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What are dimentions
& space filling curves

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What are dimentions

Machiveli
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Posted 08/16/09 - 07:38 AM: Subject: What are dimentions	#1
What exactly do we mean when we say that something (e.g. the universe) has n dimensions? Taking a naive view and going with Wikipedia dimensionality is defined as: "the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it." yet if we fill an n-dimensional space with a 1-d space filling curve then we can uniquely specify a point with a single coordinate. I guess we can claim that such curves have 'Hausdorff' dimention' but is this a property of the space or transition functions for moving about within that space (rules of the universe so to speak)? If we consider something living in such a world for example a being moving about in a 2d universe or equivalently on a curve of dimension 1 could we potentially say that the following are equivalent: (1)The state of something can be described by n-dimensional state coordinates and 'simple transition rules' (2)The state of something can be described by 1-dimensional state coordinates and 'complex transition rules' according to (1) motion can be in any dimension but you must move through consecutive numbers according to (2) motion is by jumps along the 1d path is restricted according to some complex rule. To give you all a simple concrete example: 1-d chess. If we number the squares on a chess board from 1 to 64 we can describe the movements of all the pieces in terms of mathematical operations. e.g. White pawns can move: +8(forward one row) +16 if <16 (forward 2 rows during first move) +7 or +9 if these squares are occupied by black (taking) turn into any piece if >56 But non self intersecting space filling curves are more interesting because they preserve locality (near by points in n-d space have similar coordinates in 1-d space) and can define points in real numbered space with arbitary precision. Edited by Machiveli on 08/17/09 - 01:07 AM

Cadrache
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Posted 08/16/09 - 09:29 AM:	#2
Dimentions are a bunch of letters that do not form a coherent terminology within the english or scientific language: english edition. Anyways... an interesting aspect with the 'search for God' threads a month ago is questioning whether or not we are able to define the dimension itself; or do we really only define the interaction of one dimension to another. In other words, how we define spatially is not neccessarily the whole definition of the X,Y, Z co-ordinate system. Instead it might be something akin to Bell's locality function; or a product of dimension and another unknown force.
"...There was a writer who asked why it was that when we find positive experiences we say that only the physical facts are real, but in negative experiences we believe that reality is subjective. He made an example of those who say that in birth only the pain is real, the joy a subjective point of view, but that in death it is the emotional loss that is the reality." - Tony Ballantyne, Recursion. _____________________________________________ Truth is want. - The internal state of matters. Truth is Need. - The external state of affairs.

Banno
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Posted 08/16/09 - 01:14 PM:	#3
Dimentions are presumably things said twice. Trimentions, furthermore, are things said three times, and as you will be aware, what is said three times is true. Dimensions, on the other hand, seem to be what they do. A space-filling curve provides the exact same information as a set of dimensions, but presented in a different way. It would seem that one can picture this as collapsing multi-dimensional space in on itself. Very cool, Machivell. Thanks for drawing this to my attention.
Davidson: We make maximum sense of the words and thoughts of others when we interpret in a way that optimizes agreement. Russel Morris: There's a meaning there, but the meaning there doesn't really mean a thing... Ned: Such is life

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Posted 08/16/09 - 01:42 PM:	#4
There is a rigorous definition of dimension in mathematics and there is a more common sense use of it I think the Wikipedia definition, for the latter, could be upgraded if they allow themselves some humor. "The dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it without playing tricks " Sadly they do not. When I think of dimension, I think of it in terms of a manifold. As a particular of something more general. Edited by hyena in petticoat on 08/19/09 - 12:53 AM. Reason: Illiteracy.

Cadrache
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Posted 08/18/09 - 10:23 AM:	#5
How do we define "not dimension"? I mean literally everything can be compared to numbers.
"...There was a writer who asked why it was that when we find positive experiences we say that only the physical facts are real, but in negative experiences we believe that reality is subjective. He made an example of those who say that in birth only the pain is real, the joy a subjective point of view, but that in death it is the emotional loss that is the reality." - Tony Ballantyne, Recursion. _____________________________________________ Truth is want. - The internal state of matters. Truth is Need. - The external state of affairs.

Cadrache
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Posted 08/18/09 - 10:29 AM:	#6
I do however find it interesting that merely turning a physical object around in your hands proves that the first 3 dimensions are in fact the same singular dimension...
"...There was a writer who asked why it was that when we find positive experiences we say that only the physical facts are real, but in negative experiences we believe that reality is subjective. He made an example of those who say that in birth only the pain is real, the joy a subjective point of view, but that in death it is the emotional loss that is the reality." - Tony Ballantyne, Recursion. _____________________________________________ Truth is want. - The internal state of matters. Truth is Need. - The external state of affairs.

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Posted 08/18/09 - 10:41 AM:	#7
Maybe you can define that an object have not-dimension of 3. If you cannot specify a location on it with 3 coordinates. Or do you mean dimension of 0? Headphones, ashtrays, apples, books and the letter "d" have all dimension of 0 because we treat them as a basic object. Maybe books have dimension 1 over some fixed integer. Edited by Incision on 08/18/09 - 12:05 PM. Reason: spacing

Cadrache
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Posted 08/18/09 - 10:46 AM:	#8
Nah, objects which constitute spatial phenomena. If you have a box aligned with one of it's sides along the X dimension, and then turn it, then there should exist a distortion to the side of the box such that the box attains infinite length -if conservations laws are to hold true. You see for any value X, there is an 'infinite' value y. The 1:1 relationship of an object lying along the X-dimension is in fact a 1:infinity relationship.
"...There was a writer who asked why it was that when we find positive experiences we say that only the physical facts are real, but in negative experiences we believe that reality is subjective. He made an example of those who say that in birth only the pain is real, the joy a subjective point of view, but that in death it is the emotional loss that is the reality." - Tony Ballantyne, Recursion. _____________________________________________ Truth is want. - The internal state of matters. Truth is Need. - The external state of affairs.

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Posted 08/18/09 - 10:55 AM:	#9
I don't understand ,Can you explain this: should exist a distortion to the side of the box such that the box attains infinite length

Cadrache
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Posted 08/18/09 - 11:49 AM:	#10
Law of conservation of energy states that the total energy between conversions from one form to the other is conserved. We traditionally look at this as a 2 part setup. Energy A to Energy B. Mathematically we write this as for every part of Energy 1, there exists 2 parts of Energy B. Or a 1:2 conversion rate. So if we have a value of 1 in electricity, then with the 1:2 conversion rate, you end up with 2 in magnetic energy. As an example. The ratios are off a bit. Some factors like the value of Electricity through numbers is not the same value of magnetic force. So a 1 in electricity does not equal a 1 in magnetism. In other words, a part of the difference in conversion is a difference in transfering 1 data set to a different membrane. In this case, the idea is that when the energy electricity exists - the energy magnetism does not exist within electricity, or vice versa. At the moment, I only know of 3 times when you could claim that conversion is a direct 1:1 ratio. 1. Inverse force. Unfortunately this results in a 0 value and therefore introduction of X to -X means that X no longer exists. 2. No conversion. In other words, this sort of implies a division or change of 0 in one aspect and therefore cannot technically happen. (when you talk about membranes that are equal one to the other, it most likely falls in 1 or 2.) 3. newtons law that states that no change has occured. In this case it means that the actual values do not change membranes. This seems to only happen if the subtrate (membrane) does not alter. In other words, if you change dimensions - there should logically be 2 unique occurances to the box itself. The side that was aligned on the X dimension should either physically shorten or lengthen when it is aligned with the Y dimension. Yet it doesn't.(?) This also means that if you turn it one way, then the box lengthens. If you turn it the other way to align to the Y-dimension, then it should shorten. There is a slight counter argument that mostly states that because I physically moved the object then I added energy to the system and that change in energy can make the X:Y ratio to only appear to be a 1:1 ratio. Unfortunately it is counter-argued on the basis of how much energy I input. If I put twice the amount of energy in the turning of the box on a BoxB, then that box should therefore elongate while the first one doesn't! The energy above and beyond the bare minamum to move the box should constitute a change in the length of the box if it affects the X:Y ratio.
"...There was a writer who asked why it was that when we find positive experiences we say that only the physical facts are real, but in negative experiences we believe that reality is subjective. He made an example of those who say that in birth only the pain is real, the joy a subjective point of view, but that in death it is the emotional loss that is the reality." - Tony Ballantyne, Recursion. _____________________________________________ Truth is want. - The internal state of matters. Truth is Need. - The external state of affairs.

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