The way mathematicians bend space

so how is our universe wrapped up?
it might be a flat universe
where it just sort of goes on forever
that’s what most people think

or it’s closed up in some way
such that if I keep on looking
I’ll actually eventually see the back of my head
I’ll see the back of my head
 
Poincaré was interested in what possible shapes
could the universe be wrapped up in
could it be a big hyper football?
or does it bend like a hyper bagel?

we live in a three dimensional space
a particular three dimensional space
in a fairly small bit of it
we don’t explore huge amounts
we don’t explore huge amounts

now the way mathematicians bend
three dimensional space
is actually they don’t bend it at all
they slice it into bits

and then they tell you
how conceptually you glue
the pieces together again

you work out how all the pieces
fit together edge to edge
you do a massive mathematical simplification
and you end up discovering

that you can simplify the structure