Where a Möbius strip is a two dimensional object with only one surface and one edge, a Klein bottle is a two dimensional object with a single surface, and no edges. Consider for comparison, that a sphere is a two dimensional surface with no edges, but that has two surfaces.
A Klein bottle can be constructed by taking a rectangular subset of and identifying opposite edges with each other, in the following fashion:
Consider the rectangular subset . Identify the points with , and the points with the points . Doing these two operations simultaneously will give you the Klein bottle.
Visually, the above is accomplished by the following. Take a rectangle, and match up the arrows on the edges so that their orientation matches:
This of course is completely impossible to do physically in 3-dimensional space; to be able to properly create a Klein bottle, one would need to be able to build it in 4-dimensional space.
To construct a pseudo-Klein bottle in 3-dimensional space, you would first take a cylinder and cut a hole at one point on the side. Next, bend one end of the cylinder through that hole, and attach it to the other end of the clyinder.
A Klein bottle may be parametrized by the following equations:
where and are chosen arbitrarily.
|Date of creation||2013-03-22 13:37:00|
|Last modified on||2013-03-22 13:37:00|
|Last modified by||vernondalhart (2191)|