Hyper Dimensional Philosophizing

[Oppressed]

LOL.

I just caught on to a possible confusion about the meaning of dimension that I was overlooking, even though I understand it both meanings.

What helped me to catch this was the process of simplifying the example.

What is the meaning of dimension as it is used in the phrase “one dimensional length” as I have been trying to direct us towards talking about to simplify the discussion and avoid confusion.

Ah, but what it the meaning of dimension as it is used in the phrase “the dimension of the line segment is …”?

I was so focused and thinking about the first definition, I overlooked the fact that this second definition could cause trouble because it has a very distinct different meaning, even though the two meanings have a lot of similarity.

I should have thought about that and clarified it.

The meaning of dimension as it is used in the phrase “the dimension of the line segment is …” is parameter of the space the line segment lies in, in other words, it is a quantity of length. I really should have thought of this possible misunderstanding.

The meaning of dimension as it is used in the phrase “one dimensional length” is NOT a quantity at all and can NOT be measured. You can measure quantities in this dimension. It is this meaning of dimension I’ve been talking about.

Since I began this discussion with a post about dimensions in respect to the dimension of length, the dimension of time, the dimension of mass and so on, I just clicked in the definition as appropriate to what I was discussing and forgot about the fact someone might confuse the other type of dimension was what I was talking about.

You can not measure the dimension of length but you can measure quantities in the dimension of length.

You can not measure the dimension of time but you can measure quantities in the dimension of length.

Because of the reference to dimension of length, time and so on, I automatically understood what meaning of dimension I was referring to and failed to realize that someone could make the mistake of applying the other meaning of dimension which does not apply correctly with the context of what was being discussed.

Have you tried to apply the definition I give for dimension to dimension as it is meant in “one dimensional length”?

“dimension” is a dependence of a given quantity on the base quantities of a system of quantities, represented by the product of powers of factors corresponding to the base quantities.

 

[PixyMisa]

 

[PixyMisa]

Just unpicking this:

“dimension” is a dependence of a given quantity on the base quantities of a system of quantities, represented by the product of powers of factors corresponding to the base quantities.

If we apply this definition to your original question, then the Universe has an infinite number of entirely arbitrary dimensions. This seems unhelpful.

This is also emphatically not the definition physicists or mathematicians use when discussing that sort of concept.

 

[Oppressed]

GroundStrength,

I appreciate your comment.

In response to the large number of people on the forum who state I do not know what I am talking about and that I am wrong, I am going to go into the University over the next week and talk to some professors I know. It will take a week or so, because it is not always easy getting hold of the professors I know and it is summer so I may have trouble finding them at all.

I have not discussed this in a University setting for close to 10 years. Maybe I have early Alzheimer’s and don’t remember correctly and my mind has gone to Swiss cheese and I no longer have the capacity for abstract analytical has evaporated.

I doubt this, but I shall see if when I go in person to talk with people I personally know who are very knowledgeable on this subject as to whether they understand what I am talking about and whether or not they think I am correct or incorrect.

 

[PixyMisa]

People aren't simply stating that you are wrong, they are explaining why what you are saying makes no sense, e.g.

If we apply this definition to your original question, then the Universe has an infinite number of entirely arbitrary dimensions. This seems unhelpful.

You never bother to address these points; you simply re-assert your definition, sometimes with the help of links that don't back up what you are saying.

 

[Oppressed]

Well, did find any Professors I know today, but I did touch base with Professor Greene and will have lunch with him tomorrow. If I have the same problem explaining this to him then I’ll concede ground to you.

PixyMisa,

Applying my definition does allow for an infinite number of dimensions, but not for an infinite number of dimensions of space.

Can you narrow the topic down to a single dimension that we can both agree is a dimension and then discuss some things about it?

If we begin with the simpler flat Euclidean three dimensional space, we can narrow down upon a single dimension of this space.

This gives us a single thing to focus on.

Are you willing to do that with me?

 

[PixyMisa]

Can you narrow the topic down to a single dimension that we can both agree is a dimension and then discuss some things about it?

If we begin with the simpler flat Euclidean three dimensional space, we can narrow down upon a single dimension of this space.

This gives us a single thing to focus on.

Are you willing to do that with me?

Okay, let's give it a shot:

A three-dimensional Euclidean space is made up of three independent dimensions of distance. That is, any point in the space is defined by exactly three numbers, each of which can take a value independently of the others.

 

[Jimbo07]

If we begin with the simpler flat Euclidean three dimensional space, we can narrow down upon a single dimension of this space.

This gives us a single thing to focus on.

Are you willing to do that with me?

We can get even more succinct, and start with:

|1 0| |i|
|0 1| |j|

Can we agree on what we're talking about in this case? What this is? What its dimension is? How it relates to geometry?

ETA: meh... I'll let Pixy run with it. I'll leave this post in place, but you can largely ignore it...

 

[Oppressed]

A three-dimensional Euclidean space is made up of three independent dimensions of distance. That is, any point in the space is defined by exactly three numbers, each of which can take a value independently of the others.

I agree.

Each of these three dimensions are perpendicular to each other, but the orientation of the three dimensions is completely arbitrary, so long as they remain perpendicular to each other. Thus is you select an origin point, you can define an axis representing one dimension as being directed in any direction, so long as the second axis you define is perpendicular to the first and the third axis you define is perpendicular to both the first and the second. Defining this three axes for coordinates does not change the fact that space has three dimensions, just as changing the orientation or origin has no effect on space having three dimensions.

Is this something you would agree with?

 

[PixyMisa]

There are some mathematical subtleties there that I'm not equipped to deal with right now (it's 4AM and I've got a headache ). Rather than giving you a half-answer, I'll leave this open for the other guys.

 

[GodMark2]

I agree.

Each of these three dimensions are perpendicular to each other, but the orientation of the three dimensions is completely arbitrary, so long as they remain perpendicular to each other. Thus is you select an origin point, you can define an axis representing one dimension as being directed in any direction, so long as the second axis you define is perpendicular to the first and the third axis you define is perpendicular to both the first and the second. Defining this three axes for coordinates does not change the fact that space has three dimensions, just as changing the orientation or origin has no effect on space having three dimensions.

Is this something you would agree with?

That is mostly correct so far. You need three independent axes to define a 3-dimensional space, any three independent axes will define a 3-dimensional space, and any 3-dimensional space will contain at least one set of three 3 independent axes. Flat euclidean space contains an infinite number of triplets of mutually independent axes.

Independent axes are axes in which movement along one causes no change in movement along the other, for the entire length of the axis (dx/dy = 0). Note: In general, they are not required to be perpendicular. In particular, the three axes chosen for Cartesian coordinates are not perpendicular, as at least one is a plane (or is indeterminate, residing in a plane).

The selection of an origin point is irrelevant (as it would be completely defined once the three independent axies are chosen as axis1=axis2=axis3=0).

 

[Cuddles]

Well, did find any Professors I know today, but I did touch base with Professor Greene and will have lunch with him tomorrow. If I have the same problem explaining this to him then I’ll concede ground to you.

There's a big difference between talking face to face and posting on a message board. It's entirely possible that you can make yourself clear in a conversation with constant feedback, but that won't prove anything about how you come across here. On this board, you have not made yourself clear. It's as simple as that. Getting a professor to understand you over lunch will not change that. It may change our opinions on whether you don't understand what you are talking about or just have difficulty communicating, but it will not change the fact that you have trouble getting your point across on a message board.

This is not about anyone proving anyone else wrong, it is about us trying to understand you. The problem so far has not been that we don't understand or that you don't understand, it is that you have refused to admit there could be a problem with understanding.

 

[Oppressed]

Cuddles,

I have commented before that I believe the problem here between my understanding what dimensions means and what other people on this forum understand what dimension means can be a matter of communication, either my inability to explain myself well enough to be understood or you inability to understand what I am saying or a combination of both.

But I have also said that I have not had this problem in my many past conversations, person to person, with professors of physics, mathematics and engineering as it came up in various classes over my 6 years of study. One of the points in talking about it now, person to person, with a professor, is to insure to myself that this still holds true, that when I talk about it face to face with a professor that we both understand each other and that my understanding of dimension is correct. If I can not make myself understood to a professor in a face to face conversation, then I’d want to know that as a form of self checking.

Narrowing down the topic to a specific dimension is also another way of trying to make sure we understand each other. We can begin by building a groundwork of things we agree are true about this single dimension.

Do you agree with the last couple of posts about what is said about this single dimension and its relation to the 3 dimensional space it 1 dimension of?

Well, I need to head out to the University now.

 

[Jimbo07]

. One of the points in talking about it now, person to person, with a professor, is to insure to myself that this still holds true, that when I talk about it face to face with a professor that we both understand each other and that my understanding of dimension is correct.

A potential problem with this approach (and I've fallen foul of this before, myself), is that if you present your own side, the professor may agree with you, tacitly understanding your point within a certain context. If you're not fully aware of the points of contention, you may still come away with an imperfect picture.

 

[Fredrik]

Each of these three dimensions are perpendicular to each other, but the orientation of the three dimensions is completely arbitrary, so long as they remain perpendicular to each other. Thus is you select an origin point, you can define an axis representing one dimension as being directed in any direction, so long as the second axis you define is perpendicular to the first and the third axis you define is perpendicular to both the first and the second. Defining this three axes for coordinates does not change the fact that space has three dimensions, just as changing the orientation or origin has no effect on space having three dimensions.

Please explain why this space that you describe is 3-dimensional. Which one of its many properties makes it 3-dimensional?

By the way, it's completely false that the "dimensions" of a vector space must be perpendicular to each other. There are plenty of vector spaces that are such that "the angle between two vectors" isn't even defined.

 

[Oppressed]

Jimbo07,

Perhaps you are correct, but when I was going through my classes and I discussed this topic and its relevance to the particular class, I was also normally earned an A for the class. I did talk to Professor Greene today and had no problem being understood. He did say that it was not a typical way of looking at it because most people took a narrower view looking at dimension with reference to what they were focusing on but not looking at the broader meaning of dimension as it applies across disciplines.

Fredrik,

To simplify the topic, I am focusing on the common well understood real physical flat three dimensional Euclidean space. To further simplify it, I want to narrow it down to a single dimension of length in this space, but the process of that does take considering the 3 dimensional space as well.

I believe that PixyMisa and GodMark2 both understand what space I am talking about and now we are working building common agreements over this space so that we have a this base framework we all agree is correct before getting to the nitty gritty of where we disagree over what dimension means. Maybe along the way, they will show me that I am thinking about it wrong or maybe they will come to understand what I mean. Hopefully we will all come out of it with a better understanding of Euclidian space.

GodMark2,

That is mostly correct so far.

I’d like to work that mostly part out so that we are in full agreement over what is correct.

In general, they are not required to be perpendicular. In particular, the three axes chosen for Cartesian coordinates are not perpendicular, as at least one is a plane (or is indeterminate, residing in a plane).

Is this the only area you disagree with me so far on?

I can’t seem to get some of the characters I’d like to use to work.

Let’s say we begin with Cartesian coordinates arbitrarily set in space somewhere.

Given we have three vectors originating from the origin:
A = (1*m*i, 0*m*j, 0*m*k) = 1 meter along the x-axis.
B = (0*m*i, 1*m*j, 0*m*k) = 1 meter along the y-axis.
C = (0*m*i, 0*m*j, 1*m*k) = 1 meter along the z-axis.
Such that “i” is the unit vector along the x-axis, “j” is the unit vector along the y-axis, “k” is the unit vector along the z-axis and “m” is the unit quantity of length “meter”.

Are the vectors A and B perpendicular?
Are the vectors B and C perpendicular?
Are the vectors A and C perpendicular?
Can you explain how one of the Cartesian coordinate axes can not be perpendicular to one of the other two?

 

[GodMark2]

I believe that PixyMisa and GodMark2 both understand what space I am talking about and now we are working building common agreements over this space so that we have a this base framework we all agree is correct before getting to the nitty gritty of where we disagree over what dimension means. Maybe along the way, they will show me that I am thinking about it wrong or maybe they will come to understand what I mean. Hopefully we will all come out of it with a better understanding of Euclidian space.

GodMark2,



I’d like to work that mostly part out so that we are in full agreement over what is correct.

In particular, the three axes chosen for Cartesian coordinates are not perpendicular, as at least one is a plane (or is indeterminate, residing in a plane).

Is this the only area you disagree with me so far on?

I can’t seem to get some of the characters I’d like to use to work.

Let’s say we begin with Cartesian coordinates arbitrarily set in space somewhere.

Given we have three vectors originating from the origin:
A = (1*m*i, 0*m*j, 0*m*k) = 1 meter along the x-axis.
B = (0*m*i, 1*m*j, 0*m*k) = 1 meter along the y-axis.
C = (0*m*i, 0*m*j, 1*m*k) = 1 meter along the z-axis.
Such that “i” is the unit vector along the x-axis, “j” is the unit vector along the y-axis, “k” is the unit vector along the z-axis and “m” is the unit quantity of length “meter”.

Are the vectors A and B perpendicular?
Are the vectors B and C perpendicular?
Are the vectors A and C perpendicular?
Can you explain how one of the Cartesian coordinate axes can not be perpendicular to one of the other two?

I mistyped, and didn't proofread the suggested correction. I apologize. I meant to type "Circular coordinates".

Cartesian coordinates are the example of three perpendicular axes.

Circular coordinates are an example of non-perpendicular coordinates in Eucledian space.

As Fredrik had pointed out, there can exist spaces where the concept of perpendicular is undefined. In Eucledian space (which we are constraining ourselves to), however, it is defined. But even in Eucledian space, it is not necessary for the axes to be perpendicular.

 

[Oppressed]

GodMark2,

No problem about the mistype. I make mistypes, slips and mistakes too. The important thing is to try and understand each other.

With regard to using the Cartesian coordinate system to describe the three dimensional space, are you in agreement with what I’ve said so far?

I think where we might have a real difference here, is when we shift to using cylindrical and spherical coordinates. I have wondered if there is a valid view point for lat Euclidian three dimensional space which can result in something other than three dimensions of length which are all perpendicular to each other. I am sure I have not figured out all possible viewpoints to come at this question, but so far I have always ended up with three dimensions of length which are all perpendicular to each other.

Let’s say we consider a two dimensional plane represented in Cartesian coordinates. I’ll define the following points using the letter “C” to indicate the point is in Cartesian coordinates followed by a number to indicate which point.

C1 = (+1,+0)
C2 = (+0,+1)
C3 = (-1,+0)
C4 = (+0,-1)

If you draw a straight line from point C1 to C2, then from C2 to C3, then from C3 to C4, then from C4 to C1 you get a square in two dimensional flat Euclidian space.

But what happens to the dimensions of this two dimensional plane if we switch to polar coordinates?

Given the point C = (x,y) as a general point in Cartesian coordinates, point P = (r,θ) would be the general point in Polar coordinates such that r is the directed distance from the origin to point C and θ is the directed angle counterclockwise from the x-axis to the line segment between the origin and point P. Thus the same point is identified using two different coordinate systems.

C = (x,y) = P = (r,θ)

r in the polar coordinates is a dimension of length.
θ in polar coordinates is a dimension of angle.
Equations written in polar coordinates have the dimensions of length and angle.
θ is generally considered dimensionless because it is a ratio of two dimensions, but it is truly a dimension of 1 such that it the ratio of a length/length and this does have some associated implications.

But, is the two dimensional plane now in dimensions of length and angle because we are using the polar coordinate system? I don’t think so.

The two perpendicular lengths that comprise the dimensions of the plane are the fundamental dimensions of the plane. The dimension of angle is a derived dimension from these two fundamental dimensions.

C1 = (+1,+0) = P1 (+1,0)
C2 = (+0,+1) = P2 (+1,1/2*pi)
C3 = (-1,+0) = P3 (+1,pi)
C4 = (+0,-1) = P4 (+1,3/2*pi)

Using polar coordinates, what is the distance between the points P1 and P2? How about between P2 and P3? What is the area of the square defined by 4 polar coordinate points P1, P2, P3 and P4? What are the dimensions involved to measure the area of that square, even though you are using Polar coordinates to calculate it?

 

[GodMark2]

GodMark2,

No problem about the mistype. I make mistypes, slips and mistakes too. The important thing is to try and understand each other.

With regard to using the Cartesian coordinate system to describe the three dimensional space, are you in agreement with what I’ve said so far?

I think where we might have a real difference here, is when we shift to using cylindrical and spherical coordinates. I have wondered if there is a valid view point for lat Euclidian three dimensional space which can result in something other than three dimensions of length which are all perpendicular to each other. I am sure I have not figured out all possible viewpoints to come at this question, but so far I have always ended up with three dimensions of length which are all perpendicular to each other.

If you restrict yourself to using axes measured in length, than this is true, actually, it's part of the definition of Eucledian space. But, of you choose different axes, like the ones used in polar coordinates, they are no longer required to be perpendicular each to the others.

Let’s say we consider a two dimensional plane represented in Cartesian coordinates. I’ll define the following points using the letter “C” to indicate the point is in Cartesian coordinates followed by a number to indicate which point.

C1 = (+1,+0)
C2 = (+0,+1)
C3 = (-1,+0)
C4 = (+0,-1)

If you draw a straight line from point C1 to C2, then from C2 to C3, then from C3 to C4, then from C4 to C1 you get a square in two dimensional flat Euclidian space.

You get a Eucledian plane. I say this to avoid the possible confusion between different definitions of the word 'space'.

But what happens to the dimensions of this two dimensional plane if we switch to polar coordinates?

Well, that depends on which definition of 'dimension' you're using.

Given the point C = (x,y) as a general point in Cartesian coordinates, point P = (r,θ) would be the general point in Polar coordinates such that r is the directed distance from the origin to point C and θ is the directed angle counterclockwise from the x-axis to the line segment between the origin and point P. Thus the same point is identified using two different coordinate systems.

Certainly.

C = (x,y) = P = (r,θ)

r in the polar coordinates is a dimension of length.
θ in polar coordinates is a dimension of angle.
Equations written in polar coordinates have the dimensions of length and angle.
θ is generally considered dimensionless because it is a ratio of two dimensions, but it is truly a dimension of 1 such that it the ratio of a length/length and this does have some associated implications.

Sigh. And now you're starting to use a second definition of 'dimension. I hope you don't get confused and try to use it in the same context as you did your previous mention of the word.

But, is the two dimensional plane now in dimensions of length and angle because we are using the polar coordinate system? I don’t think so.

The two perpendicular lengths that comprise the dimensions of the plane are the fundamental dimensions of the plane. The dimension of angle is a derived dimension from these two fundamental dimensions.

But you do anyway.
"Angle from reference" is one of the axes of the space defined in polar coordinates.
The unit of measure of "angle" is a derived unit. This does not interfere in any way with it's ability to be an axis of a space.

C1 = (+1,+0) = P1 (+1,0)
C2 = (+0,+1) = P2 (+1,1/2*pi)
C3 = (-1,+0) = P3 (+1,pi)
C4 = (+0,-1) = P4 (+1,3/2*pi)

Using polar coordinates, what is the distance between the points P1 and P2?

Distance is a measure, consisting of quantity and a unit. The coordinate system used to derive it has no effect on the measure itself. The value will be sqrt(2) base length units. It can be calculated from by using

d = sqrt(r1²+r2²-2r1r2cos[theta2-theta1]).

How about between P2 and P3? What is the area of the square defined by 4 polar coordinate points P1, P2, P3 and P4? What are the dimensions involved to measure the area of that square, even though you are using Polar coordinates to calculate it?

The units would be length squared, no matter what coordinate system is used in it's derivation.

You are still trying to cling to the one word "dimension", using it everywhere, without taking into account it's different meanings. Then you take two different places where you use "dimension" and say "Eureka! They must mean the same thing! They use the same word, after all.". That's why I chose the term "axis" when defining Eucledian space, it's harder to get confused with "unit".

Try expressing your ideas without using the word "dimension" itself, instead using words that have the same meaning in the context in question.

 

[Fredrik]

Fredrik,

To simplify the topic, I am focusing on the common well understood real physical flat three dimensional Euclidean space. To further simplify it, I want to narrow it down to a single dimension of length in this space, but the process of that does take considering the 3 dimensional space as well.

I believe that PixyMisa and GodMark2 both understand what space I am talking about and now we are working building common agreements over this space so that we have a this base framework we all agree is correct before getting to the nitty gritty of where we disagree over what dimension means.

Are you implying that I don't understand what space you're talking about? Of course I do.

You really should think about what property of "flat three-dimensional euclidean space" makes it three-dimensional. Doing so would (or at least should) make you see that this has nothing to do with what you've been calling a dimension.

I will tell you the answer myself. I will write E³ instead of three-dimensional euclidean space from now on, to make sentences shorter.

Geometry

Post #158 makes me believe that you understand cartesian, cylindrical and spherical coordinates. Good. Now consider this:

Let I be the function that takes an arbitrary point p in E³ to its cartesian coordinates (x(p),y(p),z(p)).

Let C be the function that takes an arbitrary point p in E³ to its cylindrical coordinates (r(p),phi(p),z(p)).

Let S be the function that takes an arbitrary point p in E³ to its spherical coordinates (r(p),phi(p),theta(p)).

In differential geometry, functions like I, C and S are called coordinate systems. Note that they are all functions from E³ into R³, the set of ordered triples of real numbers. Their "job" is to assign three numbers to each point in E³. The fact that it's always three numbers and not, say, five or five hundred, is the reason why E³ is said to be a three-dimensional manifold.

That's the definition of "dimension" that's used in geometry. If the coordinate systems of a manifold assign n numbers to every point in their domains of definition, then the manifold is said to be n-dimensional.

Note that this is a different definition than the one you gave us in #79. Do you agree that this is different? Yes or no?

Algebra

A set of vectors {v₁,v₂,...,v_n} is said to be linearly independent if the equation

a₁v₁+a₂v₂+...+a_nv_n=0

where the a's are real numbers, can only be true if a₁=a₂=...=a_n=0.

Define three vectors e₁,e₂ and e₃ by

e₁=(1,0,0)
e₂=(0,1,0)
e₃=(0,0,1)

and let v=(a,b,c) be an arbitrary vector.

Note that the set {e₁,e₂,e₃} is linearly independent, but the set {e₁,e₂,e₃,v} is not (no matter how we choose a, b and c).

Proof that the latter set is not linearly independent:

a*e₁+b*e₃+c*e₃+(-1)*v=0

This is the reason why R³ is said to be three-dimensional. The maximum number of linearly independent vectors is three. If there are n linearly independent vectors in a vector space but not n+1, then the vector space is said to be n-dimensional. The number n is called the dimension of the vector space.

Note that this is a different definition than the one you gave us in #79. Do you agree that this is different? Yes or no?

Note also that the set {e₁,e₂,u} where u=(1,1,1) is linearly independent, even though u is not perpendicular to e₁ or e₂. Vectors don't have to be perpendicular to be independent, so the definition of the dimension of a vector space doesn't in any way imply that "dimensions" are perpendicular.

You

Let me remind you of what you said in #79:

When you measure the quantity of something of the same nature and characteristics, there is a dimension to that measurement, be it length or be it apples.
...
For example [...] take the distance between two points. We can measure that distance. We can select an arbitrary unit smaller than the distance we want to measure and then count out the quantity of the units that fit along the length. The dimension is what the units consist of. You can arbitrarily define all manner of different units, but if you are measuring the distance between two points all these units consist of the same dimension.

In a similar manner, let’s say you have 1,000 apples. You want to measure how many apples you have. You define a unit of measure for the apples as 12 apples = a dozen apples. You can then count out how many apples you have in that unit of measure and say 83 and 1/3 dozen apples. You can also define as a unit of measure of 120 apples = a small gross of apples and count the apples by those units. But the real dimension of what you are counting in is not based on the units and thus the dimension in this case is apples.

This is a very good explanation of one thing that can be called a "dimension". But does it look anything like the geometric or algebraic definitions I included above? Where do you mention functions that assign coordinates to points in some abstract space? Nowhere! Where do you mention linearly independent vectors. Nowhere! So how can you still act like these three things are the same?

You have so far refused to admit that according to your definition, there are infinitely many dimensions (i.e. directions) in E³ (or equivalently, in R³). You should think that through again, because you are wrong. Every direction is a dimension according to your definition, but not according to the algebraic definition where it's the number of independent (not necessarily perpendicular) directions that counts, and not according to the geometric definition where it's the number of numbers that a coordinate system assigns to a point that counts.