Klein bottle - Interesting mathematics facts?

We all love the Klein bottle, though not many of us know what they represent or how to describe them mathematically.

Felix Klein was a German mathematician known mainly for his work in group theory ( study of algebraic structure known as groups), complex analysis ( investigates functions of complex numbers), and non-Euclidean geometry (otherwise known a hyperbolic geometry). He is best know for his conception of the Klein bottle.

A Klein bottle is an object with no inside and no outside. A continuous shape where one can travel from the starting point along the surface of the object, never crossing an edge, and make it back to the original location. It is a 1 sided object with no edges.

This is a representation of how a Klein bottle is formed.

The Klein bottle is a one sided ( you heard correctly) closed surface which cannot be embedded in 3D Euclidean space (other wise known as Cartesian space, refers to something where you have points, lines, and can measure angles and distance + Euclidean axioms are satisfied) but may be immersed ( immersion is a special type of map ... a non singular map from one manifold to another such that every point in the domain of the map, the derivative is injective linear transformation) as a cylinder looped back through itself to join with its other end from the inside.

Mathematically this is how one could represent a Klein bottle.

$x=cos\left(u\right)\ast \left(cos\right(u/2)\ast (\sqrt{2}+cos\left(v\right))+(sin\left(u/2\right)\ast sin\left(v\right)\ast cos\left(v\right)\left)\right)$

$y=sin\left(u\right)\ast \left(cos\right(u/2)\ast (\sqrt{2}+cos\left(v\right))+(sin\left(u/2\right)\ast sin\left(v\right)\ast cos\left(v\right)\left)\right)$

$z=-1\ast sin\left(u/2\right)\ast (\sqrt{2}+cos(v\left)\right)+cos\left(u/2\right)\ast sin\left(v\right)\ast cos\left(v\right)$

Polynomial description of Klein bottle :

$(x^2+y^2+z^2+2y-1)\left[\right(x^2+y^2+z^2+2y-1{)}^2-8z^2]+16xz(x^2+y^2+z^2-2y-1)=0$

Pretty picture of Klein bottle. Period.

Source : Describing a Klein bottle

Edit:

For people who are asking for practical usage of Klein Bottle, I found some good lines from Google.

No one asks a poet what a new poem “does.” The poem’s simplicity, elegance, and beauty are sufficient reasons for its existence. Aren’t the same things true for a mathematical proof?

And if you're still not convinced.