Math Question: Pythagorean Theorem

[Rade]

The Pythagorean Theorem applies to triangles made of three straight lines --------------- with two at a right angle. I have a question.

 

Is their such a Theorem that would hold if each of the three lines were a partial wave function (any type, all types) --^^^^^^^^--, and not a complete line -------------- ?

 

Note that the requirement that a right angle is met because we maintain line segments where we form the right angle. If the theorem prediction c^2 = a^2 + b^2 for the sides of the triangle does not hold for partial waves, would this represent an exception to the Phythagorean Theorem ? I cannot find anything on the topic of creating triangles out of partial waves.

[Qfwfq]

Calling it a triangle doesn't imply the same theorem must hold, despite it being defined in a way quite distinct from the usual.

 

Define the "partial wave" in such a manner that implies each one having its "length" and then figure whether there is any relation at all between the three quantities. Without this, there's not much point asking the question.

[C1ay]

Pythagoras' Theorem is just a special case of the Law Of Cosines....

[Ben]

I cannot find anything on the topic of creating triangles out of partial waves.

I am not surprised. Sorry to be blunt, but your question makes very little sense.

 

Let's try it this way (which may not be the best). Consider the plane [math]R^2[/math] and there define two points, distribution of your choice.

 

Call these points [math]A,\,B \in R^2[/math]. Then there is an infinity of curves (there is a technical definition, but we can just think of these as "lines") that connect [math]A[/math] to [math]B[/math].

 

Now there is an unexpectedly difficult piece of kit that proves that the shortest distance between these two points is what is commonly called a "straight line". I am referring, of course to the variational principle. Look it up if you dare!

 

The important point, however, is this; although every other curve that connects [math]A[/math] to [math]B[/math] may be "real" in some sense, the shortest distance - the straight line - may not actually "exist" in the same sense. In other words, it is a mathematical abstraction, and jolly good luck to it too.

 

So that, given a third point [math]C \in R^2[/math], I can, under the appropriate assumptions about their distribution in the plane, ALWAYS apply the Pythagorean whether or not they are "actually" connected by straight lines or not.

[sanctus]

Rade, to have another type of answer, if you take your partial wave connecting A and B and measure its length (not wavelength but the distance covered by an ant walking on this line/partialwave) x and then you measure the length y of the partial wave connecting B and C, you can then construct a "new (which is wrong because new supposes there was an old one)" triangle with distance A'B'=x and B'C'=y, keeping the same angles as your pseudo-triangle.

Then you get A'C' via pythagora or cosine rule...

 

 

The problem then is that you can have partial waves with diferent wavelengths and amplitudes so that many partial waves have length A'C'...

So this to show in a non-mathematical way as opposed to Ben, why it does not really work or make sense like that.

And also how do you define a partial wave? ----^^^^--- is not enough how many periods of the wave do you need, how long is the straight part in comparison to the non etc...

[Rade]

Pythagoras' Theorem is just a special case of the Law Of Cosines....

 

OK--thanks. I looked at your link and it says this:

 

Let a, b, and c be the lengths of the legs of a triangle opposite angles A, B, and C. Then the law of cosines states :

 

a^2 = b^2+c^2-2bccosA

 

b^2 = a^2+c^2-2accosB

 

c^2 = a^2+b^2-2abcosC.

 

Thus, the "Law of Cosines" by definition relates to the lengths of the legs (a, b,c)--but the Law does not specify that length has to be a straight line (perhaps this is assumed--but it is not stated). If this is an basic assumption of the Law, then perhaps the Law does not apply to waves ??--I have no idea, that is why I ask.

 

So, would the Law of Cosines hold if each length (a,b,c) was an identical wavefuntion of some sort --^^^^^^^^^^-- and not a straight line ------------------ ? Is everyone saying this is not a valid question for a mathematician :phones:

 

I understand a wave can have many different forms, so my next question is, IF the Law of Cosines can be applied to waves, would it hold true for all possible wavefuntions that are mathematically possible ? Of course this question is not valid if the Law cannot be applied to waves for the lengths (a,b,c) of the legs of the triangle.

[Rade]

So that, given a third point [math]C \in R^2[/math], I can, under the appropriate assumptions about their distribution in the plane, ALWAYS apply the Pythagorean whether or not they (points A, B, C ) are "actually" connected by straight lines or not.

OK--let me be clear--you are saying that the Paythagoren Theorem (hence the Law of Cosines)

 

a^2 = b^2+c^2-2bccosA

 

b^2 = a^2+c^2-2accosB

 

c^2 = a^2+b^2-2abcosC.

 

can be applied to any plane created "triangle" that is mathematically possible if the legs of the triangle are a wavefunction ^^^^^^^^ of any form ? If so, then, thank you, you have answered my OP question.

[Rade]

The problem then is that you can have partial waves with different wavelengths and amplitudes so that many partial waves have length A'C'...

So this to show in a non-mathematical way as opposed to Ben, why it does not really work or make sense like that. And also how do you define a partial wave? ----^^^^--- is not enough how many periods of the wave do you need, how long is the straight part in comparison to the non etc...

Yes, I understand this may be a "problem", that is why I asked my question. In short, does the Law of Cosines apply for all the many (infinite ?) "partial waves that have length A'C' as you defined ? This is not clear to me.

[lawcat]

The short answer to your question is yes, but probably not as you envision it.

 

The reason is that the theorem, whether law of cosines ot Pythagorean, applies to scalars or vectors. A wave can be represented as a vector, as magnitude at an angle from some reference point, and only in such form the theorems apply.

 

Remember how waves propagate from the origin or reference. Then a wave can be represented as magnitude of a sine wave at some phase angle. Then you can use these theorems to calculate some phenomena of two waves of different magnitudes and phases converging. Also, since these behave like vectors, rules of vector calculus apply, the dot product, cross product, gradient etc.

 

Otherwise, a wave is represented as a Fourier series and can have a final form of a Fourier transform, whether periodic or aperiodic. And those are analyzed in Fourier form in time domain or in LaPlace form using LaPlace transform in z domain. The easiest way to work with waves is graphically.

[sanctus]

Yes, I understand this may be a "problem", that is why I asked my question. In short, does the Law of Cosines apply for all the many (infinite ?) "partial waves that have length A'C' as you defined ? This is not clear to me.

By the procedure I describe you get a length A'C'. There are an infinity of partial waves which have this length (increasing frequency and lowering amplitude or vice-versa).

[Rade]

OK, thanks to all for the help. I now see how the Law of Cosines must apply to the OP question. It was not a very insightful question that I asked.

 

Does anyone know if "string theory" gives any physical meaning to combining three strings (as waves) into the geometry of a triangle ?