A photograph of a stapler is a 2-dimensional immersion of a 3-dimensional stapler. In the same way, Klein Bottles are 3-D immersions of the 'true' 4-D Klein Bottle.

Cliff's page is great. No fancy, flashing, dancing or loud bits - just stuff that makes your brain hurt.

Anyone who can take a completely abstract mathematical system, create a model of it and then sell it is a master. I could only hope to learn. And if one day we end up using a 4 dimensional version of these to power spacecraft or cities, then I can say I told you so......

At last, Acme has conquered topological and engineering frontiers to manufacture genuine glass Klein Bottles. These are the finest closed, non-orientable, boundary-free manifolds sold anywhere in our three spatial dimensions.

These elegant bottles make great gifts, fantastic classroom displays, and inferior mouse-traps. With its circle of singularities, an Acme Klein Bottle can be said to exist inside of itself -- especially handy during time-reversals.
A topologist's delight, handcrafted in glass

Cliff (left) sells Klein Bottles that are a 3-dimensional 'photograph' of a true Klein Bottle.

As an alternative to buying an Acme Klein Bottle, you can save money by just memorizing this set of parametric equations, since it defines the surface of every Klein Bottle.:

x = cos(u)*(cos(u/2)*(sqrt_2+cos(v))+(sin(u/2)*sin(v)*cos(v)))
y = sin(u)*(cos(u/2)*(sqrt_2+cos(v))+(sin(u/2)*sin(v)*cos(v)))
z = -1*sin(u/2)*(sqrt_2+cos(v))+cos(u/2)*sin(v)*cos(v)

or in polynomial form:

Yep, no doubt about it: Your Acme's Klein Bottle is a real Riemannian manifold, just waiting for you to define a Euclidean metric at every point. Acme is proud to be our universe's foremost supplier of immersed, boundary-free, nonorientable, one-sided surfaces. We make and sell Klein Bottles.

The 'What is' page is worth a read if you would like an english example of the equations above - the standout is the Job's page...