You can make a torus from a square sheet of rubber by first gluing two opposite sides to form a cylinder, and then gluing the two boundary components of that cylinder to get the torus.

Klein Bottle

Mirror image and orientability. For more information, on Klein Bottles, visit the Topological Zoo.

Acme is proud to be our universe's foremost supplier of immersed, boundary-free, nonorientable, one-sided surfaces. Pierce your ear and you'll increase your genus by one. Surely the Klein bottle has volume.

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Once uncorked, it has a lip which separates the inside from the outside. To understand why, first think of the more familiar doughnut known mathematically as a torus. Making a torus: Your Acme's Klein Bottle is a real Riemannian manifold, just waiting for you to define a Euclidean metric at every point.

For example, a loop drawn on a piece of paper in two-dimensional space has a well defined inside and outside, but a loop drawn in three-dimensional space doesn't.

It has two sides: To find out more about it see the article Inside the Klein bottle. This, in turn, causes it to have one handle.

Top N Facts About The Klein Bottle

Its inside is its outside. For the other pair of sides, however, don't identify points that are directly opposite, but points that are diagonally opposite, as shown in the picture.

Nathaniel Hellerstein a friend of this author created a Klein Bag from a sock. The same applies to the Klein bottle.

Introducing the Klein bottle

It's just one way of representing the bottle in three-dimensional space there are others too. Plug this hole with a finger or cork, and then repeatedly turn the bottle end over end until it has all mixed together.

In order to add colored water to your Klein bottle, complete the steps listed above, and then place a few drops of food coloring inside the tube at the bottom. This book's a perfect introduction to topology - for high school students through postodocs. Animation of two sided sheet runs for 12 seconds. If you slide a shape over one of the edges that had opposite points identified vertical in our picture , it reappears reappear on the opposite side, like before.