things Schmohawk.
This is a delicate scientific apparatus!
But can you help us with one of our students
who seems to be undergoing some sort of strange interaction with the configurator?
You see, we sent him into a spherically-curved 3-dimensional manifold
and now he seems to be ... well undergoing some sort of weird distortions.
GAAAAHHH!
He is hideous!
Actually that's how he looks normally.
Oh.
Now, about the distortions ...
Ach! There is nothing to worry about.
Your student is completely unharmed.
In fact, he probably doesn’t notice that anything is different
except maybe the strange glow of his own image coming from all the directions.
A.V.! Don’t go into the light!
Don't worry about a thing young lady.
This young man doesn’t see himself moving at all.
As far as he is concerned, he is always in the center of his unbounded space.
And this is similar to the way we see our own universe.
We always seem to be in the center, with no boundaries in any direction.
But as you can see, this doesn’t necessarily mean
that the amount of space in our universe is infinite.
Our universe may be finite but curved and unbounded, just like A.V.’s space.
But why does he keep getting all stretched out of shape?
What you are seeing is not A.V. getting stretched.
This is just a distortion caused by the way we are viewing his space.
Unfortunately, there is no way to project a curved space like this into our universe
without some strange distortions.
I think Mr. Moosemasher has a question.
Professor. What do you mean by distortions?
Well, let’s think for a minute about the spaces which we have created.
We started with a 2-dimensional cylindrical space.
Ah! This is a good example of a 2-dimensional space with some of the boundaries removed.
Yes. We started with a rectangular bounded space
and then connected the left and right boundaries
so horizontally at least, the space became unbounded.
But we needed a third dimension
to visualize the left and right boundaries connected together without any discontinuities.
A cylinder is a good way to visualize this connection.
But, of course, this third dimension did not really exist.
It was just a way of visualizing the unbroken connectivity of the 2-dimensional space.
Ah! But this kind of space is not distorted.
All the rules of Euclidian geometry are still valid.
The toroidal space had a more complicated connectivity.
It was also a flat rectangular space
but now the left and right boundaries were connected
and the top and bottom boundaries were connected.
Even using three dimensions
there is no way to show the connections for all four boundaries
without distorting the space in some way.
For example, we can start by forming a cylinder to connect two of the boundaries.
But then we must also connect the other two boundaries.
There is no way to do this without distorting some parts of the space.
This means that in order to display the space as a torus
the distance between some points must increase while others decrease or stay the same.
In other words, the space becomes "distorted".
But remember that the torus is only a way of visualizing the connectivity
of this 2-dimensional flat space.
If we show the space as a flat rectangle and just remember that the boundaries are connected
then no distortions are needed.
So this space is Euclidian too.
Just like the cylindrical space
it is just flat rectangular space with the boundaries connected differently.
Now, the spherically-curved 2-dimensional space was an entirely different matter.
The distortions in this space were not just a result of the way we viewed the space.
They were actually inherent in the space itself.
This is why we call this space “curved”.
In curved space, not all the laws of Euclidian geometry are valid.
Lines that start out as parallel eventually cross.
The angles of a triangle add up to more than 180 degrees.
The circumference of a circle is not equal to pi times the diameter, and so on.
When the space is very large compared to the observer, these distortions are small.
If you drew a shape on the surface of the Earth
the laws of Euclidian geometry would seem to hold perfectly well
unless the shape you drew was very very large.
This is why it took so long for people to realize that the Earth was not flat.
That’s right.
If you draw a very small circle on the surface of a sphere
it is pretty much like drawing it on a flat surface.
But as you make the radius larger, the geometry becomes more and more non-Euclidian.
In fact, once the diameter of the circle has stretched half-way around the sphere
the circle’s circumference actually becomes smaller as the radius increases.
Eventually the circle shrinks back to the original size
but now the inside has become the outside and vice versa.
Yes. The same thing will happen with a triangle.
Imagine drawing a triangle inside the circle.
If the triangle is small, it’s three angles will add up to about 180 degrees
and it will seem to behave according to the laws of Euclidian geometry
but as the triangle grows, it becomes more distorted.
It’s angles become larger and larger
until the diameter of the circle reaches half-way around the sphere.
At that point, each angle is 180 degrees and the three angles add up to 540 degrees.
The triangle now looks just like a circle.
As the radius of the circle continues to grow
the lines connecting the three vertices form another triangle
but the inside of this triangle was the outside of the original triangle and vice versa!
The same thing will happen to a square
or any other shape.
So on a curved surface, things can sometimes be quite distorted.
Ah! But here is the important point.
When we try to represent curved space as a flat surface
then we create distortions that are not actually present in the space.
Perhaps an example would help.
Let’s look at a typical map of the Earth’s surface.
Now imagine that you start at the equator and travel toward the North Pole.
As you travel north, on the flat map your image becomes more and more distorted.
When you reach the North Pole, you encounter what is called a “singularity”.
This is a point where our image becomes infinitely distorted.
The North Pole is the entire top boundary of our flat map.
In other words, this one singular point on the sphere
becomes an infinite number of points on our flat map.
Likewise, at the South Pole you encounter another singularity.
This point is the bottom boundary of our flat map.
Eventually if you continue to follow a geodesic path
you will return to your original position and your original shape on the flat map.
But remember that these additional distortions only appear
when we project the surface of a sphere onto a flat surface.
An object moving around in this curved space does not really change its shape as it moves.
It only appears to change its shape in our flat representation.
So remember class
on the surface of a sphere there are real distortions
because the laws of Euclidian geometry don’t hold
and then there are additional distortions
caused when we try to map the curved space onto a flat surface.
Any questions?
Yes Adrian.
But what about A.V.?
Oh yes! A.V.
Uh ... maybe Mr. Squeejeeopolis would like to explain what’s going on with A.V.
What do I look like? Sigmund Freud?
Actually I meant ...
Of course! You want to know why he looks so weird.
A.V. is perfectly hunky dory.
Just like the curved 2-dimensional surface of the Earth
his curved 3-dimensional space has some small distortions.
But the big distortions you are seeing
are only due to the way we must visualize his space.
Maybe you can understand better if we start with the Earth again and map it a little differently.
This time, instead of mapping the surface of the Earth onto a cylinder
and then unwrapping the cylinder
let’s start by making a small hole in the surface of the Earth at the South Pole.
Now let’s stretch the hole bigger and bigger
until the entire surface of the Earth becomes a disk.
Every point on the surface of the Earth is represented on the disk
just like every point was represented on the rectangular map
but the distortions are much weirder!
Now the South Pole has become the perimeter of the disk
and the North Pole has become a point at the center of the disk.
The equator becomes a circle half way between the North and the South Poles.
Now what would happen as you traveled from the North Pole
on a great circle route around the Earth?
As you travel south, your image becomes more and more distorted
until when you reach the South Pole
your image stretches all the way around the perimeter of the disk.
As you leave the South Pole, your image becomes less distorted
until you end up back where you started at the North pole.
Of course! Now I see!
This is exactly what is happening to A.V. except in 3-dimensions.
Aha!
You know, for a PhD college professor guy you are pretty smart.
Uh ... Thank you.
So you see class
just like mapping the 2-dimensional curved surface of a sphere
onto a 2-dimensional flat disk
the multidimensional conformal space projection configurator
maps Mr. Geekman’s 3-dimensional spherically-curved space
into the volume of a 3-dimensional sphere in our un-curved space.
Any questions?
Okay. When A.V. appears to be at the center of the sphere
it is just like our example of being at the North Pole on the surface of the Earth.
Remember how this mapped to the center of the flat disk?
Okay. Now when A.V. floats around to the opposite side of his space
it is just like our example at the South Pole
which was represented by the outer boundary of the disk.
This is why his image sometimes stretches around the outer boundary of the sphere.
Any questions?
Mr. Moosemasher?
Can A.V. come home now?
Of course! I almost forgot.
All I have to do is hit Escape.
Dang!
Schmohawk!
When was the last time you changed the oil in this thing?
There’s no place like home. There’s no place like home.
Okay. Let’s fire this baby up again!
Help me Mr. Wizard!
Mommy.
Geekman! I never thought I’d be glad to see you again!
Daddy?
Geekman! You’re geek meat!
Not Daddy?
Well, I guess that was enough excitement for one lecture.
I’ll see you all again when we present the next in my series of lectures at Why U!