Wednesday, February 16, 2011

Triple Klein Bottle

The Klein Bottle isn't mind-blowing enough, apparently. Now there's this:


That's from a website called Mathematical Imagery by Jos Leys. No, I don't know how I found that. Internet surfing lands us in the strangest places. :-)

KLEIN BOTTLE MATHS (for those who like Maths, and beware, this is about your run-of-the-mill Klein bottle, not the .... thing in the photograph above ...... )  :

The "figure 8" immersion (Klein bagel) of the Klein bottle has a particularly simple parameterization. It is that of a "figure-8" torus with a 180 degree "Möbius" twist inserted:
\begin{array}{rcl}
x & = & \left(r + \cos\frac{u}{2}\sin v - \sin\frac{u}{2}\sin 2v\right) \cos u\\
y & = & \left(r + \cos\frac{u}{2}\sin v - \sin\frac{u}{2}\sin 2v\right) \sin u\\
z & = & \sin\frac{u}{2}\sin v + \cos\frac{u}{2}\sin 2v
\end{array}
In this immersion, the self-intersection circle is a geometric circle in the xy plane. The positive constant r is the radius of this circle. The parameter u gives the angle in the xy plane, and v specifies the position around the 8-shaped cross section. With the above parameterization the cross section is a 2:1 Lissajous curve.

The parameterization of the 3-dimensional immersion of the bottle itself is much more complicated. Here is a simplified version:
\begin{align}
x & = \frac{ \sqrt{2} f(u) \cos u \cos v (3\cos^{2}u - 1) - 2\cos 2u}{80\pi^{3}g(u)}-\frac{3\cos u -3}{4}\\
y & = -\frac{f(u)\sin v}{60\pi^{3}}\\
z & = -\frac{\sqrt{2}f(u)\sin u \,\cos v}{15\pi^{3}g(u)}+\frac{\sin u \cos^{2} u + \sin u}{4}-\frac{\sin u\,\cos u}{2}
\end{align}
where
f(u) = 20u^{3}-65\pi u^{2}+50\pi^{2}u-16\pi^{3}\,
g(u) = \sqrt{8\cos^{2}2u-\cos 2u (24\cos^{3}u-8\cos u + 15) + 6\cos^{4}u (1 - 3\sin^{2}u)+17}
for 0 ≤ u < 2π and 0 ≤ v < 2π.

In this parameterization, u follows the length of the bottle's body while v goes around its circumference.

UM-m-m, OK. 

Off topic: This webpage (The Best NYC Pastrami Sandwich restaurants) is wrong. The best hot-pastrami-on-rye sandwich on Earth can be found at Harold's New York Deli Restaurant in Edison, New Jersey. ;-p

Important note: Like the website says, all in CAPS, sorry: HAROLD JAFFE, OWNER OF HAROLD'S NY DELI IN EDISON NEW JERSEY IS NOT IN ANY WAY AFFILIATED WITH
HAROLD'S DELI IN PARSIPPANY NJ OR ANY OTHER LOCATION

Trust me, he's not kidding, cuz I've eaten at both. Go to Edison, or don't go at all. The Parsippany Pastrami ain't bad, but it ain't as good as Jaffe's restaurant in Edison, unless "average" or "OK" turns you on. IMO, which I'm entitled to.

2 comments:

Pat B said...

Steve,
It looks like the recycling logo for the 21st century... way cool...

off topic, I'm thinking there must be some kind of intellectual game where you make a word or acronym out of the word verification things on comment pages and such. A hit tv show in the waiting?

Steven Colyer said...

You mean like that old military word: FUBAR ?

FUBAR = Frequently Underachieving Ballistic Arms Reduction

Sure that's what it means (smirks).

Oh, I have a MUCH better Mathematical artwork on the way. Or, um, AS good, because God forgive us we compare artworks, beauty being in the eye of the of the beholder and whatnot. :-)