Thread: Explain to me the 4th dimension.

Dependent dimension, as in, values in one affect the other. i² = -1, so a number in one complex dimension actually can affect the other. The imaginary dimension is connected to the real one. You also have cases like quantum superposition, with an n number of dimensions, but the diagonal of their values must always equal 1. Yes, there are independent dimensions, but the imaginary dimension is dependent.

The Riemann Sphere is exactly what I'm talking about - difference is, I didn't know it's name, since I got there on my own, on high school.

Basically, a system with n number of dimensions means a form has n number of perpendicular projections. The dimensions can be anything you define them to be, from statistical information to hyperspatial constructions. What we usually mean by 3d, though, is that the universe has 3 spatial dimensions.

Originally Posted by Kromoh

Dependent dimension, as in, values in one affect the other. i² = -1, so a number in one complex dimension actually can affect the other.

i² = -1 has nothing to do with dimension. That equations relies on the multiplicative structure that we add to the complex numbers on top of their additive and metric structure which is just that of a two dimensional vector space with a constant, positive definite metric over the real numbers. It confuses the issue of dimension to bring the multiplicative structure into it. The fact is that one can freely choose the real and imaginary components of the complex numbers. The dimensions are independent.

To say that a dimension is dependent on another means that you are trying to describe the space with too many dimensions. For example, if I take all the points (x, y) so that 4x + 5y = 0, then I get a line. I can describe a point with its x and y values and pretend that it's two dimensional but once I choose one, I can not freely choose the other. I can even work out the equation y = 4x/5 and then see that all points on the line will have the form (x, 4x/5). So this space has one dimension even if I want to try to describe it with two. We could say that the y dimension is dependent on the x dimension but that just means that the y dimension is an artifact of our initial description of the line. There is no such thing as a dependent dimension. If you're into math, that's very important to understand.

The 5th dimension has actually been recorded on tape. It had something to do with a soul that was stoned at a picnic.

http://www.youtube.com/watch?v=3IrJRZRZIn4

^ Serve it down brother, serve it down.

Even if y is an "artifact" of x, they are still different dimensions, and represent values of different nature. That is not so apparent in a line, but it is when dealing with conic curves, for example. Yet, they are dependent - as are mass and density, for example. One directly affects the other because of its very definition.

Originally Posted by Kromoh

Even if y is an "artifact" of x, they are still different dimensions, and represent values of different nature.

I said that the need for a second coordinate is an artifact of the means with which we constructed the line.

Originally Posted by Kromoh

That is not so apparent in a line, but it is when dealing with conic curves, for example. Yet, they are dependent - as are mass and density, for example. One directly affects the other because of its very definition.

What I am saying is most apparent on a line but if you want to do it with conic sections, fine. take the ellipse x^2 + 2y^2 = 4. this gives y =+/- sqrt(2 - x^2/2). so now having chosen x, you have two possible values of y. y is again dependant on x and it is a one dimensional space. I might as well embed it in an n-dimensional space and say that there are n-1 'dependant dimensions'. By what you are trying to say, we are surrounded with an unbounded amount of 'dependant dimensions'. The whole point of dimension is that there are no dependancies between them. What you want to say is that the second coordinate is dependent on the first coordinate when we embed a one dimensional space in a two dimensional space. saying that one dimension depends on another or that one dimensions affects another makes no sense. You are confusing dimensions with coordinates. It's an important but (not so) subtle distinction. Mathemeticians failed to make it until the late 1800's when Peano cooked up this example of a curve that covers a 2 dimensional area so that one could label each point in a two dimensional space with one coordinate. After that, they had to redefine dimension without using the number of independent coordinates and basically arrived the explanation of dimension that I gave with ants and bounded areas to do so.

I should also do the mathematicians walk of shame and say that I was wrong when I said that 1/t maps (-inf, inf) onto (-1, 1). It was actually the first thing that I thought when I woke up this morning so I must of dreamed about it. the function is |x|x/(x^2 + 1) . That was an ugly mistake.