Topology - Part 2


Uploaded by MyWhyU on 05.07.2011

Transcript:
Professor Dreamy ... I mean Schmohawk.
Adrian Scienstein, do you have a question?
I’ve been wondering professor, what exactly is space?
Well Adrian, nobody actually knows what space is.
On the smallest scales space may have very different properties
than what it appears to have on the scales that we can measure with today’s technology.
One way we can think about space is as a way of connecting up points.
In a 2-dimensional bounded space, it’s easy to visualize how points might be connected.
Now if we connect the points on the top boundary with the points on the bottom boundary
we create a 2-dimensional cylindrical space.
We can visualize this as a flat sheet of space rolled up into a cylinder.
Of course instead of connecting the points on the top and bottom
we could have connected the points on the left and right boundaries
to create a different cylinder.
Now what do you think would happen if we connected the points on the top and bottom
as well as the points on the left and right?
Uh ... Professor, Professor? Uh ... Professor!
A.V. Geekman, yes.
You would obviously have created the spherical 2-dimensional manifold
in which Mr. Moosemasher is currently trapped
like a big fat 2-dimensional rat!
When I get out of here you’re geek-meat Geekman!
Not so fast boys.
Let’s see what kind of space would be created.
First we connect the left and right boundary points.
Then we connect the top and bottom boundary points.
So we get a donut!
Mmmm ... donuts.
Actually, it's called a "torus".
Mmmm ... toruses.
This isn’t the only way we could have visualized
connecting up the top and bottom boundaries of the cylinder.
We could have stretched the cylinder around
connecting the top and bottom up like a garden hose
but this still would have created a torus.
In fact, both toruses are equivalent ways of visualizing the connection of the points.
Let’s demonstrate this space with our multidimensional configurator.
Okay Hulk.
Now throw your test object ... I mean "football".
Okay Professor.
What happened?
Well, just as in the case of the spherical space
the toroidal space in which you currently exist is unbounded.
The only difference is in the way the space is connected to itself.
What do you mean Professor?
Maybe it will make it easier to understand
if we have our projector display your toroidal space a little differently.
Just like cylindrical space, toroidal space is 2-dimensional and flat.
Our projector displayed Hulk’s space as a torus
since that is a good way of visualizing a flat surface
with the left and right boundaries connected
as well as the top and bottom boundaries connected.
However, although Hulk’s space is “unbounded” it is still quite flat.
There are no distortions as we encountered in the surface of a sphere
so all the rules of Euclidian geometry are still valid.
Let’s look at an instant replay of that last pass.
Cylindrical space has no boundaries in the horizontal or vertical directions
so the football is free to continue on its path.
That is, until it encounters something.
Yeah, something big and dense!
Hey, Professor.
How did the football team get in here with me?
Hulk is seeing images of himself like in the cylindrical space
except this time light rays from Mr. Moosemasher
travel around and strike him from the top and bottom
as well as from the left and right.
Hulk sees himself in every direction
left, right, above, and below.
Toroidal space is finite
but Mr. Moosemasher sees it as infinite in both the horizontal and vertical directions.
Okay guys. Cut that out now or I’m going to tell the coach!
Mr. Geekman.
You certainly seem to be enjoying our little demonstration.
Yes Professor.
It’s a fascinating demonstration if I do say so.
In fact, it’s too bad that the multidimensional conformal space projection configurator
is incapable of creating 3-dimensional space or I’d have volunteered myself.
Well I’m glad that you mentioned that A.V.
In fact, the configurator can create space of any number of dimensions.
And thank you for volunteering.
Gulp.
First we must return Mr. Moosemasher back to our 3-dimensional Euclidean classroom.
Okay Mr. Geekman.
Are you ready?
Beam me up Scotty.
Mr. Geekman.
You are now inhabiting a 3-dimensional finite bounded flat rectangular space.
Hey, look at me! I’m Marcel Marceau!
Don’t worry Mr. Geekman. We can give you a little more breathing room.
Just like in Hulk’s 2-dimensional cylindrical space
we can connect the left and right boundary points in A.V.’s 3-dimensional space
to eliminate those boundaries.
Cool!
We have connected the points on the right boundary to the points on the left boundary
creating the 3-dimensional equivalent to the 2-dimensional cylindrical space.
Now let’s connect the top boundary to the bottom boundary
and the front boundary to the back boundary.
Cool to the third power!
By connecting all the boundary points on the opposite sides of Mr. Geekman’s space
we have created a 3-dimensional toroidal space.
Unfortunately, we would need four dimensions to properly display this weird torus.
I see Mr. Geekman is enjoying his new unbounded space.
I’ll stand on my head to sell you a used car!
Perhaps we should try something a bit more "stringent".
Do you recall the 2-dimensional spherically curved space which we sent Hulk into?
Well, as it turns out, there is a 3-dimensional spherically curved space
which our space configurator is capable of generating.
This type of unbounded, curved space
is one of the many topologies of space allowed by Einstein’s laws of general relativity.
I’ve never actually tried this before but
Oh, what the hey!