Before we begin to think about the 3-Klein bottle, let's explore some similar two-dimensional objects. A Möbius band is constructed from a long rectangle. We attach the two short ends together with a 180-degree twist.
Imagine you are a Flatlander, and someone gives you a rubber rectangle like the one above, with instructions to connect it according to the arrows. What would you do? The rectangle itself fits into your two-dimensional world, but you cannot fathom how to attach the ends in the requested manner. A Möbius band cannot be constructed in 2-space because there is no such thing as a twist in 2-space. A rectangle with different arrows would present no problem, though you as a Flatlander would carry out the directions differently than you as a Spacelander.
A Möbius band has only one side and one edge. If you make a paper Möbius band and draw a line along it, you will return to your starting point and find that the line has crossed every part of the band. A paper band, however, is not the same as a true Möbius band. A surface is something that lines and shapes exist in, not on. Banchoff explains:
"We should think of the band as made of some porous material through which an ink-drawn figure bleeds, leaving us no way to tell on which side of the strip the figure was originally placed." (p. 194)
A Möbius band is a nonorientable surface. This means that a shape on a Möbius
band can become its own mirror image, which is not possible on an ordinary surface. A
Flatlander traveling all the way around a Möbius band returns reversed.
A Klein
bottle is another nonorientable two-dimensional surface, but it cannot be
created in 3-space. Constructing a Klein bottle requires four dimensions. Like a
2-torus, a Klein bottle can be made by gluing the edges of a
square, but the Klein bottle requires a twist.(5)
Just as a Flatlander could not imagine how to construct a Möbius band out of rectangle,
Spacelanders cannot figure out how to put together this square. An inhabitant of 4-space
would have no trouble, however.
When we represent a Möbius band in two dimensions, the line crosses itself, though
the edge of the real object does not.
In the same way, a representation of a Klein bottle can be created in 3-space, but this
model, unlike the actual object, intersects itself.
This semi-transparent rendering more clearly illustrates the intersection. (These images
are from The
Geometry Center.)
We can construct a nonorientable 3-space out of a cube just as we did a
3-torus, but this construction requires a twist in the
connection of two faces.
We connect top to bottom and left to right normally, but add a 180-degree flip when we
connect the front and back faces. We may call this space a 3-Klein bottle.
In a small 3-Klein bottle, you can look up and see your feet, to your left and see your
right side, and straight ahead and see your back. At first you might not notice that anything
is different from in the 3-torus. But when you raise your right hand, you look in
front of you and see yourself raising your left hand!
Here are views of two small 3-Klein universes.
The 3-Klein bottle produces a left-right reflection. If you travel straight ahead in a
3-Klein bottle, you arrive back at your starting point reflected: If you were
right-handed, you are now lefthanded, and your heart is on the right side of your
chest. If you take the trip again, you will return to your normal state. (Travelling in
an untwisted direction does not flip you.)
However, if you are the only thing in the universe, you will not know that you have been
reversed. Since every part of your body and mind has undergone the same flip, you
have no way to determine that anything has changed. If you raise what you think is your
right hand, your reversed retinas and brain see your right hand moving. Someone who
hasn't travelled around the 3-Klein bottle, however, sees you raise your left hand.
Only if there are other people and objects in the universe can you detect a change. But
from your point of view, it is the rest of the universe that has flipped, not you. The
stars are in the wrong places, your teddy bear is missing the wrong button eye, writing
is backwards. Your own handwriting looks normal to yourself, but to other people it
appears to be mirror writing. If our universe is a 3-Klein bottle, people returning from
a trip around the universe might see a strange but somehow familiar planet where the Earth
should be.
From Chaim Goodman-Strauss'
topology handouts
From
The
Geometry Center
From the Planet Earth Home Page
Next: Projective Space
Send comments or questions to Lisa Eckstein
