Wikipedia:Reference desk/Archives/Mathematics/2010 April 13

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April 13

Three Dimension

If P (a,b,c) and Q (x,y,z) are two points in three dimensional space, how can we find equation of line PQ.

There are many methods. See system of linear equations. -- kainaw™ 03:41, 13 April 2010 (UTC)

I think the simplest method is the vector equation: r = OP + λPQ where r is the position vector of any point on the line, OP is the vector from the origin to P, and PQ is the vector from P to Q. The parameter λ indicates how far along the line we have travelled. λ = 0 at point P, λ = 1 at point Q etc. Other people will have different preferences, and the various methods are all equivalent. Dbfirs 08:04, 13 April 2010 (UTC)

If you just need a Cartesian equation, you can eliminate λ to give: (x - x₁)/a = (y - y₁)/b =(z - z₁)/c where the co-ordinates of point Q are (x₁, y₁, z₁) Dbfirs 08:13, 13 April 2010 (UTC)

Complex logarithm

S is an arbitrary set (being a sub-set of the complex plane), and L is a function satisfying: exp(L(s))=s for every s in S. Does the continuity of L in S allow us to infer that L is also differentiable in S?

Note that if S is a domain then the answer is trivially positive.

Note also that L is differentiable in S if and only if:

For every s in S there exists a number n (intended to constitute the value of L'(s)), such that for every ε > 0 there exists a δ > 0 such that every x in S which satisfies $0<|x-s|<\delta$, satisfies $\left|{L(x)-L(s) \over x-s}-n\right|<\varepsilon$.

77.124.141.144 (talk) 13:21, 13 April 2010 (UTC)

By "convergent series", do you mean "convergent sequence"? Michael Hardy (talk) 15:33, 13 April 2010 (UTC)

yes. 77.124.141.144 (talk) 16:00, 13 April 2010 (UTC)

If S is arbitrary, it doesn't necessarily make sense to talk about whether or not L is differentiable - you want the domain to be open. In this case, try proving that on each connected component of S, L is given by some fixed branch of the log function (making it holomorphic). If not, you should be more clear about what you mean by "differentiable." 136.152.163.70 (talk) 16:29, 13 April 2010 (UTC)

L is differentiable in S if and only if:

For every s in S there exists a number n (intended to constitute the value of L'(s)), such that for every ε > 0 there exists a δ > 0 such that every x in S which satisfies $0<|x-s|<\delta$, satisfies $\left|{L(x)-L(s) \over x-s}-n\right|<\varepsilon$.

77.124.141.144 (talk) 17:33, 13 April 2010 (UTC)

If L is continuous at s then for every ε>0 there exists δ>0 such that |L(s)-L(x)|<ε for all |s-x|<δ, which means there's a δ for which L(x) is always in the same branch as L(s), so in that neighborhood L is holomorphic and satisfies that definition you provided for differentiability. Rckrone (talk) 18:48, 13 April 2010 (UTC)

How can you determine that "L is holomorphic" when L is unnecessarily defined in a domain (=connected open set)? 77.124.141.144 (talk) 20:17, 13 April 2010 (UTC)

Ok to be more accurate L is the restriction to S of a function f that's holomorphic in a neighborhood of s (f can be set equal to an appropriate branch cut of the log function inside the disk around s). That satisfies the definition you provided for L being differentiable in S at s. Rckrone (talk) 20:51, 13 April 2010 (UTC)

Thank you. 77.124.141.144 (talk) 21:27, 13 April 2010 (UTC)

Reuptake of a question 2d travel distance and energy expenditure vs. 3d - ramblings

Hi )again(

I looked up an old question I once posted and saw that I wanted to pose some insert extra questions and explanations to my point that was never answered.

I have pasted the original question and answers/comments here after the line... can someone help me to elaborate on my last questions/postulations everything once more.

I'm still in the process of rereading and trying to pose a better question with the 'old' comments in mind, but till then I only want to ask you for a bit help to explore the subject more.

The headline could also be 4d vs 3d creatues

Thanks....

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About 2 D creatures traveling on a 2 D plane seen from a 3 D point of view I've got this strange theory that I conjured out of thin air som 15 years ago, have there been anybody else than me thinking about this stuff.

I postulate hereby that C a 2D creature's fastest way (like in shortest/less resource expenditure) of traveling (in an XY plane) from point A to point B would be to travel in one direction (X or Y) (not distance AB) until it would be at the distance AB - (A - (X or Y)) = B and then the distance B.

This way the creature C would only have to change direction of travel (90 degree change) once.

Try to think about this and and tell me how much further would 2 D creature have to travel compared to a 3D (capable) creature, my answer is squareroot 2 (appr. : 1.4) times more.

Reason: while the 2D creature could change direction of travel (X or Y) for every second or mm (or whatnot) that it went, this would take even more time/resources ..stop/change direction etc.... than it would to just travl the distance in the X direction and then after that in the Y direction....one start/stop/start/stop

Relate this to a 3D creature like us in a 4D world....and the answer would be 0.5 times the value of pi.....and bring in the double slith photon/electron experiments and other relativistic points... anybody got any thoughts about this...?

Could this be elaborated a bit more? [Azalin1999]

Uh. Perhaps you should phrase your question better. Are A and B points or lines? You use them as both. What's X and Y? And if it travels from A to B in a straight line, it doesn't ever need to change its direction. --BluePlatypus 18:05, 15 March 2006 (UTC) I think maybe I see what he's getting at, but it doesn't really have anything to do with 2 vs. 3 dimensions. It sounds to me like he's describing the Manhattan distance. At least in the first part where he's talking about 2 dimensions. If you compare the Manhattan distance (let's call it m) with the ordinary, Euclidean distance (d), between any two points in a Euclidean plane, it's not too difficult to show that . But I have no idea where he's getting the factor from for three dimensions. In three dimensions, you have . More generally, in n dimensions, . Nor do I see what it has to do with quantum mechanics or relativity. Chuck 18:40, 15 March 2006 (UTC) Hi again I'm the one who phrased the question. A little note about myself. Never did score high in maths and equations but I'm really curious about this 'theory'. Much thanks for the link to the 'manhattan distance' article, I think I remeber reading about it some years ago. My native tounge is Danish so that's my excuse for the inaccurate phrasing.

I originally made some drawings on a XY crossed paper to illustrate the theory. And I had a really hard time to explain my arguments for those I presented it to (mostly indiferrent friends). And I still have difficulties atm. because the thing I would like to verify is that it would be possible to apply something like this 'manhattan distance' to distances in a 3D world seen from the perspective of a fourth dimension. Which should be possible. Basically like --BluePlatypus went about distances in an euclidian plane...

The distance 1/2 Pi was derived from projecting a imaginary path (distance traveled) through 4D space compared to what a 3D creature would have go by not using the 4th dimenson...like a 2D creature can't (see, thus) travel the path that a 3D creature can. In regards to the 'Manhattan Distance' this would be akin to a race between a cab an a helicopter going over the cityblocks.

Too bad your friends are indifferent; this stuff is fun! As Chuck wrote above, yes, you can define Manhattan distance in 3D, and you don't even need a 4D perspective to do it. I'm not sure where the pi is coming from. As for bringing in modern physics like quantum mechanics and relativity, let's just say that they're not really compatible with a Manhattan metric. If you lived in a world where the physics was affected by a Manhattan metric, you would easily be able to experimentally determine the coordinate axes you lived on, whether you're a 2D or a 3D creature. But we've never observed any such special directions in space. Melchoir 19:30, 15 March 2006 (UTC) That's an understatement! :) If all directions weren't equivalent, you wouldn't have conservation of angular momentum, and that'd really yank out the carpet from under physics-as-we-know-it. --BluePlatypus 20:08, 15 March 2006 (UTC) OK, I kind of see where you're going here. First of all, it's important to remember that the factor in 2 dimensions is a maximum; it doesn't hold for all pairs of points. For example, if you're travelling from (0,0) to (5,5), the person constrained to taxicab geometry has to travel 10 units, but the person who can travel in any direction only has to travel units, and the person limited to taxicab geometry has to travel times farther in that particular case. However, if you're traveling from (0,0) to (7,24), the Euclidean traveler has to travel 25 units, but the taxicab traveler only needs to travel 31 units--just 1.24 times further than the Euclidean traveler, which is considerably less than . And of course, if you're going from (0,0) to (13,0), both people travel the same distance. The analagous situation in three dimensions, it seems to me, is someone who can travel in any one of six orthogonal directions (which we might call "north," "south," "east," "west," "up," and "down," but it's important to remember here that these are idealized directions where all north-south lines are parallel with each other and do not meet, as are the east-west lines, and also the up-down lines; and not the directions as defined on the earth, where the "north" lines all meet at the north pole, the "south" lines all meet at the south pole, and the "down" lines meet at the center of the earth). In this case the maximum factor between the distance of the taxicab traveler and the distance of the Euclidean traveler would be , for example going from (0,0,0) to (5,5,5) the taxicab traveler has to move 15 units, as opposed to the Euclidean traveler's units. But as before, that's a maximum: in going from (0,0,0) to (3,4,12) the taxicab traveler goes 19 units to the Euclidean traveler's 13, but . It's possible to choose a pair of points where the taxicab traveler's distance is exactly times as long as the Euclidean traveler's, but that's only because such a pair of points can be found for any number x such that , which happens to satisfy. The "car vs. helicopter" analogy is an interesting way to think about it, but at the same time it's misleading because you don't actually have to move in the 3rd dimension for this to happen. You can have one traveler who can move in any direction, and one traveler who is limited to moving in a set of orthogonal directions, and still have both of them confined to a plane, so it's not really an issue of one person being constrained to two dimensions while another can move in three (or three vs. four, or four vs. five, etc.) Chuck 20:33, 15 March 2006 (UTC) You're Chuck making some good points. But what if we assume that in our (unlimited) reference 3D space we are looking down on the 2 plane (which could be from any arbitary reference point/perspective) let's say head on, straight down.

Let's asume some more - Our little 2D creature can only observe (look) through either the X or the Y, and thus only observe it's destination when its in line with either the x axis of the destination or th Y axis, whichever comes first, depending on which route it took.

From the 3D POV this would satisfy my squareroot 2 clause about a 3D being would be able to go the shorter distance right? Because it can sort of observe the direct line of vision...X & Y combined.

Now if the 2D creature were able to in some way observe the destination it could of course just travel the direct route. But this would only happen after it had moved all the way along either the X or Y axis to be in line with the destination. If this 2D thing were sentinent the percieved distance the 2D being traveled would be? the squared X+Y which is less than the distance it really took if a 3D being measured...also there are 2 routes that are the same distance but starting with etiher going out of the X or the Y axis...

Relate this to a 3D world...(damn crunching)hmm non-euclidian geometry..........Spherical trigonometry

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Thanks again

fractals and harmonics

One phenomenon of a harmonic is that it consists of multiples of the base radio or audio signal only at a higher frequency. Thus the shape of a harmonic of a sinusoidal wave form will be sinusoidal and the shape of a harmonic of a triangular wave form will be triangular. The difference between fractals and harmonics then seems to be the difference in what happens in space and what happens in time. Can anyone elaborate or point me to an article that elaborates on this idea? 71.100.1.192 (talk) 22:55, 13 April 2010 (UTC)

Fractal antenna might be of interest, since that combines both fractals and radio waves. StuRat (talk) 05:52, 14 April 2010 (UTC)

equations that scale and fractals

No matter how large or small the value of x above zero in the equation x^(4/3) one sees the same curve. How is this phenomenon related to fractals? 71.100.1.192 (talk) 23:39, 13 April 2010 (UTC)

It's rather like saying that all parabolas are geometrically similar (which they are). All fractals show some form of similarity at countably infinite discrete magnifications, though none are continuously similar (with uncountably infinite magnification similarities) like the parabola. For some fractals there is exact similarity at repeated levels, for others the similarity is not precise. From a mathematical point of view, structures like coastlines and (woody) trees might be described as "pseudo-fractals" because they are similar at only finitely many magnifications, and the similarity is far from exact, with different structures at each similarity level. Perhaps an expert on fractals can check my claims. Dbfirs 20:55, 14 April 2010 (UTC)