Surfaces in higher dimensions

[geordief]

What is (unless there is more than one definition,perhaps?) the mathematical  definition of a surface in a general sense?

I am interested to know in the context of intrinsic curvature but feel I need to get this concept well understood first.

For example must  a  mathematical "surface" in a 4-D space be 2-dimensional (like a skin) or is it 3-dimensional (like a volume)?

If it is 3-dimensional,what defines it as a surface?

[studiot]

Hello geordie,

Yes there is more than one definition of a surface in Mathematics, and yes they exist in high dimensions and are called hypersurfaces.

 

To best understand this start with 2 dimensions.

A surface is a two dimensional plot or graph in three dimensions, of the function z = f(x,y) or f(x,y,z) = 0

 

Another definition is that a surface is a connected set of locally flat points.

Such a set can be a subset or subspace of any number of higher dimensions, but is still a two dimensional manifold.

I will draw some (hopefully helpful) diagrams when I have time.

[studiot]

Have you lost interest here?

[geordief]

No, not at all. I was  waiting for  you to show me the diagrams but ,yes I would have a follow up question or two  in the mean time if that's OK.

When you say "connected set of locally flat points" does "connected"** have a special ,technical meaning- or just joined  by some relation? And "flat" , is that defined by a derivative(s) being constant at the point?

 

And are all the hypersurfaces 2-dimensional then? (I was imagining somehow they would be "one dimension down" from  the space they were*in -that is a mistaken idea ,I suppose?)

 

*ie I was imagining that a "surface" in spacetime might be  a 3-dimensional set of all  points that shared the same time according to an arbitrary point of reference -and that that might be generalized.

**You used the concept of "connection" earlier in another thread

http://www.scienceforums.net/topic/101620-intrinsic-curvature/

"The relationship between two charts is called a connection. This is usually established by the 'parallel transport rule'".

 

Are you using the word in the same way in both contexts?

[geordief]

On 11/4/2017 at 11:42 PM, studiot said:

Have you lost interest here?

Just quoting you simply in case you didn't notice that I answered you just above  and to bring your attention to it ;-)

[studiot]

Sorry I haven’t done any significant posting for a while, but the crap new forum software doesn’t play nice with my style of posting and I’ve been too busy to cope with that recently.

I’m an applied mathematician which means I concentrate on getting results and leave the burdensome task of making sure the theory is ship shape and watertight to the pure mathematicians and trust them to do that.

So here’s the deal.

I’m offering a unique mix of geometry, topology and mathematical analysis, blended to serve a particular purpose. After al  these are branches of the same subject, not separate subjects on their own.

You have presented a good opening post and explained your need and goals clearly.

A topological view is too general; a geometric one to restrictive and analysis is required because you want to do calculus on surfaces.

Intrinsic curvature is firmly placed in the classical differential geometry of Euler.

So I am going to start with a few simple ideas, some classical, some modern, and develop them to a situation where you have a useful and understandable object (surface) you can do calculus on.

 

Euclid said

Quote

A point is that which hath no part

By which we mean today that

0)      A point is of zero dimensions or is a 0 dimensional space.

1)      A line of one dimension or is a 1 dimensional space.

2)      A surface is of two dimensions or is a 2 dimensional space.

3)      A solid is of three dimensions or is a 3 dimensional space.

I will return to this (0) (1) (2) (3) theme a few times from different points of view., but first a theorem.

 

If we take R to mean all the real numbers and p to mean any one particular positive integer we call the result R^p space, which has dimension p.

R⁰ has exactly one point

For any p > 0, R^p has equal numbers of points

Thus R^m has the same number of points as Rⁿ

Or we can establish  a one-to-one mapping between any R^m and any Rⁿ

 

This is the underlying reason that allows us to write down (and sometimes solve) functions and equations in Mathematical Physics and do ‘calculus' on them to find and compare properties like curvature.

So

0)      R⁰, a zero dimensional space is a point

1)      R¹, a one dimensional space is a line
Notes
A line in this sense has no ends and may be straight or curved, looped or open and extended.
Parts of a line have ends and are called a line segment or just a segment.

2)      R², a two dimensional space has no boundaries, just as a one dimensional line has no ends

3)      R³, a three dimensional space, is the normal everyday space we live and do Physics in and again has no boundaries.

 

Back again to my 0,1,2,3 from a different point of view.

0)      A point is indivisible.

1)      A point divides a line into two parts.



Note the line is considered to extend indefinitely in both directions.

2)      A line divides a surface into two parts

 

3)      A surface divides 3 dimensional space into two parts.

 

Summary

an (n-1) dimensional space divides an n dimensional space into two parts.

We call an (n-1) dimensional space an (n-1) dimensional hypersurface.

 

But can an (n-2) space divide an n dimensional space?

 

Well a single point can’t divide a plane – you can always go around it

A line can exist in 3 dimensional space, but you can always go around it.

So no, spaces of lower order than (n-1) can’t divide n space and the reason why we single out (n-1) space for special treatment.

 

This last bit introduced some new ideas,

That we can embed a lower dimensional space in one of higher dimension,

Or equivalently we can select a part of a space to create a complete space of lower dimension.

This selection is then called a subspace.

So

 

The point in FIG1 is a zero dimensional space embedded in a one dimensional space

The line in FIG2 is a one dimensional subspace embedded in a two dimensional space

The surface in FIG3 is a two dimensional subspace embedded in a three dimensional space

 

But

There is also something different between FIGs 2 and 3 and FIG 1.

 

FIG2 contains a looped line as well as an extended one  and FIG3 a closed surface, as well as an extended one.

 

This brings us to the topological definition of a surface.

 

“A surface is a two dimensional boundary between a solid object in a 3 dimensional space”

 

As this is a work in progress

We shall see in the next instalment that this is too general for our purposes, and find out that what we need to retain from topology is connectness and compactness to enable continuity and therefore calculus

 

We will also revisit the 0,1,2,3 again to introduce parameters.

 

[geordief]

1 hour ago, studiot said:

Sorry I haven’t done any significant posting for a while, but the crap new forum software doesn’t play nice with my style of posting and I’ve been too busy to cope with that recently.

I’m an applied mathematician which means I concentrate on getting results and leave the burdensome task of making sure the theory is ship shape and watertight to the pure mathematicians and trust them to do that.

So here’s the deal.

I’m offering a unique mix of geometry, topology and mathematical analysis, blended to serve a particular purpose. After al  these are branches of the same subject, not separate subjects on their own.

You have presented a good opening post and explained your need and goals clearly.

 

 

thanks. That was a lot of work for you

 

I have followed  your post  well enough so far I think

 

Just a small question (it might not be the right time to answer) :Does that  topological definition of a surface applying to 3D space  not  seem to preclude its applicability in  4D spacetime ?   I was  under the possibly naive impression that topology did have relevance to GR..   

[studiot]

21 hours ago, geordief said:

thanks. That was a lot of work for you

 

I have followed  your post  well enough so far I think

 

Just a small question (it might not be the right time to answer) :Does that  topological definition of a surface applying to 3D space  not  seem to preclude its applicability in  4D spacetime ?   I was  under the possibly naive impression that topology did have relevance to GR..   

No there is nothing wrong with the question, it is a good one.

Further I do have you intended goal as the ultimate destination (hopefully over the weekend).

Topology is concerned with the next step connectedness and also the continuity of mappings.

However not all are differentiable and we need to select manifolds that are differentiable. That is why topology is too wide as it includes those.

 

I hope you liked the method of studying the simpler lower dimensions and extending both the results and differences to higher ones.

 

Meanwhile this Wikipedia article is more accessible than most on the subject.

 

https://en.wikipedia.org/wiki/Real_coordinate_space

[geordief]

I feel comfortable with  your posts (having gone through them  a few times).

 

The Wikipedia article  is much harder and  I have doubts that I can  digest too much of it even though  it felt familiar and comfortable  at first

 

I have been trying trying to understand the affine space section (in that article). Is the (or a)  point there that ,if we  choose an origin in  the "parent" space then no point in  the affine space  has  fixed connection with that point?

Your quote"I hope you liked the method of studying the simpler lower dimensions and extending both the results and differences to higher ones."

Yes ,indeed I like the approach of starting with  the dimensions we are used to  and then (hopefully) generalizing. Baby steps  hopefully will lead to a wider perspective.

 

 

 

 

Edited November 11, 2017 by geordief
Edited my thought on affine space

[studiot]

I'm afraid that the Wiki article is one of the easiest I have seen, which is why I chose it.

Keep at it and hopefully enough will come clear.

 

Very quickly affine is what happens when we add a constant to linear and arises from the difference between linear and straight line.

The definition of linear is that for any function f(x) multiplied by a constant af(x) = f(ax)

If we add another constant to x this is not satisfied.

My sketch shows this is equivalent to moving the origin to  a point C

Look at my diagram and try a few straight lines and sketching in the effect of

y = a(mx + c) and  y = m(ax) + c

When we move the origin to a new point c it is an affine transformation

 

[geordief]

4 minutes ago, studiot said:

I'm afraid that the Wiki article is one of the easiest I have seen, which is why I chose it.

Keep at it and hopefully enough will come clear.

 

Very quickly affine is what happens when we add a constant to linear and arises from the difference between linear and straight line.

The definition of linear is that for any function f(x) multiplied by a constant af(x) = f(ax)

If we add another constant to x this is not satisfied.

My sketch shows this is equivalent to moving the origin to  a point C

Look at my diagram and try a few straight lines and sketching in the effect of

y = a(mx + c) and  y = m(ax) + c

When we move the origin to a new point c it is an affine transformation

 

thanks.

 

I am going out now ,so I will have to  look at this later as all I will have is a smartphone phone  until this evening and that is a bit limiting.

 

I will keep at that Wikipedia article....

[fourth dimension]

Rotation in four-dimensional space.

https://youtu.be/vN9T8CHrGo8
The 5-cell is an analog of the tetrahedron.

https://youtu.be/z_KnvGGwpAo
Tesseract is a four-dimensional hypercube - an analog of a cube.

https://youtu.be/HsecXtfd_xs
The 16-cell is an analog of the octahedron.

https://youtu.be/1-oj34hmO1Q
The 24-cell is one of the regular polytope.

https://youtu.be/w3-TqPXKlVk
The hypersphere is an analog of the sphere.

[geordief]

20 hours ago, fourth dimension said:

Rotation in four-dimensional space.

https://youtu.be/vN9T8CHrGo8
The 5-cell is an analog of the tetrahedron.

https://youtu.be/z_KnvGGwpAo
Tesseract is a four-dimensional hypercube - an analog of a cube.

https://youtu.be/HsecXtfd_xs
The 16-cell is an analog of the octahedron.

https://youtu.be/1-oj34hmO1Q
The 24-cell is one of the regular polytope.

https://youtu.be/w3-TqPXKlVk
The hypersphere is an analog of the sphere.

I am  not sure to make of those representations. Perhaps they are mathematically/geometrically consistent but it is too hard for me to follow the analysis of how they are constructed.

 

I would ,though be interested (if only from a pedagogic point of view) whether the surfaces/hypersurfaces Studiot  has described can be shown graphically in a similar way (or even as part of the tesseract)

 

Even if they cannot ,that in itself might  be worth knowing. 

 

At the present time .I am especially interested in any surfaces  in  3D+1 spacetime .Previously I have asked whether such mathematical surfaces can be created by setting spatial (or perhaps temporal) values to a constant  but got no specific answer at the time.

@Studiot :I don't want to get ahead of myself and am looking forward to the next installment

Edited November 12, 2017 by geordief

[studiot]

I am going to start this instalment with a quick look at the modulus function.

 

 

There is a difficulty at x = 0 where there is a sharp corner.

There are two tangents.

If we approach from A the tangent is AA'
But if we approach from B the tangent is B’B

 

The function y = |x| is not differentiable at 0 because of this.

Functions which are not differentiable at some point are not smooth.

In topological terms y = |x| is a 1D manifold, but not a differentiable manifold.

When we step up through the dimensions we want differentiable manifolds.

This is important because tangents and radii of curvature define perpendicular lines and we cannot accept competing tangents and radii.

Note that as we step up through the dimensions tangent lines become tangent planes and ‘tangent’ solids and…… also called tangent hyperplanes.

 

So what sort of manifold do we want, stepping up to 2D surfaces?

 

A football surface or a brick surface?

 

 

Topologically speaking the football surface are equivalent to the brick surface.

The surfaces are each 2D manifolds that can be seamlessly converted into each other in true blue topological fashion.

Note that I did not say that the football or brick themselves are 3D manifolds – We will examine that next time.

 

Whilst the surface of the football is differentiable at every point and therefore smooth, the edges of the brick act like the origin in the modulus function.

So the brick is not a smooth manifold and must be excluded from consideration.

OK so now it s time to get out your stamp collection to examine the football more closely and then step up to Fig7.

 

Fig6 is a 2D boundary round a 3D object in 3D space.

Fig7 is a 3D boundary around a 4D object in 4D space.
 

So are your stamps licked and ready?

The stick one onto each rectangle on the football, until you have covered it entirely.

Each stamp is a small segment of 2D space that can be stuck on to form an impenetrable barrier surrounding the ball in 3 space.

There are no gaps, although overlap is permissible.
The inside of the ball is now disconnected from the outside and this surface is curved in 3 space in such a way that it can be detected by 2D ‘flatlanders’ living on the ball

(Newcastle United players?)

This is intrinsic curvature – which we will explore in detail next time.

In just the same way we as can surround (a portion of) 3D space with a 2D surface we can surround (a portion of) 4D space with a 3D hypersurface, if we follow the same rules.

We have achieved you first goal which is to reach 4D space so I think this is a good place to stop and get into print.

Next time we start to look into the more serious mathematics of 2D curvature in 3D and see how we can step it up to 3D in 4D.

 

[geordief]

Are we  going to set  the spacetime  distance from an origin at any constant (or simply any s) and  examine  the 3D mathematical objects created when we set  any of the 4 variables  to a constant (similarly to how   one does partial differentiation)?

 

(I am quite OK  with manifolds needing to be differentiable) 

 

 

[fourth dimension]

Separate cells of the 5-cell (pentachoron).

Separate cells of the 8-cell (tesseract).

[fourth dimension]

 

Separate cells of the 16-cell (hexadecachoron).

Separate cells of the 24-cell (icositetrachoron).

Edited July 25, 2018 by fourth dimension