Relation between dimensions

[Razee01]

I wish I could ask these questions to Dr. Michio Kaku. But may be any body can answer.

A dot is from zero dimension And the only that zero dimension contains.
All possible projections of a dot at the same time on the next higher dimension create a line (1d).
Same way all possible projections of a line at the same time on the next higher dimension create a circle (2d).
And so on all possible projections of a circle at the same time on the next higher dimension create a sphere (3d).

So do all possible projections of a sphere at the same time on the next higher dimension create another 4d object?

And if so, is our universe is just one of the possible projections of the universe? And as our universe is three dimensional, at the same time do all possible projections of the universe make one 4d universe?

What is the unit of our universe? Is it zero dimensional?

Do we have any reason to measure zero dimensional object as we measure 1d/2d/3d system?

As we feel we live in a 3d Universe, We can Understand and feel all the three dimensions. Then why we don't have any real example of any 1d or 2d universe? Because as we know an Atom or even an electron cloud is a 3d system.
Is there any possibility that our senses are a compilation of all single and two dimensional systems, and thus we can sense only 3d universe but any 2d universe?
If so, vice versa is it possible that creatures of a 4d universe also can not sense the existence of our 3d world and they only know theoretically our 3d Universe should exist?

[Mordred]

Think of dimensions as an independant variable. A variable that can change without changing any other such as x y and z or under 4d time in dimensionality  of length via ct.

[studiot]

Intriguing thoughts. Welcome Razee.

+1

 

How are you using the word "projections"?

In projective geometry there are more points on the projective plane than on the Euclidian/Cartesian plane for instance.

For comparison of planes the extra point is known as the point at infinity and is considered appended to the plane in question.

Here are a couple of philosophical points to conside as well.

You cannot have a 'point' without something that is 'not point' or 'no point'.
So you need something additional to zero dimensions ie a null dimension to complete the set.

6 hours ago, Razee01 said:

What is the unit of our universe? Is it zero dimensional?

This would be the 'point at infinity' in projective geometry.

[michel123456]

8 hours ago, Razee01 said:

I wish I could ask these questions to Dr. Michio Kaku. But may be any body can answer.

A dot is from zero dimension And the only that zero dimension contains.
All possible projections of a dot at the same time on the next higher dimension create a line (1d).
Same way all possible projections of a line at the same time on the next higher dimension create a circle (2d).
And so on all possible projections of a circle at the same time on the next higher dimension create a sphere (3d).

So do all possible projections of a sphere at the same time on the next higher dimension create another 4d object?

And if so, is our universe is just one of the possible projections of the universe? And as our universe is three dimensional, at the same time do all possible projections of the universe make one 4d universe?

What is the unit of our universe? Is it zero dimensional?

Do we have any reason to measure zero dimensional object as we measure 1d/2d/3d system?

As we feel we live in a 3d Universe, We can Understand and feel all the three dimensions. Then why we don't have any real example of any 1d or 2d universe? Because as we know an Atom or even an electron cloud is a 3d system.
Is there any possibility that our senses are a compilation of all single and two dimensional systems, and thus we can sense only 3d universe but any 2d universe?
If so, vice versa is it possible that creatures of a 4d universe also can not sense the existence of our 3d world and they only know theoretically our 3d Universe should exist?

Think of this: for a point to exist, it must...exist.

So, in reality a point is already 1 dimension (Time). A line is 2 dimensions (1 space + 1 time =2), An area is 3 dimensions ( 2 space + 1 time=3), and a sphere is 4D (3D+1T)

You cannot have a point without time. And Time does not come after the sphere, Time is there right from the beginning.

[Strange]

2 minutes ago, michel123456 said:

So, in reality a point is already 1 dimension (Time). A line is 2 dimensions (1 space + 1 time =2), An area is 3 dimensions ( 2 space + 1 time=3), and a sphere is 4D (3D+1T)

I don't think that redefining everything in this way is helpful. A point can be defined (as a mathematical abstraction) even in 4D space-time. So saying a point is a line and a line is a surface is just confusing.

8 hours ago, Razee01 said:

So do all possible projections of a sphere at the same time on the next higher dimension create another 4d object?

Yes, you can create a 4D sphere (confusing called a 3-sphere!) and a 5D sphere, etc. (And the same for any other shape; there can be higher dimensional equivalents.)

[geordief]

22 minutes ago, michel123456 said:

Think of this: for a point to exist, it must...exist.

So, in reality a point is already 1 dimension (Time). A line is 2 dimensions (1 space + 1 time =2), An area is 3 dimensions ( 2 space + 1 time=3), and a sphere is 4D (3D+1T)

You cannot have a point without time. And Time does not come after the sphere, Time is there right from the beginning.

Is it more correct to say "a point is in one dimension" etc? Or "a point illustrates one dimension" ?

Must we distinguish between mathematical points and physical points?(having heard tell of "point particles", but not understanding the subject)

Edited May 11, 2018 by geordief

[Strange]

1 hour ago, geordief said:

Is it more correct to say "a point is in one dimension" etc?

 It is probably more accurate to say that a point has zero dimensions. That’s kinda the definition. 

It can be a point in any number of dimensions - in a line, in a plane, in an N-dimensional space ...

Edited May 11, 2018 by Strange

[michel123456]

1 hour ago, Strange said:

 It is probably more accurate to say that a point has zero dimensions. That’s kinda the definition. 

It can be a point in any number of dimensions - in a line, in a plane, in an N-dimensional space ...

What I want to say is that if you erase Time, you cannot even imagine a point. The concept of a point in space standing out of time (somehow eternal) is wrong IMHO. In fact what we see as a point in space is a section of a line extending in Spacetime.

[JohnMnemonic]

Funny coincidence, that there's a new thread about dimensions, becase I'm working right now on a model of relativity in 5D - X, Y, Z + scale dimension + time flow rate as freqency of cycles... Of course, it's just my own funny concept, because I'm not a professional physicist

 

Quote

What I want to say is that if you erase Time, you cannot even imagine a point. The concept of a point in space standing out of time (somehow eternal) is wrong IMHO. In fact what we see as a point in space is a section of a line extending in Spacetime.

Points in space exist physically, as Planck's lenghts - which defines the volume of a photon. Planck's time is the shortest moment of time - time, in which photon is passing a single Planck's lenght. Those are the basic units of 4D space - points of physical dimensions.

But if we want to use different units - like meters and seconds, then photon has to be described as a wavefunction - and not as a point in space.

Edited May 11, 2018 by JohnMnemonic

[studiot]

41 minutes ago, michel123456 said:

What I want to say is that if you erase Time, you cannot even imagine a point. The concept of a point in space standing out of time (somehow eternal) is wrong IMHO. In fact what we see as a point in space is a section of a line extending in Spacetime.

I think that is a metaphysical point you are making.

It certainly disagrees with Euclid.

The first of the Fundamental Definitions, which precede event the axioms, is

A point is that which hath no part.

An old fashioned way of saying zero dimension.

For those interested the second fundamental definition is

A line is a breadthless length.

 

Of course, the ancient Greeks disagreed on definitions

 

Aristotle:

A point, a line and a surface all define the prioor by means of the posterior, a point is an extremity of a line, a line is an extremeity of a surface and a surfeace an extremeity of a solid.

 

Proclus gave us

A monad having position.

The modern view of a point is as the limit of some process.

 

 

 

[JohnMnemonic]

Quote

The modern view of a point is as the limit of some process.

I would define a point, as the smallest unit of a measured value - like distance in space

In quantum physics, points (units) of physical space are defined by Plank's units - physical points in dimensional space...

[StringJunky]

33 minutes ago, studiot said:

I think that is a metaphysical point you are making.

It certainly disagrees with Euclid.

The first of the Fundamental Definitions, which precede event the axioms, is

A point is that which hath no part.

An old fashioned way of saying zero dimension.

For those interested the second fundamental definition is

A line is a breadthless length.

 

Of course, the ancient Greeks disagreed on definitions

 

Aristotle:

A point, a line and a surface all define the prioor by means of the posterior, a point is an extremity of a line, a line is an extremeity of a surface and a surfeace an extremeity of a solid.

 

Proclus gave us

A monad having position.

The modern view of a point is as the limit of some process.

 

 

 

Physically, wouldn't a  point be the smallest unit of space?

[Strange]

1 hour ago, michel123456 said:

What I want to say is that if you erase Time, you cannot even imagine a point. The concept of a point in space standing out of time (somehow eternal) is wrong IMHO. In fact what we see as a point in space is a section of a line extending in Spacetime.

The coordinates (x,y,z,t) define the location of a single (zero-dimensional) point in space-time.

If you have a line in space-time then it is a line: a one dimensional entity. A point in space could well be represented by a line (its world-line) in space-time. (which could be of finite length, or infinite.) On the other hand, it could just be a point in space-time as well.

[geordief]

 

3 hours ago, Strange said:

 

It can be a point in any number of dimensions - in a line, in a plane, in an N-dimensional space ...

So ,in an N-dimensional space one can embed objects of any dimensionality such as  n-1,n-2 .....2,1,0  ?

And the point is just the object with the least dimensionality

Just mathematically,I suppose. 

[Strange]

5 minutes ago, geordief said:

 

So ,in an N-dimensional space one can embed objects of any dimensionality such as  n-1,n-2 .....2,1,0  ?

And the point is just the object with the least dimensionality

Just mathematically,I suppose. 

Yes, yes and ... yes!

Points, lines, etc are mathematical abstractions with no physical reality.

[JohnMnemonic]

Quote

Points, lines, etc are mathematical abstractions with no physical reality.

But Planck's lenght is not a mathematical abstract, but a measurable volume of physical space which should be treated as an indivisible point.

Edited May 11, 2018 by JohnMnemonic

[geordief]

26 minutes ago, Strange said:

Yes, yes and ... yes!

 

Not a sneaky reference to our Molly?

https://www.scienceforums.net/topic/114679-most-overrated-novels-of-all-time/?do=findComment&comment=1051974

 

[Strange]

36 minutes ago, JohnMnemonic said:

But Planck's lenght is not a mathematical abstract, but a measurable volume of physical space which should be treated as an indivisible point.

There is no reason to think the Planck length is indivisible. 

18 minutes ago, geordief said:

Not a sneaky reference to our Molly?

https://www.scienceforums.net/topic/114679-most-overrated-novels-of-all-time/?do=findComment&comment=1051974

 

Sadly not.

Not even, "When Harry Met Sally"

[JohnMnemonic]

Quote

There is no reason to think the Planck length is indivisible. 

Of course, science has to assume all possible scenarios. However at this time, it can be used as a indivisible point of space for most of observable physical processes (if not all of them). If we will somehow discover even smaller volume in the future, then we will simply use it as even more precise definition of a point. My point is, that point doesn't have to be always an abstract term. For now, Planck's units can be treated, as definitive and precise values in experimental physics and mainstream theories.

Edited May 11, 2018 by JohnMnemonic

[studiot]

2 hours ago, JohnMnemonic said:

I would define a point, as the smallest unit of a measured value - like distance in space

In quantum physics, points (units) of physical space are defined by Plank's units - physical points in dimensional space...

 

Well there you have an immediate problem.

Is -1 smaller than zero?

Or is zero the smallest unit of some measured value?

 

Of course in quantum theory the 'smallest unit' gives rise to the uncertainty.

 

 

[Strange]

4 minutes ago, JohnMnemonic said:

However at this time, it can be used as a indivisible point of space for most of observable physical processes (if not all of them).

Why? 

[studiot]

1 hour ago, geordief said:

So ,in an N-dimensional space one can embed objects of any dimensionality such as  n-1,n-2 .....2,1,0

Or indeed an N dimensional object.

I think you mean of dimension less than or equal to N.

 

1 hour ago, StringJunky said:

2 hours ago, studiot said:

 

Physically, wouldn't a  point be the smallest unit of space?

Well according to Euclid a point occupies zero space (space = volume as Mordred likes to say)

[geordief]

20 minutes ago, studiot said:

Or indeed an N dimensional object.

I think you mean of dimension less than or equal to N.

 

 

I see.Would you actually use the term "embed" in that case?

A bit redundant?

[studiot]

23 minutes ago, geordief said:

I see.Would you actually use the term "embed" in that case?

A bit redundant?

 

An embedding is technically an injection

An injection is a mapping from a domain set to a co-domain set (they may be the same set or different sets)

If every member of the domain is mapped to (injected) to a member of the co-domain and vice versa the mapping is said to be onto.
This is the case when embedding an N dimensional object in an N dimensional coordinate system.

If every member of the domain is mapped to a member of the co-domain, but not all the codomain members are used the mapping is said to be into.
This is the case when an n dimensional object is embedded in an N dimensional coordinate system (n < N).

So yes, though you would probably not bother with the distinction.

Edited May 11, 2018 by studiot

[JohnMnemonic]

Quote

Is -1 smaller than zero?

Or is zero the smallest unit of some measured value?

I think, that -1 is actually "bigger" than 0.

 0 is not a unit. We can make a unit from any measured value - even candies or apples  But then 1 apple becomes the unit of apples. 0 apples, means that there's no apples to count (measure).

While speaking about XYZ dimensions, then 1 and -1 define distances  in opposite directions, from the center of a frame (point 0). So, it''s the distance, which becomes the unit.

1 hour ago, Strange said:

Why? 

Because we don't know about anything, what would be smaller, than a photon...