Question About Dimensions

[Luminal]

Never mind.

Edited September 10, 2008 by Luminal
Please delete.

[Bignose]

Why is it possible to easily formulate a geometrical and mathematical description for 4+ dimensional realities (and even model them visually in programs), yet as far as we know nothing of the sort exists?

 

This isn't right. Consider a spherical particle. It can have a location which would be located in a 3 dimensional space. It can then also have a velocity, which would have a "position" in a "velocity" space. Since velocity can be in any of the three spatial dimensions, you have 3 velocity dimensions. So, that's a total of 6 dimensions.

 

You can do the same with accelerations, so that's a total of 9 dimensions.

 

You can also allow the particle to rotate in any of the three dimensions, so now we're up to 15.

 

You can included rotational acceleration, so that 18 dimensions.

 

Now, what if you allow the particle to have different sizes? The volume of the particle is another dimension, so we're up to 19 dimensions.

 

19 dimensions by considering something as simple as a a spherical particle that rotates and comes in different sizes. That I can say pretty confidently "exists".

 

This doesn't even consider more than one particle.

 

If we consider N particles located in space with a certain velocity (i.e. ignore accelerations, rotations, and size) we can write an equation known as the Liouville equation:

 

[math]\frac{\partial{\rho}}{\partial{t}} + \sum_{i=1}^{N}( \frac{\partial{\rho}}{\partial{x_i}}\dot{x_i} + \frac{\partial{\rho}}{\partial{v_i}}\dot{v_i})=0[/math]

 

This equation determines the probability that particle i (where i goes from 1 to N) will be located in the volume of space [math]dx_i[/math] with a velocity in the range [math]dv_i[/math].

 

This equation describes a function in 6N dimensions. That is, if there are 100 particles, this is a function of 600 dimensions. I think you can think of many, many situations where there are 600 particles. It is very, very easy to imagine a situation beyond just 3 dimensions.

 

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awww man, I typed all that and now "never mind".... oh well.

[Quartile]

I think Luminal is referring specifically to spatial dimensions.