Wikipedia:Reference desk/Archives/Mathematics/2008 October 20

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October 20

On hyperchains

I've often heard it said (most recently by xkcd in the comic-off with the New Yorker) that it is impossible to loop things through one another in $n>3$ dimensions. Assuming this is true, and that I understand the definition (that two objects are linked if they are not intersecting but cannot be rigidly translated and rotated to an arbitrary separation without intersecting), why doesn't this generalization of the torus work?

${\begin{aligned}s(w)&:=1-(w/c)^{2}\\sb^{2}&=z^{2}+\left({\sqrt {x^{2}+y^{2}}}-sa\right)^{2}\end{aligned}}$

Admittedly, it tapers to 0 thickness near $w=\pm c$, but what's wrong with the fundamental idea of interlocking two of these so that shifting them in w can't free them because one gets too small? --Tardis (talk) 02:48, 20 October 2008 (UTC)

I think that the sentence it is impossible to loop things through one another in $n>3$ dimensions refers to special situations, most likely to objects that are immersed in $3$ dimensions, meaning that you can unlink them in the larger dimensional space (this has an immediate application in comix). Also, two closed curves (that is embeddings of S^1 ) in the $n>3$ dimensional space cannot be linked, as a simple transversality argument shows. Otherwise, in general, there are of course examples. --PMajer (talk) 07:07, 20 October 2008 (UTC)

You may find Knot theory#Higher dimensions interesting. In particular, all knots in more than 3 dimensions are trivial and likewise links (as PMajer says). To get interesting knot theory in higher dimensions you need to use higher dimensional knots (eg. knotting 2-spheres rather than circles (1-spheres)). --Tango (talk) 10:50, 20 October 2008 (UTC)

It's the "rigidly" bit that's giving you trouble. In the comic he's talking about flexible ropes, and the corresponding topological object is the homeomorphic image of a circle, up to ambient isotopy. In other words, you can unlink any two flexible loops of rope (provided they're either very thin or very stretchy) in four dimensions. Black Carrot (talk) 19:35, 20 October 2008 (UTC)

Reflection

Hello. What is the equation of the reciprocal function reflected upon y = 2x? I reflected the asymptotic lines (x and y axes) of $y={\frac {1}{x}}$ upon the reflection line and sketched the new function. It looks like a hyperbola, which puzzles me after reading the article. Thanks in advance. --Mayfare (talk) 03:39, 20 October 2008 (UTC)

The graph of $y={\frac {1}{x}}$ is a hyperbola - to see its formula in a more recognisable form, rotate it through 45 degrees to get $x^{2}-y^{2}=2$. So it is not surprising that it is still a hyperbola after reflection. Gandalf61 (talk) 05:44, 20 October 2008 (UTC)

I forgot to mention that the equation is in the form of 12x² + bxy + cy² + d = 0; $b,c,d\neq 0$. Sorry. --Mayfare (talk) 01:31, 21 October 2008 (UTC)

If I understand your query, you wish to write an equation for those points (x',y') that are reflected images of all points (x,y) satisfying a certain equation F(x,y)=0. So, first you need to write a system for the reflection of the form x=ax'+by', y=cx'+dy'; then you can write your equation as F(ax'+by',cx'+dy')=0. To find a,b,c,d you can use the fact that if (x,y)=(1,2) then (x',y')=(1,2), and if (x,y)=(2,-1) then (x',y')=(-2,1). If you plug these values of x,y,x',y' into x=ax'+by', y=cx'+dy' you will get 4 linear equations whence, you can determine a,b,c,d. --PMajer (talk) 16:23, 21 October 2008 (UTC)

Help

Does 5+0=908235482947578679402? —Preceding unsigned comment added by Banna ant (talk • contribs) 16:24, 20 October 2008 (UTC)

No. ~ mazca ^t|c 16:35, 20 October 2008 (UTC)

It does modulo 908235482947578679397. --Tango (talk) 18:59, 20 October 2008 (UTC)

He specified equality, not congruence. 81.187.252.174 (talk) 12:40, 22 October 2008 (UTC)

Also modulo 7 if one wants to go down that road. Dragons flight (talk) 19:44, 20 October 2008 (UTC)

Yeah, I decided minusing 5 was easier than finding a smaller number that worked! --Tango (talk) 20:17, 20 October 2008 (UTC)

or 1288959887 or 100660949533. (Wow.) —Tamfang (talk) 02:43, 21 October 2008 (UTC)

Also: true, modulo 1. Eric. 131.215.45.87 (talk) 22:39, 20 October 2008 (UTC)

Yeah. Or you could interpret the numbers in the equations in base 0.5014 (approx), see Answer to Life, the Universe, and Everything#Base_13. – b_jonas 16:44, 26 October 2008 (UTC)

the mathematician's personality.

without meaning to start a debate, would you say excelling in maths would be associated with developing a certain type of personality? I mean that the same person, Joe, 17, is considering studying maths intensively and studying literature. If he does study maths, and excels, will he develop a certain type of personality concurrently? (On average, and versus the "control" of doing the other thing). Thanks! —Preceding unsigned comment added by 94.27.161.108 (talk) 19:53, 20 October 2008 (UTC)

That's a very difficult question to answer. There are certainly personality traits associated with mathematicians (although the correlation isn't very strong - there is plenty of variation), but I don't know how you would work out whether they become mathematicians because of their personality or they developed their personality because they became mathematicians (or, quite possibly, a bit of both). As you say, to get an accurate comparison you need to compare with a control group, and that would require choosing people's careers at random rather than letting them choose, which isn't likely to happen (and if it did, there would be a massive selection bias caused by only certain types of people being willing to surrender their free will in such a way). --Tango (talk) 20:07, 20 October 2008 (UTC)

While there is a lot of variety in people that study math, I have noticed in my peer group (graduate students in math) that the people who excel are usually interested in math. I don’t think that’s as trivial an observation as it sounds though, because I’ve met plenty of people that are studying math and seem to not be interested in it.

In fact, one of the major purposes for REU’s in the U.S. seems to be to help undergraduate math majors figure out whether or not they actually would enjoy doing math for a living, before they make the decision to commit several years of study toward that goal. GromXXVII (talk) 10:59, 21 October 2008 (UTC)

I think maths appeal to people who like a sense of the absolute in the universe. That there are answers that does not depend on the political environment. 122.107.147.49 (talk) 08:53, 21 October 2008 (UTC)

The universe? What has mathematics got to do with the universe? Algebraist 10:39, 21 October 2008 (UTC)

We have a whole article on the subject: Universe (mathematics). Our universe just isn't the same as the one discussed on the Science ref desk! --Tango (talk) 14:39, 21 October 2008 (UTC)

My own experience from doing mathematics, rather that merely reading mathematics, is that it improved my ability to concentrate and my patience against other people. I think there is no such thing as an impatient and short-tempered mathematician. Bo Jacoby (talk) 10:49, 21 October 2008 (UTC).

John Forbes Nash was a Nobel prize winning mathematician who developed schizophenic personality. Cuddlyable3 (talk) 09:47, 22 October 2008 (UTC)

I think it's generally agreed that he would have been schizophrenic no matter what. Black Carrot (talk) 19:48, 22 October 2008 (UTC)

Yes, and some say that mathematics even helped him in the long painful period of his desease. For sure he gave contributions to mathematics also after he recovered. Great mathematician, and great man. --PMajer (talk) 21:13, 22 October 2008 (UTC)