As soon as I saw the box from Meghan in the porter’s lodge, I knew that there was a closed, non-orientable, boundary-free manifold in Wadham. Despite my birthday not being for another four days, not opening it at that point would have been pointless and superfluous. After all, it is better to have a Klein Bottle on display than a Klein bottle which you know to be in a box. I trust that Meghan will understand.
As you are like to find in the office of a particularly cool mathematician, it is a genuine Klein Bottle: such as you would get if you could glue the edges of two Mobius strips together. While that is not actually possible in three dimensional space, the Klein Bottle is a three-dimensional cross section of that higher dimensional object. Imagine, for a moment, a hair elastic twisted into a figure-eight shape. In three dimensions, you can do that without having it intersect itself. If you were to draw that figure-eight hair elastic, however, or take a photo, it would look as though it intersects itself. The same is true of a Klein Bottle embedded in three dimensional space. Note that even if our universe really does have ten spacial dimensions, or more, as postulated by string theory, there are still only three of them unfurled enough to put parts of a glass Klein Bottle in.
Invented by Felix Klein – a German professor of mathematics – in 1882, a Klein Bottle has only one side (no inside and outside like a balloon), yet also no rim or lip (like a bowl or an open wine bottle). It’s the only gift I’ve ever received that I printed off an encyclopedia article about, for use in explaining to guests. You can also tell people it’s a work of modern art.
Many thanks Meghan, for furnishing me with what may be the geekiest thing I have ever owned. Like surviving through a battle in which your friends died, getting a Klein Bottle creates a commitment to live the rest of your life in a certain spirit. It’s also dramatically quieter than my rock tumbler used to be.
It should also be noted that the literature that comes with Acme Klein Bottles is extremely funny.
We at Acme Klein Bottles strive to create the finest nonorientable surfaces and hope that you will be satisfied with your new Acme manifold. For this reason, we are pleased to offer this UNCONDITIONAL GUARANTEE complete with these conditions:
1) We unconditionally guarantee your Acme Klein Bottle to be free of any defects in workmanship or workwomanship for a period of ONE YEAR following purchase. If you aren’t satisfied with your Acme Klein Bottle — for any reason — just return it for a refund or replacement. You pick up shipping charges.
2) We guarantee safe arrival. If your Klein Bottle arrives broken, call or send email and we will immediately send a replacement.
3) We slightly guarantee your Klein Bottle for THREE MONTHS against any cracks or breakage, whether due to earthquakes, clumsy undergrads, or greasy fingers. Just mail us a fragment and $10, and we will send a replacement.
4) We warrant each Acme Klein Bottle for a period of FIVE YEARS to be absolutely free of any magnetic monopoles. If you discover one, contact us immediately and we will refund your purchase price right after claiming the Nobel Prize.
5) Furthermore, we guarantee for TEN YEARS that any polyhedron spanning your unbroken Acme Klein Bottle will have about as many edges as the sum of its vertices plus faces.
6) We further warrant for ONE MILLION YEARS that within a Euclidean plane, the square of a right triangle’s hypotenuse will equal the sum of the squares of the two remaining legs.
In addition, Acme’s provides this exclusive LIFETIME GUARANTEE: We guarantee that you will live your entire lifetime, or double your money back.
Acme’s unconditional guarantee has the condition that we do not warrant any Klein bottle against the actions of cats, ferrets, or axolotls. We will NOT BE HELD RESPONSIBLE for any incidents relating to these beasts of burden whatsoever in any form or spatial dimension.
All other warranties, express and implied, are null and void except during total solar eclipses. Purchaser shall have the option at his, hers, or its sole discretion, to try to collect on this guarantee. Guarantee void if a substantial portion of the Klein bottle leaks into the 4th dimension. The big print giveth and the small print taketh away.
I’m going to assume that your pet ocelot falls under the beasts they will not be held responsible for.
Meghan
George? He’s harmless, as anyone who has read the poem I wrote about him will know.
You claim that your Klein bottle is “embedded in three dimensional space.” In fact, it is actually immersed in it.
Captain. My liege!
We all thought you were dead! No mere mortal could have survived being differentiated so many times. You truly are f(x)=e^(x)!
I can say without equivocation that the blog comments have reached an unprecedented level of geekiness.
A Klein Bottle cannot be embedded in 3 dimensions, but you can immerse it in 3-D. (An immersion may have self-intersections; Embeddings have no self-intersections. Neither an embedding nor an immersion has folds or cusps.)
Here’s a good short introduction to topology, for those wondering what the big deal is.
I can’t begin to understand what the hell the bottle symbolises but I still think it looks very cool.
This WikiPedia article explains it pretty well.
Bathsheba Sculpture LLC
Welcome. I’m an artist exploring math and science in sculpture, and this is my gallery and storefront.
Here you’ll find my sculptures and math models, handheld geometry made of 3D printed metal. Plus the pocket mini versions – all the art for a fraction the price!
The warranty on my Klein Bottle states:
There is now apparently “Overwhelming” Evidence For Magnetic Monopoles.
Notably, the evidence for magnetic monopoles exists at “low temperatures in ‘Dirac strings’ within a single crystal of Dysprosium Titanate.”
I don’t think my Klein Bottle contains Dysprosium Titanate, and it is generally kept around room temperature and pressure.
Classic case of science journalists overblowing a mundane result. Yes, connected quasi-monopoles are interesting. they are visible in any conducting medium. But there’s a HUGE difference between a quasi-monopole that is at the end of a finite-length, shielded dipole and a true monopole that actually violates the magnetic divergence-free condition.
In solar physics we call such things “unipoles” to distinguish them from the infinitely harder-to-find “monopoles”. Unipoles are all over the surface of the Sun, because the conductive interior hides the field lines that connect opposing unipoles.
It is disingenuous at best and downright deceptive at worst to call the HZB result “evidence for magnetic monopoles”, because it ain’t.
The only plausible true magnetic monopole detection ever was still in Blas Cabrera’s instrument at Stanford in the 1980s. It was never replicated, so it is unknown whether they exist but are extremely rare (and Cabrera was just lucky) or whether his detector glitched.
The elusive monopole
ELECTRICITY and magnetism are two sides of the same coin. There is supposed to be a deep symmetry between them. So why, when we see lone electrical charges such as electrons or protons, do we never see a lone magnetic pole – a monopole?
They are almost certainly out there: monopoles plays a pivotal role in the widely accepted “grand unified theories”, for example. GUTs suggest that the four forces of nature arose from one superforce that existed until moments after the big bang.
The trouble is, our universe is thought to contain no more than one of these “GUT monopoles” for every 1029 protons and neutrons in atomic nuclei: if there were any more than that, our most sensitive searches would have found one.
And it’s not like we can make them for ourselves. GUT monopoles are thought to be so massive that we don’t have powerful enough particle accelerators to create any. Not that this seems to bother physicists too much. Joe Polchinski, a theorist at the Kavli Institute for theoretical physics in Santa Barbara, California, has said that the existence of monopoles “seems like one of the safest bets that one can make about physics not yet seen”.
https://m.youtube.com/watch?list=PLt5AfwLFPxWJeBhzCJ_JXdaIXi_YJl7Bh&v=-k3mVnRlQLU
Klein Bottles with Cliff (extra footage)