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From the Fun Fact files, here is a Fun Fact at the Advanced level:

Klein Bottle

Figure 1
Figure 1

A Klein bottle is a surface with a very strange property. A surface is any object that is locally 2-dimensional; 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 3-dimensions, so you need at least 4-dimensional space to do this. See Figure 1 for a sketch of what it might look like, if you allow it to self-intersect.

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 non-orientable, just like Projective Planes.

Mathematicians like to classify surfaces (meaning they try to understand what are all the possible 2-dimensional 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 3-dimensional objects (called manifolds) look like, and we do not know which one of those objects our 3-dimensional universe is, either!

How to Cite this Page:
Su, Francis E., et al. "Klein Bottle." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

Keywords:    topology, 2-manifold
Subjects:    topology
Level:    Advanced
Fun Fact suggested by:   Francis Su
Suggestions? Use this form.
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