Figure 1

A Klein bottle is a surface with a very strange property.
A surface is any object that is locally 2dimensional;
every part looks like a piece of the plane.
A sphere and a torus are surfaces, and they have 2 sides:
you can place a red ant and a blue ant on the sphere in
different places and never have them be able to touch each
other (put one on the "inside" and one on the "outside").
But the Klein bottle is a
surface with no "inside" and "outside"; it has just one
side! It is like a Mobius band but it also has
no "edges"! It is what you get when you glue two
Mobius bands along their edges. You cannot do this in
3dimensions, so you need at least 4dimensional space to
do this. See Figure 1 for a sketch of what
it might look like, if you allow it to selfintersect.
Presentation Suggestions:
Draw the standard picture of Klein bottle, and explain
how the object can live in four dimensions instead of three.
What students find fun about this Fun Fact is the exercise
of understanding 4 dimensions better.
The Math Behind the Fact:
The Klein bottle has another property. It is nonorientable,
just like Projective Planes.
Mathematicians like to classify surfaces
(meaning they try to understand what are all the
possible 2dimensional surfaces). We actually now know
the answer to this question. If we were living 500
years ago and didn't know the shape of the surface of the
earth, then knowing all possible surfaces would at least
limit the possibilities.
But we still don't know what
all possible 3dimensional objects (called manifolds)
look like, and we do not know which one of those
objects our 3dimensional universe is, either!
How to Cite this Page:
Su, Francis E., et al. "Klein Bottle."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
