Spatial dimensions

[Genady]

4 hours ago, Markus Hanke said:

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. 

Since the question mentions numerical distances, I assume the manifold in question is endowed with a metric.

[studiot]

4 hours ago, Markus Hanke said:

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. 

 

8 hours ago, Genady said:

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.

I rather think we are talking at cross purposes here.

 

My first thought in reply to the op was to note that you need more than the bare manifold to establish dimensions.

In the model I originally proposed the manifold appears to correspond to the bare set of elements with one important condition.

The condition that every element is of the same type.

You also need at least two elements to have a metric.

 

But this is still not enough to define or allow swansont's 'spatial dimensions'.

In order to create the neighbourhoods necessary to support the maths involved you need to be able to order the elements of the manifold, ie the axiom of choice is required.

 

Anyway I thought I was trying to discuss ways of determining the 'spatial' dimensions of the manifold when I posted my little question about non euclidian space.

I wanted to get this one over quickly as it is the simplest case I could think of.

So yes I am aware that there are points on the Earth, close to the poles, where one or the other journey is just not possible so to be more specific,

The start point should be somwhere near the equator (anywhere will do) - I cannot see any points that would lead to the two journeys ending up in the same place.

They would of course do so if the surface of the Earth was flat.

This demonstrates that the surface of the Earth requires at least another dimension.

 

Hopefully you both agree with this and we can move on to exploring what happens in more general dimensions.

Much more insight is gained by studying stress paths in 1 , 2 and 3 dimensions and finding out ( as Hamilton did) that there is no 4 dimensional equivalent.

I think the next dimensionality that supports this is 8, but I might be wrong about that.

[Genady]

4 minutes ago, studiot said:

I cannot see any points that would lead to the two journeys ending up in the same place

Any point 50 miles south of equator.

 

5 minutes ago, studiot said:

In order to create the neighbourhoods necessary to support the maths involved you need to be able to order the elements of the manifold, ie the axiom of choice is required.

According to the definition of manifold, it already has necessary structures, specifically, it has neighborhoods defined and, moreover, homeomorphic with Euclidean neighborhoods. It does not necessarily have a metric.

[studiot]

12 minutes ago, Genady said:

Any point 50 miles south of equator.

 

You are correct, that was a symmetry I had overlooked. +1

13 minutes ago, Genady said:

According to the definition of manifold, it already has necessary structures, specifically, it has neighborhoods defined and, moreover, homeomorphic with Euclidean neighborhoods. It does not necessarily have a metric.

Whose definition?

My reference maths dictionary gives

"A collection of points of a set,"      for a bare manifold     as per Cantor's regularisation of Riemann's original useage.

It goes on to add descriptive terms to narrow this down to different kinds of manifold eg for differential topology and so on.

 

 

I draw your attention to the comments I already posted about set theory.

In particular the 12 lines above the beginning of paragraph 96 on page 162.

[Genady]

41 minutes ago, studiot said:

Whose definition?

The definition that I knew and have seen elsewhere is consistent with the definition in Manifold - Wikipedia:

Quote

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, 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.

As you see, in this definition, manifold has sufficient structure for defining dimensions based on a continuous 1-1 correspondence, as per 12 lines above the beginning of paragraph 96 on page 162.

[studiot]

So Wikipedia contradicts itself  - so what ?

you used the words dimension and space.

Surely this thread is not about the definition of a manifold ?

 

Note that Wikipedia acknowledges many kinds of manifold and  dimension:

affine, euclidian box-counting, Minkowski, correlation hausdorf (which I originally asked you about) amongst others.

 

All I said was your initial specification was incomplete without further information.

The fact that you  are now referring to inherited characteristics demonstrates that this was reasonable.

 

You also asked how to determine the number of dimensions of a particular 'space'.

Quite a reasonable request so why are you now wasting discussion time quibbling definitions rather than discussing concrete procedures offered ?

 

[Genady]

2 minutes ago, studiot said:

All I said was your initial specification was incomplete without further information.

Agreed.

 

2 minutes ago, studiot said:

discussing concrete procedures offered

If the 'space' has structure of 'wikipedia-defined manifold', then the procedure provided in MTW works (shown here: https://www.scienceforums.net/topic/132322-spatial-dimensions/?do=findComment&comment=1249405).

A remaining question I see is, is there a procedure that works for a space with less structure? I don't know answer to this question.

 

[TheVat]

Am trying to search what MTW is.   Can't see from the page screenshots.  A Google search found Mathematics Their Way, but that is a text for K-2, so probably not.  A citation would be helpful for some who are following this thread.

[Genady]

Just now, TheVat said:

Am trying to search what MTW is.   Can't see from the page screenshots.  A Google search found Mathematics Their Way, but that is a text for K-2, so probably not.  A citation would be helpful for some who are following this thread.

Oh, I am sorry, I thought I cited it fully on the first mention. It refers here to Gravitation by Misner, Thorne, Wheeler. I have 2017 edition.

[TheVat]

Thanks.  You probably cited it in a related thread and I missed that.

I will borrow it from a nearby college.

[Markus Hanke]

18 hours ago, Genady said:

A remaining question I see is, is there a procedure that works for a space with less structure?

I’m sorry, maybe I’m being a bit thick here, but I’m still not 100% sure what ‘structure’ were are talking about exactly. The definition says it locally resembles a Euclidean space - so can we assume the presence of a connection and a metric, or just a connection, or neither?

[Genady]

1 hour ago, Markus Hanke said:

can we assume the presence of a connection and a metric, or just a connection, or neither?

I think, neither. We assume that

Quote

closeness is defined but cannot necessarily be measured by a numeric distance.

(Topological space - Wikipedia).

[joigus]

Skimming through this very interesting topic, with special emphasis on Genady's quote from MTW. The first thing that came to my mind is that @Genady's question is, I think, equivalent to,

Is there any way to define dimension --in pure mathematics-- that can be considered more primitive* than counting coordinates?

A connection requires a differentiable structure. For that you have to have to be given your space in terms of equations, whether implicit or explicit, parametric, etc. From there, derivatives allow you to give a sense to the concept of "moving" (vectors). The so-called tangent space. Metric and parallel transport (connection) can be introduced independently. In spaces given in this way, you can start talking about dimension long before you have a metric or parallel transport. Eg, in thermo you have systems defined by eq. of state like f(p,V,T)=0. Even though you don't have any meaningful metric, or parallel transport (although you could talk if you want about a tangent space consisting in the different thermodynamic coefficients); you do have a dimension, which in the example is = 2.

MTW's criterion is, I think, based on the topological notion of space. A topological space is basically a set with an inclusion operator on which you can define interior, exterior, and boundary.

Quote

Lebesgue covering dimension or topological dimension of a topological space is defined to be the minimum value of n, such that any open cover has a refinement in which no point is included in more than n+1 elements, if a space does not have Lebesgue covering dimension m for any m, it is said to be infinite dimensional.

This I have found in https://u.math.biu.ac.il/~megereli/final_topology.pdf, which in turn I've found by googling for "dimension of a topological space".

It seems that this criterion is somewhat different from MTW's, but they agree in that they're both topological. I must confess I'm a tad out of my depth with these "coverings" and "refinements of a covering" in topology.

I must also say I'm always baffled by these questions when no analitic example (ie, using coordinates) is allowed. How are you even given a space when no coordinates are allowed?

* Relying on fewer assumptions, that is.

[Genady]

1 hour ago, joigus said:

How are you even given a space when no coordinates are allowed?

For this, I can think of a space with different dimensionality in different parts. It would be a topological space, but not a manifold. 

[studiot]

How many dimensions does a (straight) line have ?

Most would straightaway say 1

Yet that line is a space of elements that require two pieces of information to specify them.

[Genady]

Just now, studiot said:

two pieces of information

Which?

[studiot]

1 hour ago, Genady said:

Which?

The line is a space of line segments.

Each of which require two pieces of information for example beginning and end points or midpoint and length.

 

Yes it is also a space on individual points (which have zero dimension by themselves) which only require one piece of information.

[Genady]

5 minutes ago, studiot said:

The line is a space of line segments.

Each of which require two pieces of information for example beginning and end points or midpoint and length.

 

Yes it is also a space on individual points (which have zero dimension by themselves) which only require one piece of information.

Ok.

[joigus]

1 hour ago, Genady said:

For this, I can think of a space with different dimensionality in different parts. It would be a topological space, but not a manifold. 

Can you actually provide the example?

Otherwise it's like... "I'm thinking of a space... you know. It's fractal here and non-fractal there, non orientable...", and so on. You see what I mean? Which one is it?

Meanwhile, in coordinate land, say...

f₁(x,y,z)= x+y²-1=0

f₂(x,y,z)=2x+y-cos(xyz)=0

The implicit function theorem guarantees there are ranges of x, y and z that make this a 2-dim analytic manifold, at least for certain values of x, y, z. Then you can worry about metric, angles, and parallel transport. Thus we would have a metric space with parallel transport, so we can talk about infinitesimal distance, angles, and distance along a path. But dimension is there at the very beginning.

[Genady]

6 minutes ago, joigus said:

Can you actually provide the example?

I'll try.

Take a surface of a cone and attach a line to the tip. This space is two-dimensional 'under' the tip and one-dimensional 'above' it. No coordinates can pass the tip.

[Markus Hanke]

Are we not perhaps overthinking this a little? After all, the quote given is from MTW, which is a book about spacetime. My feeling is that what the authors had in mind was a physical space, as well as physical procedures to determine its dimensionality. The more abstract notions mentioned here are all good and well in pure maths, but they don’t necessarily relate to GR.

[Genady]

3 hours ago, Markus Hanke said:

Are we not perhaps overthinking this a little? After all, the quote given is from MTW, which is a book about spacetime. My feeling is that what the authors had in mind was a physical space, as well as physical procedures to determine its dimensionality. The more abstract notions mentioned here are all good and well in pure maths, but they don’t necessarily relate to GR.

Yes. Although they might become relevant at the Planck's scales.

[joigus]

On 9/6/2023 at 6:10 PM, Genady said:

I'll try.

Take a surface of a cone and attach a line to the tip. This space is two-dimensional 'under' the tip and one-dimensional 'above' it. No coordinates can pass the tip.

Oh, OK. I was confused by your words "you are given a space of unknown geometry". You must be given something. You must depart from some assumption.

I think that may be at the root of why some of us have thought you were talking about physics.

10 hours ago, Markus Hanke said:

Are we not perhaps overthinking this a little?

I've been known to do that... 

[Genady]

1 hour ago, joigus said:

I was confused by your words "you are given a space of unknown geometry".

Yes, I realize now that instead of unknown geometry I should've said, arbitrary geometry. Also, instead of smooth space I should've said, topological space. Well, this discussion has clarified not only the answers but also the question. At least, to me. Thanks to everyone.

[joigus]

On 9/7/2023 at 11:08 PM, Genady said:

Yes, I realize now that instead of unknown geometry I should've said, arbitrary geometry. Also, instead of smooth space I should've said, topological space. Well, this discussion has clarified not only the answers but also the question. At least, to me. Thanks to everyone.

This is actually a question that's very close to my heart, so I'll be looking forward to derivations into both pure mathematics as well as physics, as Swanson and Hanke have suggested.