Klein bottle

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 2 and identifying opposite edges with each other, in the following fashion:

Consider the rectangular subset [-1,1]×[-1,1]. Identify the points (x,1) with (x,-1), and the points (1,y) with the points (-1,-y). 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:

x ={acos(u)(1+sin(u))+rcos(u)cos(v)0u<πacos(u)(1+sin(u))+rcos(v+π)π<u2π
y ={bsin(u)+rsin(u)cos(v)0u<πbsin(u)π<u2π
z =rsin(v)

where v[0,2π],u[0,2π],r=c(1-cos(u)2) and a,b,c are chosen arbitrarily.

Title Klein bottle
Canonical name KleinBottle
Date of creation 2013-03-22 13:37:00
Last modified on 2013-03-22 13:37:00
Owner vernondalhart (2191)
Last modified by vernondalhart (2191)
Numerical id 12
Author vernondalhart (2191)
Entry type Definition
Classification msc 54B15
Related topic MobiusStrip