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May 8

Möbius Strips

Why do amazing things happen when Möbius strips are cut in half? —The preceding unsigned comment was added by Metroman (talk • contribs) 00:48, 8 May 2007 (UTC).[reply]

Möbius Strip =).--Kirby♥time 02:33, 8 May 2007 (UTC)[reply]

The answer requires a study of topology.

If we look at an ordinary cylindrical strip, such as that given by

x^{2}+y^{2}-4=0,\quad -1\leq z\leq 1,\,\!

we find that it has a bottom edge (at z = −1) and a top edge (at z = 1), which is unremarkable. If we slice it along z = 0, we get two similar strips, each half as tall as the original.

Instead take that original strip and slice it along the line x = 0, y = 2, giving a ribbon we can unroll to look like a long rectangle. If we give it a half-twist along its length and glue the cut ends back together, we create a different topology. Now we have only one edge. Stranger still, the surface has lost topological orientability. Now if we slice the twisted and glued ribbon — the Möbius strip — along its original center line, we do not separate the surface. See the article and its links for more discussion, and try a Web search for "Afghan bands". --KSmrq^T 02:44, 8 May 2007 (UTC)[reply]

Mollusc Theory

Please can somebody explain this newton theory in simple terms and what it means to everyday dimensions? If there is evidence that it is based on scientific fact and not a poor 'theory of everything' notion? Also if there is connections with Micheal Angelo symetry based work and also DaVinci and Newton I hear. I know very little but have come accross it in my work recently and need enlightment on all of the above I have heard about, so anyone please feel free to knock yourself out...... —The preceding unsigned comment was added by Harveyvics (talk • contribs) 14:55, 8 May 2007 (UTC).[reply]

Is there a relation between the heading Mollusc Theory and the question? By "this Newton theory", do you mean Newton's laws? In that case this is the wrong reference desk; you should pose the question at Wikipedia:Reference desk/Science. If you mean Calculus, then this is the right desk, but we won't be able to help you much with these questions. Science is not about facts, but about discovering patterns in observations; the latter are the only "facts", but they are not "scientific" facts as such. Patterns that conform to observations are called "laws"; such laws are only good as long as it lasts – until new observations are made that cannot be explained with the existing laws. Then some genius (such as Isaac Newton) may come along and show us a better law. Mathematics – the discipline to which calculus belongs – is also not concerned with "scientific facts" (whatever that may be), but is about the hard consequences of precise definitions and assumptions – independent of the question whether these assumptions have a meaning in reality. I'm unaware of connections to Michelangelo and Leonardo da Vinci. If, after reading the articles linked to above, you have more specific questions concerning a mathematical topic, please do not hesitate to come back here. --Lambiam ^Talk 18:14, 8 May 2007 (UTC)[reply]

I'm wondering if this question has something to do with the golden ratio, which is often mentioned in connection with the spiral of a mollusc's shell, and other things in nature, and at times various artists have been interested in this number (I don't know specifically about Da Vinci or Michelangelo)? - Rainwarrior 04:54, 9 May 2007 (UTC)[reply]

After taking a look, I saw that Da Vinci is specifically mentioned in the golden ratio article as well. - 06:17, 9 May 2007 (UTC)

What's the formula?

What formula will let me convert the first number to the second number, if I know the following pairs? 7 converts to 16, and 0.75 converts to 3, and 0.36 converts to 2. I'm looking for the best fit, assuming it's a rather straight line. If it's wonky, then I guess we need some sort of parabolic or nonlinear formula.--Sonjaaa 22:46, 8 May 2007 (UTC)[reply]

A friend answered for me... y = 1.3 + 2.1*x

Thanks!--Sonjaaa 23:12, 8 May 2007 (UTC)[reply]

For those who have similar problems, you want to treat the information given as ordered pairs. Assuming the pairs are linear, all you have to do is find the slope of two points and substitute a point into the line equation y = mx + b to find b. Splintercellguy 00:27, 9 May 2007 (UTC)[reply]

The formula your friend gave you is the line of best fit for the given data. The fit is not perfect, because the three points do not lie on a perfectly straight line when you plot them on graph paper, but it is close. In general, infinitely many formulas will give a perfect fit, so you want a simple formula and perhaps not always a perfect fit – if you know the data already has an error, it is silly to want to have a perfect fit. In this case, in which the data are almost on a straight line, the line of best fit may be the best choice. However, three data points is not a large number, and two of these are close together, and then already because of that the three points are almost collinear, so it should be no surprise that you get a good fit with a linear equation. If you can tell us more about the source of the data and the purpose of the formula (interpolation?), we may be able to give a better answer. --Lambiam ^Talk 03:51, 9 May 2007 (UTC)[reply]

Vectors

Hi,
Just a quick question, if you have two vectors such that a.a=c.c does this mean that they are equal? Many thanks, --Fir0002 23:41, 8 May 2007 (UTC)[reply]

Assuming by "." you mean the dot product, the answer is no. What can be said however, is that they have the same norm, since an inner product gives rise to a norm by taking the square root (this can always be done, because of the properties of inner products). Phils 00:35, 9 May 2007 (UTC)[reply]

In a one-dimensional real vector space with the dot product u·v = u₁v₁, the answer is already no. For example, let u = (−2) and v = (+2). For a three-dimensional example, let u = (12,15,16) and v = (0,7,24). Although we might not have anticipated the one-dimensional failure, in three dimensions our intuition should have alerted us, because the dot product of a vector with itself reduces three numbers to one; something must be lost. --KSmrq^T 04:24, 9 May 2007 (UTC)[reply]

Just a quick side point, in terms of combining 3 numbers into 1 number and something must be lost, he was talking about real numbers, not, say positive integers. It's trivial to have a injection from vectors with integral components into the whole numbers (no loss of information) using exponentiation of prime factors. I suppose also possible with vectors of rationals, etc, using Gödel Numbering. I'm not sure the best way to express the idea that combining 3 reals into 1 cannot be done without loss of info. Seems like it was a very basic theorem in topology or something about continuous mappings between spaces. It's been too long since I've been in a theory class. I'm trying not to confuse the issue, I was just saying "combining three numbers into one cannot be done without loss of information" is not true for all kinds of numbers. But even if such an mapping is possible, the utility of such an encoding may be quite narrow, and more than likely is not related to matrix algebra.

~~To reiterate KSmrq's point, something MUST be lost when mapping 3 reals to 1.~~ If we know additional info about the mapping, we may be able to prove that for any number type, mapping a multidimensional vector of such numbers to a single number cannot be done without loss of information.

My apologies if I was not able to explain this in an understandable way. I generally don't feel like commenting after someone like KSmrq because the man (or woman?) seems to know what he's talking about and (in some cases) any additional comment I may give may confuse the issue. I'm even wondering if this posting is helpful. I hope it is.Root⁴(one) 16:41, 9 May 2007 (UTC)[reply]

You can certainly combine real numbers into one real number: just interleave the digits. π, e, and φ become 321.176411188520983.... This isn't bijective (since choosing 0.999... or 1.000... affects the result), but it does not lose information. What you can't have is a continuous (or at least smooth?) mapping that doesn't lose information. --Tardis 19:48, 9 May 2007 (UTC)[reply]

Wow, I see. Thanks for correcting me. Root⁴(one) 20:27, 9 May 2007 (UTC)[reply]

We don't need smooth: you can't continously inject Rⁿ into R^m for n>m. One proof is that such a map would give a continuous injection of S^m into R^m, which is impossible by Borsuk-Ulam.

Of course, none of this changes the fact that KSmrq's original point is a very sensible intuition, especially for linear algebra where dimension tends to be well-behaved, but like any intuition it needs to be checked. Algebraist 21:28, 9 May 2007 (UTC)[reply]

The dot product, being symmetric and bilinear, is equivalent to a quadratic form, so is both continuous and smooth. But, yes, sometimes our intuition gets upset; see Peano curve. --KSmrq^T 23:04, 9 May 2007 (UTC)[reply]