Negative dimensions?

[Chaotic42]

Ok, so the area of a square is x²
The volume of a cube is x^3
The volume of a 4 dimensional hypercube is x^4

Would the volume of a -3 dimensional cube be 1/(x^3)?

If x is 3m, would the volume of the cube be (1/27) cubic meters?
Are there negative dimensions?

 

[rjain]

No, there aren't negative dimensions. You'd need an ordered set with a negative size in order for that to be possible. Actually, I don't know too much about pure (non-metric) topology, so I don't know how to express this in those terms. Then again the OP is talking about areas and volumes, so let's assume he requires a metric.

 

[MrDudeMan]

think about what you are asking...the volume of a -3 dimensional....volume of a -3...so the volume of something that doesnt exist?

 

[Mingon]

You would (iirc) have to use J numbers for negative numbers you wish to square. so you substitute -3 for 3J and your answer would be 9j² But its been a while since I have done these so I could be wrong - or you might do it different stateside.

 

[rjain]

Mingon: Huh? J numbers? You mean imaginary numbers? That's usually represented using i = +sqrt(-1) on both sides of the pond. But I have no idea what that has to do with this discussion. Where are you getting the idea that you can just replace a factor of a number with its square root? He's not squaring anyway. His example question was about cubes.

 

[ZeroNine8]

Basically, negative dimensions don't exist and are meaningless for standard discussions of volume, area, etc. Typical dimensions are expressed in some form of magnitude and direction. Whether you use polar, spherical, cartesian, or other coordinate frames, you reference some magnitude with some direction, either angles or positive/negative along an orthogonal axis.

This being said, I don't think you can have a 'negative' dimension in the way you're speaking of, due to the fact that magnitudes are always positive and, therefore, so are their products. If you think of a 1m cube in 3d cartesian space, no matter what quadrant(s) the cube lies in, the dimensions are still 1m per side. The point of reference for defining the space does not change the nature of objects in that space.

I guess you can mathematically claim that there are negative dimensions and somehow show negative volumes or something like that, but it would be meaningless to anything physical as far as all that I've ever heard of.

 

[Lynx516]

ZeroNine you say they have to have a magnitude and directon how then do non integer dimensions work and beleive me they exist.

 

[rneff50]

What? You can have non integer vectors (magnitudes with directions).

 

[Lynx516]

Does not work you have to be able to describe teh item with a non integer number of cohordinates.

 

[Mingon]

Mingon: Huh? J numbers? You mean imaginary numbers? That's usually represented using i = +sqrt(-1) on both sides of the pond. But I have no idea what that has to do with this discussion. Where are you getting the idea that you can just replace a factor of a number with its square root? He's not squaring anyway. His example question was about cubes.

A freindly response as ever I did say its been a while.... (10years+)

A good book to read if you get a chance is Imagining Numbers (Particularly the Square Root of Minus Fifteen) by Barry Mazur
Below is a little quote.

In pictorial terms, if a square has equal sides of 4 inches, then its area will be 16 inches. But in order to find the square root of a negative number, you have to imagine a negative area, with four sides of negative length. It's an impossibility, but, as Mazur argues, a mind-bendingly creative one

The point is that the formula is the same whether the numbers are negative or positive, by substituting the - for something else (e.g. i to keep you happy) you can work it out and then just substitute - back for i afterwards. I know it sounds strange but I am pretty sure it works. Again try it, see what you get and then comment.

So a cube of -3 sides is -27^3



Edit> But then again my brain is fried from teaching 45 8 year olds today so I could be (and probably are) totally wrong, not to worry Rjain will jump on it straight away

 

[Matthias99]

Originally posted by: Lynx516
Does not work you have to be able to describe teh item with a non integer number of cohordinates.

Objects with noninteger dimension (such as fractals) work just fine -- but they have to exist as part of an n-dimensional space. It doesn't make sense to talk about a "2.5-dimensional space", just like it doesn't make sense to talk about a "-3 -dimensional space". You can't have half of a coordinate.

 

[ZeroNine8]

Originally posted by: Lynx516
ZeroNine you say they have to have a magnitude and directon how then do non integer dimensions work and beleive me they exist.

Does not work you have to be able to describe teh item with a non integer number of cohordinates.

I'm not even sure what you're trying to say here... are you talking about 1/2 dimensions?

The number of coordinates (component vectors) used to completely describe something in 3D space is 3, for 2D it is 2, it doesn't matter what coordinate system you are using as long as the dimensions you are working within are orthogonal. I don't see how integer/non integer has anything to do with it, as the component vectors are simply a magnitude and direction, typically with the direction noted as + or -. As for half dimensions, I don't know that I believe that they exist, as the only way I have encountered them are in mathematical equations that are simplified models of full dimensional systems. If you maintain that these 'non integer' dimensions truly exist, and not just as a mathematical construct in a formula, please direct me to some info on their existance.

Edit: I just want to clarify, I've got no problem with the math being used for negative dimensions and half dimensions, I'm just saying that these things do not exist as such (to my knowledge), rather they are only mathematical ideas with no physical counterpart. I think this was the original question, or at least my understanding of it.

 

[Pudgygiant]

If it was a negative dimension, wouldn't you expect the volume of a -3 dimension cube to be negative?

 

[Mingon]

Edit: I just want to clarify, I've got no problem with the math being used for negative dimensions and half dimensions, I'm just saying that these things do not exist as such (to my knowledge), rather they are only mathematical ideas with no physical counterpart. I think this was the original question, or at least my understanding of it.

Whenever I teach negative number to children I always say to them they are only used in scales but dont refer to quantaties, as you say an actual physical object cannot have a negative dimension, but a scale can refer to a number as negative e.g. volts, direction etc. Alot of children have problems with the concept.

 

[Chaotic42]

Originally posted by: Pudgygiant
If it was a negative dimension, wouldn't you expect the volume of a -3 dimension cube to be negative?

Well, I suppose it depends. x^-3 is just 1/x^3.

 

[ZeroNine8]

Originally posted by: Chaotic42
Ok, so the area of a square is x²
The volume of a cube is x^3
The volume of a 4 dimensional hypercube is x^4

Would the volume of a -3 dimensional cube be 1/(x^3)?

If x is 3m, would the volume of the cube be (1/27) cubic meters?
Are there negative dimensions?

Volume is usually calculated as L*W*H (length, width, height) for 3d solids, thus if a negative dimension solid existed, you would probably multiply (-L)*(-W)*(-H) to find the volume. For this cube you speak of, I would think that the volume would be (-x)^3, not x^(-3), as I can't see any reason the negative would end up on the exponent from this formulation. Interestingly enough, negative dimensional shapes would have positive areas and negative volumes if they existed. The exponent does not directly refer to the number of dimensions, it is only a simplification of the special case for volume of a cube (L*W*H) where all sides are equal. If someone knows differently, I would be interested to find out how negative dimensions are handled when it comes to volume, area, etc.

 

[rjain]

non-integral dimensions are fractal dimensions. They aren't quite the same thing as we are discussing. A 2.5-dimensional fractal is a 3-d object which has a fractal boundary (or is it the boundary itself that is 2.5-dimensional?).

 

[rjain]

Originally posted by: Mingon
The point is that the formula is the same whether the numbers are negative or positive, by substituting the - for something else (e.g. i to keep you happy) you can work it out and then just substitute - back for i afterwards. I know it sounds strange but I am pretty sure it works. Again try it, see what you get and then comment.

Um. No.
By your logic, i^2=-1^2=1, and there's no i in the result.
i^3 = -1^3 = -1. no i again. The correct result is -i.

I think you're just confusing the issue. I can't see any situation in this question where square roots of negative numbers would apply. You don't take the square root of a positive number to find the volume of a cube. You just cube the length of the side of the cube.

edit: fix quoting

 

[ZeroNine8]

isn't this j or i you're using the sqrt(-1)? if so, then i^2 != -1^2

and how did i get brought up anyway, like rjain said, you don't do any square roots in finding volume with normal dimensions, why is it an issue now?

 

[DrPizza]

Originally posted by: ZeroNine8
isn't this j or i you're using the sqrt(-1)? if so, then i^2 != -1^2

and how did i get brought up anyway, like rjain said, you don't do any square roots in finding volume with normal dimensions, why is it an issue now?

Okay, since many of us have taken classes that deal in fractional dimensions (fractals), we're pretty satisfied that the mathematics exist. I'd bet that you could derive equations in the same manner for negative dimensions, (although I'm not about to think about it and get a headache...)..

What I'm going to propose we do next, after we settle the issue of negative dimensions is go on to....











imaginary/complex dimensions. What's the volume of a cube in the sqr(2)+5i dimension?
As silly as it sounds... just think how silly many equations were considered, until one day an Einstein or someone came along and decided they worked great for describing a new theory in physics.

Maybe... this is an area that someone could actually develop, get their name attached to, and get a doctorate for. (I'll think this one over this weekend... I call first dibs)

 

[rjain]

Eh? fractal dimensions are being used to define... what?

Oh! the boundary can be considered to be a space, as it has infinite size, right? But since you can't exactly move with the same freedom you can in the space with an integral number of dimensions more, you can't claim that the space has that many dimensions. Cool. :light:

Good books about this stuff?

Edit: so there would be some range of coordinates that are forbidden, right? You'd define some axes and then place some limits of their extent... but what limits would those be? It would still have to be an infinite subspace...

 

[RossGr]

Originally posted by: Mingon
You would (iirc) have to use J numbers for negative numbers you wish to square. so you substitute -3 for 3J and your answer would be 9j² But its been a while since I have done these so I could be wrong - or you might do it different stateside.

Electronic Engineers use j as the imaginary component in circiut analysis.

 

[Mingon]

Yeah the memories going in my old age, i or j we used cant remember which, but it was 10years ago at least, come to think of it I cant remeber the context very well. Oh well I must have thrown it out dreamcatcher style, to make room for other things.

 

[RossGr]

Originally posted by: ZeroNine8

Originally posted by: Lynx516
ZeroNine you say they have to have a magnitude and directon how then do non integer dimensions work and beleive me they exist.

Does not work you have to be able to describe teh item with a non integer number of cohordinates.

I'm not even sure what you're trying to say here... are you talking about 1/2 dimensions?

The number of coordinates (component vectors) used to completely describe something in 3D space is 3, for 2D it is 2, it doesn't matter what coordinate system you are using as long as the dimensions you are working within are orthogonal. I don't see how integer/non integer has anything to do with it, as the component vectors are simply a magnitude and direction, typically with the direction noted as + or -. As for half dimensions, I don't know that I believe that they exist, as the only way I have encountered them are in mathematical equations that are simplified models of full dimensional systems. If you maintain that these 'non integer' dimensions truly exist, and not just as a mathematical construct in a formula, please direct me to some info on their existance.

Edit: I just want to clarify, I've got no problem with the math being used for negative dimensions and half dimensions, I'm just saying that these things do not exist as such (to my knowledge), rather they are only mathematical ideas with no physical counterpart. I think this was the original question, or at least my understanding of it.

You can have a non orthoganal coordinate system which spans a space, the definiction of the dimension of a space specifies that the basis vectors be lineraly independent, not orthoganal. It is not clear how, in this context you could have a space with negitive dimensions.

 

[ZeroNine8]

I was using this definition of orthogonal:

N mutually orthogonal vectors span an N-dimensional
vector space, meaning that, any vector in the space can be
expressed as a linear combination of the vectors. This is
true of any set of N linearly independent vectors.

The term is used loosely to mean mutually independent or well
separated. It is used to describe sets of primitives or
capabilities that, like linearly independent vectors in
geometry, span the entire "capability space" and are in some
sense non-overlapping or mutually independent. For example,
in logic, the set of operators "not" and "or" is described as
orthogonal, but the set "nand", "or", and "not" is not
(because any one of these can be expressed in terms of the
others).

which basically means each coordinate is linearly independent of the others.