Spatial dimensions

[Genady]

Imagine that you are given a smooth space of unknown geometry. What kind of constructions would you use to figure out the number of dimensions of this space?

[swansont]

2 minutes ago, Genady said:

Imagine that you are given a smooth space of unknown geometry. What kind of constructions would you use to figure out the number of dimensions of this space?

You’d have to have something in the space to be able to observe behavior in order to tell.

[Genady]

10 minutes ago, swansont said:

You’d have to have something in the space to be able to observe behavior in order to tell.

Right. Let's see. If we can mark points with a marker, and we have a piece of a line (a ruler with no marks), will it be enough?

[swansont]

17 minutes ago, Genady said:

Right. Let's see. If we can mark points with a marker, and we have a piece of a line (a ruler with no marks), will it be enough?

What is the marker? A line is not a physical object.

[Genady]

1 minute ago, swansont said:

What is the marker? A line is not a physical object.

It's a Mathematics forum 

[swansont]

Fair enough

[Markus Hanke]

9 hours ago, Genady said:

Imagine that you are given a smooth space of unknown geometry. What kind of constructions would you use to figure out the number of dimensions of this space?

You don’t need a metric for this, but I think you need to know at least what kind of connection your manifold is endowed with, for this to be possible at all (someone please correct me if I’m wrong).

Given a connection, you can use geodesic deviation - you analyse all the possible ways that geodesics can deviate in the most general case (no Killing fields) on your manifold. From this you can deduce the number of functionally independent components of the Riemann tensor and torsion tensor, which straightforwardly gives you the dimensionality of the manifold. 

This is independent of any metric; it even works if there’s no metric at all on the manifold. Practically doing it may, however, be pretty cumbersome.

Anyone know of an easier way?

[studiot]

11 hours ago, Genady said:

It's a Mathematics forum 

I'm so glad you guys had this conversation whilst I was considering my answer.

Like swansont, I originally thought you meant physical space and how you would measure the dimensions by physical observation.

But 'space' in Mathematics is different from 'space' in Physics.

 

Whatever the space you are referring to you would need to have more detail to proceed.

 

In pure mathematics, space refers to a master or container set for several sets which make up the space. These are not subsets, since the nature of the elements in each is different.
You need at least set of elements or points, a set of axioms and a set of relations between the elements.

This gives rise to different mathematical spaces eg geometric space, hausdorf space, phase space and so on.

These rules will enable the mathematical determination of both the meaning and number of dimensions of that space.

 

[Genady]

3 hours ago, Markus Hanke said:

Anyone know of an easier way?

Yes, a simpler procedure is described in MTW on page 10. While they just describe it, clearly and 'obviously', I posted this question hoping that we can arrive to it or other(s) and clarify on the way what the minimal requirements and assumptions are. 

 

1 hour ago, studiot said:

Whatever the space you are referring to you would need to have more detail to proceed.

The starting point is that the 'space' is manifold.

Since each manifold has a definite dimensionality, the question is well defined in this case. 

 

I am glad to have you guys participating.

[studiot]

I meant to say +1 to Markus for his addition before.

 

51 minutes ago, Genady said:

The starting point is that the 'space' is manifold.

Since each manifold has a definite dimensionality, the question is well defined in this case. 

Mathematically ?

 

see here since it was Cantor who I think first defined manifold.

 

 

 

BYW I'm still not sure if you mean find out the dimension

by observation in which case I would recommentd looking for shadows or considering the commutativity of rotations

or by theory in which case I would refer you to the topological idea of shrinking a fundamental area to a limit as described in Needham's book we have been discussing.

 

Edited September 3, 2023 by studiot

[swansont]

3 hours ago, studiot said:

Like swansont, I originally thought you meant physical space and how you would measure the dimensions by physical observation.

The title does say spatial dimensions…

[Markus Hanke]

2 hours ago, Genady said:

Yes, a simpler procedure is described in MTW on page 10. While they just describe it, clearly and 'obviously', I posted this question hoping that we can arrive to it or other(s) and clarify on the way what the minimal requirements and assumptions are. 

Good and interesting point, I hadn’t remembered this remark from MTW.

I remember from topology that proving that the boundary of some n-dim region on a manifold to be a (n-1)-dim region, requires the notion of homeomorphisms. But I don’t know what exactly needs to be in place for this to work. 

2 hours ago, studiot said:

commutativity of rotations

Nice one!

[TheVat]

A mathematician named Klein
Thought the Möbius band was divine.
     Said he: If you glue
     The edges of two,
You'll get a weird bottle like mine.

 

[Genady]

Following this definition (Manifold - Wikipedia):

Quote

an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space.

@Markus Hanke: Since homeomorphism is a part of the definition, it does not require anything more to be in place for that to work. Right? If the boundary of n-dim region in Euclidean space is (n-1)-dim region, the homeomorphism caries this property into the manifold.

@studiot: Dimensions are defined, through the homeomorphism, the same way as dimensions in Euclidian space, i.e., the mapping of neighboring points has to be continuous.

Here is the procedure described by MTW:

[Genady]

4 hours ago, TheVat said:

A mathematician named Klein
Thought the Möbius band was divine.
     Said he: If you glue
     The edges of two,
You'll get a weird bottle like mine.

 

[Genady]

8 hours ago, studiot said:

I would refer you to the topological idea of shrinking a fundamental area to a limit as described in Needham's book

Could you please point to the page in the book for this?

(I mean, the page number)

[TheVat]

There is decomposition of a manifold by triangulation.  That works on a 3-manifold, but not always on a 4.   And there are fractal manifolds, which I don't know much about.  Maybe not workable given the boundary of a Mandelbrot set.  If the boundary of a manifold is one dimension lower, then....hmm.  

There is also handle decomposition, which takes balls.  Sorry.

[Markus Hanke]

11 hours ago, Genady said:

Since homeomorphism is a part of the definition, it does not require anything more to be in place for that to work. Right?

Yes, it would appear so…but I’ll defer to what the real mathematicians here have to say on this.

[Genady]

This discussion, especially the reference provided here (https://www.scienceforums.net/topic/132322-spatial-dimensions/?do=findComment&comment=1249400) emphasized the importance of continuity assumption regarding coordinates, e.g., in these statements from MTW:

and

 

[studiot]

Suppose two people started at a point on the earth's surface:

the first travelled 100 miles north then turned and travelled 100 miles east.

the second travelled 100 miles east then turned and travelled 100 miles north.

 

Would they be in the same place at the end ?

 

 

 

[Genady]

20 minutes ago, studiot said:

Suppose two people started at a point on the earth's surface:

the first travelled 100 miles north then turned and travelled 100 miles east.

the second travelled 100 miles east then turned and travelled 100 miles north.

 

Would they be in the same place at the end ?

 

 

 

Depends on the starting point.

[studiot]

In what way

[Genady]

3 minutes ago, studiot said:

In what way

There are points such that if the two people start their journeys (as described) in these points, they will be in the same place at the end.

There are other points. If the two people start their journeys in these points, they will not be in the same place at the end.

[Genady]

To make it more precise:

There are points such that if the two people start their journeys (as described) in one of these points, they will be in the same place at the end.

There are other points. If the two people start their journeys in one of these points, they will not be in the same place at the end.

[Markus Hanke]

9 hours ago, studiot said:

Would they be in the same place at the end ?

On manifolds with curvature, covariant derivatives do not, in general, commute; thus, again in general, they wouldn’t end up at the same place.

But answering this question requires that your manifold is endowed with a connection.