4D existence

[whiteonriceboy]

I'm not trying to start an argument here , but why can we assume that there is a tetraspace at all? No one has ever bumped into a plane with trees and people and stuff on it before, have they? Lines, planes, points, our dimension, and tetraspace all are sensible ideas, but it seems ridiculous to think that they actually exist, doesn't it? People and trees don't just randomly get knocked into Emily's world very much at all, now do they?

I like to wonder about the possibilities of another dimension alltogether as much as anyone in this forum, but thinking sensibly, it can't really be.

[pat]

Sure, we have no proof that there's any way to get out of our three-dimensions.

For my part, I think the 4-D speculations about the creatures and what things look like and how things behave is good exercise though for helping to visualize other higher-dimensional things. There are countless (well, technically "countable") things in mathematics and physics which could be better visualized if we were more fluent with higher-dimensional spaces.

[RQ]

We have 0 proof that there are 4 Dimensional universes. That is up with God or whatever/Whoever/however created the universes. These are just analogies to our extended 3D universe to a 4D one.

[Geosphere]

but why can we assume that there is a tetraspace at all?

The same reason that SpiderMan stories are not about the kid who sits behind Peter Parker in homeroom.

Because if we didn't have that focus, there would be no reason for the existance of the medium. I.E., no Spiderman stories, and no tetra forums.

[jinydu]

There are countless (well, technically "countable") things in mathematics and physics which could be better visualized if we were more fluent with higher-dimensional spaces.

To mathematicians, a very important example of that would be the ability to visualize complex-numbered graphs in their full glory.

To visualize an "ordinary" real-numbered function, f(x), requires only two dimensions. One dimension (usually the x-axis) represents the different possible values of x. The other dimension (usualy the y-axis) represents the corresponding values of the "output", f(x). Thus, we can construct graphs of functions like:

f(x) = x
f(x) = x^2
f(x) = e^x
f(x) = sin x
etc.

However, there is a constraint. Both the input, x, and the output, f(x), must be real numbers.

Mathematicians often work with a more general class of numbers, the complex numbers. These can be expressed in the form a+bi, where a and b are real numbers and i is the square root of -1. As can be seen, it takes two real numbers to determine the "position" of a complex number. Thus, it takes 2 dimensions to represent a complex number. This is still acceptable.

The problem comes when we try to graph a function in complex numbers. It takes 2 dimensions to represent the input, x, and another 2 dimensions to represent the output, f(x). That's 4 dimensions total, (sadly) more than we have available to us.

Since we can only visualize 3 dimensions, we are forced to restrict either x or f(x) to be a real number, thus eliminating one of the dimensions. Just as the prisoners in Plato's Cave (http://faculty.washington.edu/smcohen/320/cave.htm) could only see 2D shadows of real 3D objects, we can only see 3D shadows of 4D complex functions. The actual 4D shape generated by a complex function is made up of an infinite number of 3D slices. We are able to view each of these slices, but never at the same time. The best we can do is program our computers to show us each of these slices, one by one. But even then, it is only a partial victory, since we see a 3D object changing, when the actual 4D shape never changes.

If we could be like Emily (a hypothetical 4D being), we could see these complex graphs in their entirety all at once. In the Cave allegory, it would be like the prisoner who stepped out of the cave and saw things in their true, 3D form. But in this case, the true form of these complex graphs is in 4D.

[Keiji]

There's a flaw in your idea there: You say that output and input are both counted as dimensions. What about a computer monitor? The input for it is x and y, the output is r,g and b. To see that we would therefore "need 5 dimensions". Yet, we see it simply as an image.

[pat]

And, to some extent, that same sort of idea can work for any two-dimensions in and three- (or fewer) dimensions out function. We could graph complex numbers with darker colors meaning more negative real-part and lighter colors meaning more positive real part with more reddish meaning more negative imaginary part and more greenish meaning more positive imaginary part.

But, it introduces some arbitrary biases. Why not make the imaginary the red-green? Why does the input get to be spatial but the output has to be chromatic when technically, they're both spatial? Can you intuitively rotate this around to look at it from different angles?

[Keiji]

Um, no, if you used red for imaginary and green for real, then you would have full red = 1i, full green = 1, full yellow = 1i+1, full black = 0.

That way is better than using hue for imaginary and luminosity for real, because if luminosity is maximum or minimum, hue doesn't affect the color.

[Geosphere]

Step a few places back.

Color IS binary. Colors are varying wavelengths. Wavelength is frequency. Frequency is pulse. Pulse is binary. We break it into 3 primary colors, but its really bandwidth zones.

[Keiji]

Frequency is not pulse. It can be sampled, but it cannot be recorded exactly.

[jinydu]

In any case, using colors to show complex numbers is still only a partial victory. It has the advantage that it can show complete information about the graph within the bounds of the screen. However, it is still not as good as having a graph with four spatial axes. I think that if we took a picture of the graph, rotated the graph, then took another picture, other people wouldn't recognize the two pictures as showing the same object. On the other hand, Emily would probably have no trouble seeing the graph from any angle.

[Geosphere]

It can be sampled, but it cannot be recorded exactly.

Why?

[Keiji]

Because anything in the real world has infinite detail, but stored numbers have finite capabilities.

[Keiji]

Bump.

Can we get back on topic now?

[PWrong]

There is a different way you can graph a function of complex numbers.
You have two graphs. On one graph you draw some points. Then the function changes the points and draws them on the other graph. You don't see the whole 4D shape, but you can get a good idea of what the function does by drawing a grid, a circle, or a smily face.

There was a site with an example of this,
http://www.dcs.gla.ac.uk/~bunkenba/CC/CC.html, but it doesn't seem to be working anymore.

[pat]

And, if you graph complex functions like that, you may be able to get the gist of them. But, there's still no intuitive way to rotate them around at all.

For example, suppose that you had two simple functions of a real variable x with real constants a and b such that a² + b² = 1. Your two "functions" are:

f(x) = x²
g(x) = (a/2) ± √( bx + ( a²/4 ) ) + (b/a²)( (a/2) ± √ ( bx + ( a²/4 ) ) )²

Assuming that I did my calculation correctly, you should be able to plot both of those as you would any function of a real variable. Once you look at those two graphs, you will probably immediately suspect that one is simply a rotation of the other.

Now, if you kept the same real constants a and b but switched to a complex variable x, I think you'd be hard-pressed with either the chromatic-view or the two-independent-planes-view to have any idea that the two graphs were related.

[PWrong]

That second funtion looks pretty similar to √x, except translated a bit. I guess the translation makes it harder, but you're right either way. I can't see much connection with x^2 with the two planes method, although I don't know if there's a chromatic-view anywhere.

But isn't it reasonable to expect that even basic visualisation skills would be harder to achieve in 4D? I expect 4D life would evolve to cope with it, so Emily would be more intelligent than us, and if we could send humans into 4D, they would have to be a well-trained mathematician and genius. Anyway, it's quite difficult for most people to visualise a three dimensional function.

Speaking of complex numbers, I've been wondering about how to integrate complex functions. I know a little bit about integrating the actual functions, but I'm confused as to how this applies to areas.

I was trying to work out the area of the mandelbrot set using calculus. I found the area after one iteration, because it's just a circle, but after the 2nd iteration it becomes a kind of elliptic thing. Basically, I'd like the area inside the curve |Z^2 + Z| = 4 for complex Z. I can't find it anywhere on the internet, because I don't know the correct term for it. Do you know how to do that sort of thing?

[pat]

It is reasonable to expect that visualization in 4-D would have some difficulties not present in 2-D. My big problem with the methods mentioned above is that they have arbitrary discontinuities.

With chromatic viewing, we're clearly saying that some dimensions are spatial while others are chromatic, when in fact, we're trying to show them all as spatial. You cannot intuitively rotate such things so that the chromatic and spatial mix.

With the two-planes method of graphing complex functions, you can't intuitively rotate across planes. It's less of a break than the chromatic case. But, it's still arbitrary. I mean, given a function f( a + b i ) = x + y i, why not graph a, b, and x in three-space and y on a line? Or, why not graph a, b, x, and y each on a separate line?

As for integrating with complex variables, I think you're going to run into problems with the expression | z² + z | = 4 because the absolute value sign brings in a discontinuity. But, I'll think about it some more.

[pat]

Here's an example of how much different it is viewing a function in two-dimensions versus as a chromatic one-dimensional representation or as a pair-of-lines representation.

I suppose, with some practice, one might be able to work reasonably well with the chromatic representation. The big problem with it though is that we only have a finite chromatic range, but we could extend our paper to any size needed (in theory). I cannot picture "darker than black" as well as I can picture "taller than this whiteboard".

[PWrong]

As for integrating with complex variables, I think you're going to run into problems with the expression | z² + z | = 4 because the absolute value sign brings in a discontinuity. But, I'll think about it some more.

I don't think it's discontinuous when it's complex. If you substitute z=x+yi, then you get |(x+yi)^2+(x+yi)|=4

You could solve it by separating the real parts from the imaginary parts, then using pythagorus. You could then integrate y with respect to x and find the area. In practice though, I get a huge implicit equation.

(x^2 +y^2+x)^2 +(2xy +y)^2 = 16

I did manage to differentiate it, and eventually I brought it down to an explicit equation, but I still can't integrate it. Polar form doesn't work very well either. I'm sure there must be an easier method for them.

[pat]

Let w = z² + z. Then, you're looking to integrate where: w w^* = 4² (where w^* is the complex conjugate of w). I don't think that's a smooth function of w.... (or am I on crack?)

From the sounds of what you have above, you're integrating/differentiating as if it were two functions of two real variables. Integrating/differentiating as complex numbers isn't quite the same.

If your equation were just: z² + z = 4, then the integral of it would be (1/3)z³ + (1/2)z² = 4z + w for some constant w.

But, that doesn't work very well once we throw in conjugation.

[mghtymoop]

gravity and the fact that acceleration increases mass is all the proof we need that a fourth dimension must exist

[PWrong]

From the sounds of what you have above, you're integrating/differentiating as if it were two functions of two real variables. Integrating/differentiating as complex numbers isn't quite the same.

The two real variables are just the real and imaginary parts of z.
The conjugates just cancel out to the sum of the squares of each component.

w w^* is just Real(w)²+Imag(w)², which is close to what I had. I did make a mistake though. I'll go through the whole process now.

|z²+z|=4

|(x+yi)²+(x+yi)|=4

|(x²+2xyi-y²)+(x+yi)|=4

rearranging, this gives
|(x²+x-y²)+(2xy+y)i|=4

The absolute value is the sum of the squares of the real and imaginary components, so
(x²+x-y²)²+(2xy+y)²=4²
That's what I should have had before.

I found some more information about it at
http://mathworld.wolfram.com/MandelbrotSet.html
Apparently my equation is a "lemniscate".

Incidentally, to everyone, this might be my last post for a while. My mock exams are in two weeks, and I should have started studying weeks ago. I'm going for a maths degree at UWA . Wish me luck, and I'll come back when exams are over.

[trill]

like many things in science, this is theoretically possible but we do not know if it is true until it is proved--even if possibly there is no known way to prove it.

[RQ]

like many things in science, this is theoretically possible but we do not know if it is true until it is proved--even if possibly there is no known way to prove it.

Correct! Except, not only is there no known way to prove the existence or nonexistence of a fourth dimension, there isn't one.

However, the fact that gravity accelerates and increases mass proves nothing, but that it is a property of the universe. There are an infinite number of dimensions: 1,2,3,4, and that is fundamentally true, but there can be no assertable proof that any such other parallel or nonparallel universes exist. Due to the anthropic principle, the only universe we know for sure exists with respect to our perception of existence, is this one.

"This means we can all be a figment of somebody's imagination?"

-Bender

[PWrong]

Hi everyone, I'm back. The mocks went well.
Don't mind me saying this, but what are you three on about? (mghtymoop, trill and RQ)

About my mandelbrot set problem, I've discovered it might not be possible. I didn't realise there were some functions that don't have an integral at all. I think the mandelbrot lemniscates are probably like them. So much for finding the area.

I'm working on a new ridiculously difficult project now, which I might describe in another post later. It's about the "tetration" function, that came up a few threads ago.

[jinydu]

Hi everyone, I'm back. The mocks went well.
Don't mind me saying this, but what are you three on about? (mghtymoop, trill and RQ)

About my mandelbrot set problem, I've discovered it might not be possible. I didn't realise there were some functions that don't have an integral at all. I think the mandelbrot lemniscates are probably like them. So much for finding the area.

I'm working on a new ridiculously difficult project now, which I might describe in another post later. It's about the "tetration" function, that came up a few threads ago.

I'm quite sure tetration would be normally called an "operation", not a "function", since it is analogous to addition, multiplication and exponentiation, and those are usually called operations. On the other hand, you could argue that addition is a multivariable function: f(x,y) = x + y.

But let me guess: You're working on expanding the tetration (a tetra b) so that it is defined for all real values of a and b (except maybe a = b = 0).

I would recommend two sites to look at:

http://home.earthlink.net/~mrob/pub/mat ... eal-hyper4
http://www.tetration.org/Tetration/index.html

For an even more ambitious goal, try extending the entire triadic operation to the real numbers (Yikes! An operation halfway between addition and multiplication).

[jinydu]

I think it would also be helpful to try to understand the Triadic Operation graphically. First, try graphing the following functions:

1) z = x+y
2) z = x*y
3) z = x^y
4) z = x tetra y
etc.

I'm sure that can actually be done, since each graph requires only 3 dimensions. However, I don't know what computer program will do it, although I think someone on this forum might know.

Now, try to imagine what it would be like if we could combine these 3D graphs into a 4D graph, placing addition in the realm w=1, multiplication in the realm w=2, exponentiation in the realm w=3, tetration in the real w=4, etc. Assemble everything together (including filling in the portions where w is not a positive integer, as in my previous post), and I think you would get a truly remarkable 4D graph. But I'm not sure if its possible...