Strange

That is obviously nonsense. F`or example, it is quite straightforward to calculate the ratio of radius to circumference of a circle on the surface of a sphere. Or the angles of a triangle. The OP seems to have missed out on several centuries of non-Euclidean geometry. ("No, I understand all that. I just can't do anything with it.")

You seem to be confusing facts (flames, gravity, evolution, etc) and the theories that attempt to explain them. The former incontrovertibly exist (you don't really need to "prove" facts). The latter are never proven. It is the only model that fits all the evidence. If you have an alternative model, please feel free to present it, the predictions it makes and how well they match observation. Neither of these are relevant to expansion. Perhaps you should learn a little bit more about the theory you are criticising?

Which technical sense is that?

So, how do you know something is being missed? That isn't what I said.

Why do you think religion has any effect on science? That is just nonsense. People were not savages 1,000 years ago (by any reasonable definition of savage). People have been engaged in rational and evidence based study of the world for many thousands of years. (Euclid and Eratosthenes come to mind immediately.) Science has developed incrementally over thousands of years. The development of scientific methods is much earlier than that. Even if you think that science began with Francis Bacon you are about two centuries out. Then why are you constantly making anti-science comments?

If you have any evidence to support that (or an alternative explanation for all the evidence supporting an expanding universe) perhaps you could present it. It might be more convincing than baseless assertions of your opinion.

Do you have any reason for suggesting that?

You keep making it sound like it is other people's fault that you are not able to work out what this function is. (Rather than a "phobia" I assume no one else cares about this function that bugs you so much. Why would anyone else waste any time on it?) Interesting statement from someone who can't do the math. Given that, how do you known anyone is missing anything?

Why not go back to the author of the simulation and ask him? He clearly knows. I'm not sure why you think anyone else is going to take the time to reverse engineer it for you.

They are "collected" by an antenna. In older (analog) receivers a variable capacitor is one way of filtering out the frequencies of interest. That is a common frequency for FM but it is not used by all receivers. You have missed out the entire demodulation stage Which, for digital and spread spectrum signals, can be pretty complex!

I disagree with several of your assumptions. Firstly, even if lengths are quantised at some scale (so that things cannot be any smaller than that size) I don't see any reason why that should imply anything about the extent of the universe. Also, the statement that "everything that has a beginning must have an end" is clearly wrong. For example, take the natural numbers: they have a beginning (0 or 1, depending on definition) but no end. The same could apply physically: if the universe is infinite, then you could go on a journey that has a start (your home, for example) but has no end.

What does that mean? We are discussing whether the universe is infinite in extent. Where does "increasing scale" come into it?

On the other hand, some things might just be impossible. In any timescale. I think Anonymous is being hopelessly naive.

Black holes are smaller than the universe, so I'm not sure how that is supposed to work. Stars have less mass than the universe so, again, I'm not sure how that is supposed to work. Black holes only expand when matter falls into them. After a while there is no more matter nearby and they stop expanding. All the matter falls into the singularity (as far as we know). None of this appears to be similar to the universe we see around us. To be more specific, the singularity at the start of the universe is, by definition, in our past. The singularity in a black hole is in the future (once you have crossed the event horizon). The universe is nothing like a black hole (and never was). Also, we have an explanation for the expansion of the universe (General Relativity). What does your idea add to that?

That is a completely different thread which I had not seen. It doesn't, at first glance, appear to have any connection to the subject of this thread. (Other than referencing the same animation.) But I see it does answer the questions about which plane is which (my guess was wrong). I don't have the time (or interest) for trying to work out the relationship between those angles. But I am surprised that it is so much of a challenge. It looks like it should be fairly easy.

Representations of vectors. I was not aware that was what you were attempting to define (your first post doesn't say that). That looks like a fairly straightforward problem in trigonometry. The upper number is how far round the circle you are. I'm not sure what the lower number is. What is the "angle between the longitude and tangent planes"? I assume the "longitude plane" is the green one? And the "tangent plane" is the yellow one? (Even though it isn't at a tangent.) What group of functions? I find that hard to believe. In your initial post this was all about direction.

That is one way of describing what a vector means. Another is to define it in terms of three (or whatever) lengths in the coordinate system, for example [x, y, z]. You can equally well define it in terms of three angles, [a, b, c]. There are functions to convert between these (and presumably other) representations. It still isn't clear to me what other representation you are trying to define. If you could define it, then it would be straightforward to define the function mapping from it to one of the others (which is, presumably, the function you are grasping for). In the first description (magnitude and direction) the direction is described [somehow] in terms of the three (or whatever) axes of the coordinate system. While the magnitude is a scalar. (I think that is what you are saying.) If you turn this around and make the "direction" a scalar, then you will need three values for the "magnitude". So you end up with something like [x,y,z,theta]. But this seems redundant to me. On the other hand, it does remind me of the homogeneous coordinate systems we use in 3D graphics where a vector in 3D space is represented by [x,y,z,w], where w is a sort of scaling factor (if you normalise the vector, it becomes the magnitude). This is used because it simplifies the maths by allowing you to use the same format for all matrix transformations. I guess you could come up with an equivalent representation using angles (homogeneous polar coordinates?). That might be close to what you are looking for?

I'm not sure what context you are thinking of here. But there isn't really a place where electric current interacts directly with the sound wave. There are two three main components for amplifying sound: a transducer to convert mechanical movement to changes in electric current or voltage (a microphone), an electronic circuit that will use those small changes to control a larger voltage or current change (an amplifier), and a transducer to convert that back to mechanical movement (speaker or headphones). There are many types of microphone. They can operate by, for example, changing resistance and hence the current through a circuit. Or it can move a coil in a magnetic field, or change the spacing in a capacitor. The amplifier consists, at the simplest level, of a transistor where the changing input current on the "base" terminal controls how much current flows through the "collector" and "emitter" terminals. The transistor has a gain which defines how much larger the output current change is that the input. Real amplifiers are much more complex for reasons of efficiency and accuracy. The speaker is typically a coil in a magnetic field: the changing current through the coil causes the speaker to move. The current is larger than that from the microphone and so the movement of the speaker is greater and, therefore, louder. Edit: Just saw "radio" in the title. Not sure what part of the question relates to radio ... Radio is transmitted as electromagnetic waves. You end with two systems like I described above. At the transmitter, instead of the speaker, you have an antenna (crudely speaking). And then you have a receiver where, instead of the microphone, you have another antenna which picks up the weak radio signal from the air. (This is all horribly simplified but I hope it helps. Or, at least, allows you to ask some more questions!)

I do! At a recent tradeshow I was talking to the guys from Everspin who produce modern magnetic memory devices. Again, combining many of the advantages of different memory types: non-volatile, low-power, fast access times, random access. But density seems to be a problem (compared to DRAM, anyway).

As I say, I think you can treat the input an output of the cos function as just "values". But as soon as you try and treat the angle as being related to direction, then it is the angle (or direction) relative to some coordinate system. I think the problem is that no one understands what you are trying to do well enough to write it down in mathematical terms.

I kind-of see what you are getting at. But the reason a length doesn't require a coordinate system is because it is a scalar value. But, I'm not sure that is even true: in order to define a length, you need to specify the two end points, which requires a coordinate system. The very fact that you have units (metres, parsecs, inches) implies a coordinate system. Similarly, there seem to be contexts where you can treat an angle as a purely scalar thing, for example the phase of a signal. But then it isn't defining a direction. (Actually, that isn't true either. The angle defines a direction in a 2D phase space and sometimes it is easier to convert your signals to that coordinate system.) As soon as you want to talk about direction, then you are no longer referring to just the magnitude of the angle, but its relationship to the basis of the coordinate system.

http://www.livescience.com/50096-chameleons-color-change.html But why is this in Speculations?

That reminds me of the riddle: An explorer walks one mile due south, turns and walks one mile due east, turns again and walks one mile due north. He find himself back where he started. Where is he?

The math is the theory. If the theory is different then how can the math be the same? If you are not simplifying the math then you are not simplifying anything. (Except, perhaps, the story you tell yourself to pretend you understand.)

Can you show that the results calculated in this way match experiment to the same high accuracy as the standard method?