Why is kleinbottle not orientable. I am not familiar with topology.
So let’s talk about chirality first so that definition is covered. Your left and right shoes are chiral mirror images of one another, since they are clearly like one another, but there’s no way to rotate a right shoe to turn it into a left shoe and vice versa. Another example, this time of a 2D chiral object, would be a spiral. A spiral spins either clockwise or counter clockwise, and no rotation in a 2D space can change that. You need to rotate the spiral in a 3rd dimension to get it to become its mirror image. You might do the same to a shoe, but you’d have to rotate it in a 4th dimension since it’s a 3D object.
So a good test of orientabilIty is this: take a lesser-dimensioned chiral shape and traverse it along the shape of choice. If there exists no traversal which can make the chiral object look like its mirror image, then the shape is orientable. This can also be said as the shape having clockwise and anti-clockwise as distinct directions. Both the Möbius strip and the Klein bottle are non-orientable because they can convert lesser-dimensional chiral objects into their mirror images simply by traversing those objects along their surface in the right fashion.
You can imagine tracing a path along a Klein bottle to see that it only has one side. To get more precise than that requires some topological context. If you slice it down the middle it turns into two Möbius strips and an orientation of the Klein bottle would induce an orientation of the strips, which are non-orientable. Alternately it has zero top integer homology, which you can get from looking at a triangulation. The orientable double cover of a Klein bottle is a torus, which is connected (if it were orientable, the double cover would be two disconnected Klein bottles).
Even if it might be the shape of the universe? How could that be bad!
Have you looked at the universe lately?
So that guy obssessed with making and storing glass klein bottles under his houses crawlspace might be creating thousands of universes? Neat!
Cliff Stoll
He didn’t make them, btw. He ordered a giant batch years ago, so the unit price was acceptable.
He also has quite an interesting life story
Wow, discovering and ultimately unmasking the hacker is a great story.
And from the end of the Wikipedia article:
Stoll sells blown glass Klein bottles on the internet through his company Acme Klein Bottles. He stores his inventory in the crawlspace underneath his home in Oakland, California, and accesses it when needed with a homemade miniature robotic forklift.
I want to know more about this!
E: found it! https://www.youtube.com/watch?t=111&v=-k3mVnRlQLU
I believe you may be mistaken on the claim he didnt make them. Theres a good few videos floating around of him blowing the glass on numberphile. The Wikipedia just says he keeps the stock in his house.
That’s not Cliff blowing the bottle.
Here’s the video where he tells the story on how he had to comission thousands of them:
Cool! Thank you for digging up that video I appreciate it :)
I’m no astrophysicist but I’m guessing we know about enough to determine that every shape might be the shape of the universe
I guess we are volume shaming now?
