Cabinet of Curiosities - Day 14

Today's curiosity is a Klein Bottle.

It doesn't look like much does it? But, trust me, it's going to blow your mind. 

You see, this is a 3D 'shadow' of a 4D object that has no edge or boundaries and where its inside is its outside. Consequently, it contains itself.

I did warn you.

 

I'm pretty sure you know what a Möbius Loop is. You take a strip of paper, give it a half-twist, then tape the ends together. What you end up with is a one sided, one edged surface.

A Klein Bottle takes this concept one step further.

In 1882, Felix Klein imagined sewing two Möbius Loops together to create a single sided bottle with no boundary. You take a rectangle and join one pair of opposite sides - you'll now have a cylinder. Now join the other pair of sides with a half-twist. That last step isn't possible in our universe, unfortunately, because it requires a a fourth dimension so that the surface can pass through itself without a hole.

Therefore, a true Klein Bottle can only exist in four dimensions but we puny humans can only perceive three. So, just as a shadow of a cat is a 2D representation of a 3D cat, so my Klein bottle is a 3D 'shadow' of a 4D Klein Bottle. And the photo on this blog is a 2D representation of my 3D bottle. 

With me so far?

Nevertheless, my 3D bottle is still pretty weird. For a start, it has just one surface and no edges. If you were an ant you could walk from the outside to the inside and vice versa without encountering any kind of barrier - it's all one continuous surface. It contains itself.

See? I told you it would blow your mind. 

I'm going for a lie-down now. I'll let  this video explain it better than I ever could.