Wikipedia:Reference desk/Archives/Mathematics/2006 October 19

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October 19

Big numbers

Ok, the sequence goes: ones, tens, hundreds, millions, billions, trillions... what's next? --JDitto 06:28, 19 October 2006 (UTC)

Quadrillion, apparently. See Names of large numbers. – AlbinoMonkey ^(Talk) 06:30, 19 October 2006 (UTC)

Whoa! Talk about quick response! Thank you for answering my question. --JDitto 06:36, 19 October 2006 (UTC)

Too bad it was not a sequence ; 1, 1000, 1000000 could have done it (OEIS A060365) :

"Search: 1, 10, 100, 1000000 = I am sorry, but the terms do not match anything in the table. " [1] -- DLL ^{.. T} 18:43, 20 October 2006 (UTC)

you forgot thousands --WikiSlasher 07:45, 25 October 2006 (UTC)

Geometric growth explained

Very good explanation of exponential growth ... another phrase used loosely and i suspect wrongly in the financial community is "geometric growth". I am not seeing that in wiki so far, so i think that would be a positive contribution. Thank you.

Mikeym1 15:53, 19 October 2006 (UTC)

Wouldn't that be the same thing? —Bromskloss 16:08, 19 October 2006 (UTC)

Our article on geometric progression notes that a geometric progression with common ratio $\notin \{0,\pm 1\}$ shows exponential growth. — Lomn 20:26, 19 October 2006 (UTC)

A spiral function

Is there any nice parametrized function that gives a "tight" spiral when applied to integers? That is, I want this:

21 22 23 24 25 26
20 7  8  9  10 27
19 6  1  2  11 28
18 5  4  3  12 29
17 16 15 14 13 30
36 35 34 33 32 31

That is, f(1)=(0,0), f(2)=(1,0), f(3)=(1,-1), f(4)=(0,-1), and so on (obviously it would be two functions, one for x and one for y). I'd appriciate it if someone atleast pointed me in the right direction :) 83.250.208.83 20:36, 19 October 2006 (UTC)

When I was coding an Ulam spiral generator, I too looked for such a thing. I ended up needing a "square sine\cosine" function (that generates a square instead of a circle), which proved to be too difficult. In the end, I just adopted a much simpler walker algorithm (that is, you have a cursor that walks the plane in one of the four directions, and you just make it turn left\right after N moves). I'm not sure what are your purposes with this, but if you can implement the walking, I'd suggest you to do it instead of the parametric. ☢ Ҡi∊ff⌇↯ 20:58, 19 October 2006 (UTC)

Well, looking at the lengths of the "sides" (straight length segments of the spiral), you're getting 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6... So, if you want to figure out what layer of the spiral a particular number is in you can make a comparison against the nearest sum of natural numbers (x2). Once you can do that efficiently, it shouldn't be too difficult to solve the rest of the problem. - Rainwarrior 00:21, 20 October 2006 (UTC)

This node Re: Spiraling integers shows a simple formula for the inverse of this function (which maps from the positions to the integers), see the whole thread for the complete discussion. – b_jonas 11:38, 20 October 2006 (UTC)

Alright, I've been playing with this idea and now I can present you to... the square sine function!

${\mbox{sinsq}}(x)=\tan(x)\operatorname {sgn}(\cos(x))((\operatorname {sgn}(\cos(2x))+1)/2)\ +\ \operatorname {sgn}(\sin(x))((\operatorname {sgn}(-\cos(2x))+1)/2)$ read that as "sinsk"

Where sgn is the very useful sign function. The square cosine is, just like with the originals:

${\mbox{cossq}}(x)={\mbox{sinsq}}(x+\pi /2)$ (read it as "cosk")

Now that you have these available, it shouldn't be too hard. All you have to do is tweak a spiral formula out of it. You could do this by using the a ceiling\floor functions and\or modular arithmetic. Also, remember that you'll have to work out the proper angles giving the radius as well (each new spiral layer has an extra amount of squares: $L_{1}=1$, $L_{2}=3^{2}-1=8$, $L_{3}=5^{2}-3^{2}=14$, $L_{4}=7^{2}-5^{2}=24$ etc. There are other little issues you'll have to deal with it, but I'll leave that to you.

But I still think you should use the "walking" algorithm I suggested earlier. It just can't get better than that, and you could easily turn it into a function, albeit not a mathematical one. But that depends of what you are trying to do, exactly. Anyway, good luck. ☢ Ҡi∊ff⌇↯ 13:57, 20 October 2006 (UTC)

Those are great answers! I think I'll be able to work something out from that. Thanks! 83.250.208.83 02:26, 21 October 2006 (UTC)

Impossible mathematical shape?

I found this drawing while surfing Wikipedia: http://upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Autocunnilingus.jpg/651px-Autocunnilingus.jpg

Please just ignore the lady in the drawing and look at the curved object on the shelf behind her. I swear that once in a math book I saw an illustration of this shape, where the tube at the bottom becomes the hollow space at the top, and I have forgotten what mathematician was being talked about or what the name of this object is. I think I might have been reading a Keith Devlin book with illustration plates...

Unfortunately the illustration has a solid line where the bottom curved tube meets the hollow space so I'm not sure if it's the same thing but it is very similar.

Can you help me find more information about that object? Is it really physically impossible to make? Please email EMAIL REMOVED because I am new to Wikipedia and might not find my way back to the right page here to check answers... thanks!

Looks like a Klein bottle. Fredrik Johansson 21:11, 19 October 2006 (UTC)

Here are a couple Calvin Klein bottles: [2] :-) StuRat 22:28, 19 October 2006 (UTC)

It is indeed a Klein bottle (that's a point on the geek-scale right there, seeing a picture of that and noticing the Klein bottle in the background ;). A Klein-bottle is a four-dimensional object, so it's not possible to build a proper one in our lame three-dimensional world, but when you die and go to the after-life (which is, as we all know, four-dimensional) you'll be able to build it. Cheers! Oskar 22:35, 19 October 2006 (UTC)

It's quite a cute pun, given the subject matter. Dysprosia 07:08, 20 October 2006 (UTC)

The central image is obviously an allusion to Ouroboros, the serpent that encircles the World with its tail in its mouth. We may assume that the object in the background is intended to resonant with that concept and excite topologists, who might otherwise find the main image, a mundane torus, uninteresting. And yes, the background object is one of the standard immersions of a Klein bottle — a torus with a twist — in 3-space.

Take a cylindrical strip, snip it apart, give one end a half twist, and glue it back together. This produces a Möbius strip, used by stage magicians in a trick known as the Afghan Bands. The original strip had two sides, but the twist has joined them into one. Likewise, the original strip had two edges, both circles, but the twist has also joined those into one.

If we curl the original strip inward and glue its two edges together, we get a torus. The Möbius strip has only one edge, but it is twice as long, containing both edges of the original strip. We can glue this one edge to itself, connecting the same points as before, to seal the edge and produce a Klein bottle. Konrad Polthier has an article with lovely illustrations.

Like the Möbius strip, the Klein bottle has only one side. That is, it is a “closed” surface (no longer having an edge) for which there is no distinction between inside and out.

However, we cannot physically produce a Klein bottle in 3-space without having it intersect itself somewhere. That said, Cliff Stoll makes nice glass models, and sarah-marie belcastro and others have made softer models — by knitting. --KSmrq^T 08:06, 20 October 2006 (UTC)