Okay, so here's the skinny on the Klein Bottle, which headlines on the home page. At first glance, it appears to be a
relative of another strange looking object, the glass retort (see photo). I actually had one of these as a kid (6th grade)
when I used to perform random acts of chemistry in my parents basement in Copiague, New York. With a shape that resembles
the head of an anteater or aardvark and a long glass snout, the primary purpose of this piece of lab glassware is to heat
a substance and extract something by distillation; the long glass nose catches the steam vapor and cools it enough to
render it back to liquid. I bought one for the main purpose of heating mercuric oxide or crushed cinnabar and extracting pure
mercury. As a kid, I though that mercury was the absolute coolest thing and I had some in a stoppered bottle that I used to
play around with. Of course, we now know how dangerous exposure to mercury can be and we'd never let kids play with it today,
but back then they actually used to put this in thermometers...the kind we put in our mouths! Oh well, considering this and
other things I did as a kid, it's a wonder I'm still alive.
Anyway, back to the Klein Bottle. This is more than a retort gone wrong. It's the best attempt (and maybe the ONLY
attempt) at rendering a 3 dimensional instance of a double mobius loop....
WAIT A SECOND! A double Mobius Loop? Okay, you weren't really expected to know what a Mobius Loop was. Hey, that's what
this site is for, to enlighten and entertain and educate...(E cubed). I would explain this, in my own words, but in the interest
of accuracy, I'll use the definition as provided by the makers of this item, the Acme Corporation (Yes, that really is their
name...I guess they're doing more than just providing items of mass destruction to Wile E. Coyote)
Ever hear of a Möbius Loop -- a one sided, one edged surface? Give a strip of paper a half-twist, then tape the ends together. It's one
side and one boundary, with delightful properties dear to mathematicians.
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In 1882, Felix Klein imagined sewing two Möbius Loops together to create a single sided bottle with no boundary. Its inside is its
outside. It contains itself.
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, sad to say. A true Klein Bottle requires 4-dimensions
because the surface has to pass through itself without a hole.
It's closed and non-orientable, so a symbol on its surface can be slid around on it and reappear backwards
at the same place.You can't do this trick on a sphere, doughnut, or pet ferret -- they're orientable. |
A Klein Bottle is locally 2-dimensional ... every small patch follows the laws of 2-dimensional Euclidean geometry. In this
sense, a Klein Bottle is a 2-dimensional manifold, and its inside is the same as its outside. But although it's a 2-D manifold,
it can only exist in 4-dimensions!
Alas, our universe has only 3 spatial dimensions, so even Acme's dedicated engineers can't make a true Klein
Bottle.
A photograph of a stapler is a 2-dimensional immersion of a 3-dimensional stapler. In the same way, our glass
Klein Bottles are 3-D immersions of the 4-D Klein Bottle. Our Klein Bottle is a 3-dimensional photograph of a "true" Klein
Bottle.
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.)
We represent a Klein Bottle in glass by stretching the neck of a bottle through its side and joining its
end to a hole in the base. Except at the side-connection (the nexus), this properly shows the shape of a 4-D Klein Bottle.
And except at the nexus, any small patch follows the laws of 2-dimensional Euclidean geometry.
Contrast this with a corked bottle -- say, a wine bottle. It has two sides: inside and outside. You can't
get from one to the other without drilling a hole or popping the top. Once uncorked, it has a lip which separates the inside
from the outside. If you make the glass arbitrarily thin, that lip won't go away. It'll become more prominent. The lip divides
one side of the bottle from the other. So an uncorked bottle is topologically the same as a disc ... it has two sides, separated
by a boundary -- an edge.
But a Klein Bottle does not have an edge. It's boundary-free, and an ant can walk across the entire surface
without ever crossing an edge. This is true of both theoretical Klein Bottles and our glass ones. And so, a Klein Bottle is
one-sided.
A Klein Bottle has one hole. This, in turn, causes it to have one handle. The genus number of an object is
the number of holes (well, it's more subtle than that, but I'm not allowed to tell you why). Other genus-1 objects include
innertubes, bagels, wedding rings, and teacups. A wine bottle has no holes and so is genus 0. (The genus
of a human being is difficult to define because it depends on what you consider a hole -- I'd estimate most people have a
genus of 0 to 4, slightly higher when yawning. Pierce your ear and you'll increase your genus by one.)
As an alternative to buying an Acme Klein Bottle, you can save money by just memorizing this set of
parametric equations, since it defines the surface of every Klein Bottle.:
x = cos(u)*(cos(u/2)*(sqrt_2+cos(v))+(sin(u/2)*sin(v)*cos(v)))
y = sin(u)*(cos(u/2)*(sqrt_2+cos(v))+(sin(u/2)*sin(v)*cos(v)))
z = -1*sin(u/2)*(sqrt_2+cos(v))+cos(u/2)*sin(v)*cos(v)
or in polynomial form:
Yep, no doubt about it: Your Acme's Klein Bottle is a real Riemannian manifold, just waiting for you to define
a Euclidean metric at every point.
Acme is proud to be our universe's foremost supplier of immersed, boundary-free, nonorientable, one-sided
surfaces. We make and sell Klein Bottles.
For more information, on Klein Bottles, visit the Topological Zoo or the Geometry Center. Or click here to see a diagram on how to make one in Japan Notice that topologists simulate Klein Bottles ... but ACME makes 'em in glass!
