Dimensions and Directions.

[thief]

Thief here....

I'm a toolmaker. I deal hands on with '3d' all day.

One point has no size, or form and is only a location device.

A second point is required, as location is relative.

A third point does not demonstrate height.

Triangles, squares, circles,etc, etc,etc...are plane geometries and have no height.

Spheres, cubes, cones, etc,etc,etc, all have points above or below a given plane. Counting you points ..one, two, three, does not describe geometry as 3d.

Previous posts didn't appear to show this.

The fourth dimension is movement.

[throng]

Thief here....

I'm a toolmaker. I deal hands on with '3d' all day.

One point has no size, or form and is only a location device.

A second point is required, as location is relative.

A third point does not demonstrate height.

Triangles, squares, circles,etc, etc,etc...are plane geometries and have no height.

Spheres, cubes, cones, etc,etc,etc, all have points above or below a given plane. Counting you points ..one, two, three, does not describe geometry as 3d.

Previous posts didn't appear to show this.

The fourth dimension is movement.

 

Well, the thing with one point is it has no location, anywhere, nowhere - irrelevent.

 

A second point would be placed in relation, but can either point be relative to the other's 'non-location'? Consider two points without a third point of observation. There's a distance of ?. What is a single distance without relative?

 

With three points and and an angle get relativity, a relationship between distance and angle, creating area, which is consequential.

 

And four points would enclose volume or 3D vacuum.

 

So, dimensions are like locations, points. Vacuum requires four 'location points' (0D).

 

The simplex tetrahedron is the only object that contains exactly equal proportions, and once five locations exist it is inevitable that there will be relative distances.

 

Thats my understanding of dimensions....

 

Edited February 1, 2009 by throng
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[thief]

Thief here...

Plane geometry lacks height.

The first point has no size.

The second point designates length only, and that length is terminal.

A third point cannot add height.

A fourth point is required for height...but may not be in the same plane as the first three.

The sphere or a circle can be dimensioned by two points ( one length ), and this may be where confusion steps in.

Movement can be noted only when any one line (a to b ) is measured at some different length, at various time.

Edited February 1, 2009 by thief

[throng]

Thief here...

Plane geometry lacks height.

The first point has no size.

The second point designates length only, and that length is terminal.

A third point cannot add height.

A fourth point is required for height...but may not be in the same plane as the first three.

The sphere or a circle can be dimensioned by two points ( one length ), and this may be where confusion steps in.

Movement can be noted only when any one line (a to b ) is measured at some different length, at various time.

 

 

The missing ingredient is angle, I mean if you have 3 points alligned, is that described as 1D or 2D? Since it has length and angle (180 deg) it has 2 degrees of measure.

 

So I say area does not define 2D, any three locations does.

 

Therefore 2 points is a distance of 1D but a line of infinite points is 2D.

 

How can 2 definitions (length and angle) apply to 1D? 1D means one measure?

 

I mean angle is a measurement.

[thief]

Thief here...

For a moment or two...think like an artist.

On a flat surface...any number of points, any number of angles, any number of contours, let your imagination go as far as you can.

All of the geometry is "plane".

The circle is unusual, as it can be designated in size by the length of one line from center point to boundary (radius), or one line through center (diameter).

Angles are a study almost to themselves (trigonometry), but without that point above or below the plane, all that you see is flat 2d.

With the addition of any one point above or below the plane, 3d takes hold.

Circles becomes cones...or spheres. Circles can also become cylinders.

 

Movement is noted when the position of a figure, is not congruent to the position of another figure. That is to say, the measure between one position is not fixed to the position of a reference, and periodic measuring will note increase or decrease of distance.(time)(4d)

High velocity equations indicate distortion to simple geometry.(physically)

[ajb]

Of course one can't say if zero dimensions is reality, and by no means can it be perceived, and since it will never be perceived, it alone speaks of it's existance.

 

One can question if any dimensions are real, because we are applying the mathematical theory of geometry to model nature. Zero dimension is just as real in that sense.

 

So again, if we are dealing with any physical theory that describes nature using a manifold (or similar "classical" geometry) then the notion of a point is clear. An element of the underlying topological set.

 

It might as well be God.

 

We should leave God out of this.

 

 

E=m because Planck derived constants and experiments confirm.

 

S as a function of U. A unit in which nothing can happen, so a description of one dimension.

 

We want a singularity on which to base the Grand Theory Of Everything and seek Higgs boson, T=0 and immaterial concepts without mass. Might as well be God.

 

Of course E is the singularity that has been so long known, we measure change, but a changing singular thing.

 

 

A singularity and a point are not to be identified directly.

 

A singularity is a point that when you approach it some quantity diverges (loosely as you may need to be much more technical than this).

 

This point exists as a "point" independent as it existence as a singularity.

 

In general relativity, a singularity usually refers to a point at which the curvature diverges. (This is a very lose "hands on definition" and again, you may need to be much more technical that this in practice).

 

 

So naturally the laws of physics lie on a very basic singularity, E, the truth. Which is 'I don't know'.

 

'All that is not known', zero dimensional?

 

Now you have lost me!

[throng]

Another thing is fractal measure.

 

If we have a line of 10cm it has a finite length yet infinite locations exist, so it is also 'endless'.

 

Theoretically, the line is also longer than 10cm, and when it is, an angle is necessary and the line becomes an infinite loop as a 2D shape.

 

Within the infinite loop lies a finite area, and within the area infinite locations exist, so the area is larger than it's perimeter, and by angular direction, becomes a finite volume surrounded by a continuous surface area.

 

So if we say volume contains infinite locations and is therefore 'larger' than its boundry, volume would somehow become continuous and 'surround' a new, as yet unknown, finite phenomena.

 

Also, area is not a fractal of volume, a line is not a fractal of either area or volume, and a point is not a linear fractal, so when we say 'an infinite number of locations' we actually refer to angular references.

 

Only three locations can be a finite length, and that is true of area as well. Only four locations make finite volume? Five locations makes a relative distance.

 

When five locations exist volume becomes continuous and 'surrounds' finite relativity?

Edited February 6, 2009 by throng

[ajb]

For spaces like [math]R^{n}[/math] the fractal dimension* is equal to the Hausdorff dimension. In fact, you should use the discrepancy between the two to define a fractal.

 

 

* I believe there are many ways to define this.

[throng]

One can question if any dimensions are real, because we are applying the mathematical theory of geometry to model nature. Zero dimension is just as real in that sense.

 

Hmmm... it's more like dimensions are a representation to express the relationships between locations, like distance or angle.

 

So again, if we are dealing with any physical theory that describes nature using a manifold (or similar "classical" geometry) then the notion of a point is clear. An element of the underlying topological set.

 

I guess points are a devise that can be used in relation to other points. Two points really only give finite definition to each other, so we make distance the finite element.

 

 

 

A singularity and a point are not to be identified directly.

 

A singularity is a point that when you approach it some quantity diverges (loosely as you may need to be much more technical than this).

 

This point exists as a "point" independent as it existence as a singularity.

 

In general relativity, a singularity usually refers to a point at which the curvature diverges. (This is a very lose "hands on definition" and again, you may need to be much more technical that this in practice).

 

I think the main thing is, a single lone point is not relative to anything else, and any two points are identical, a point can not used to measure anything else, so we can only invent definitions, depending on context.

[ajb]

Hmmm... it's more like dimensions are a representation to express the relationships between locations, like distance or angle.

 

In the sense that they are the minimum number of coordinates needed to describe a point.

 

I guess points are a devise that can be used in relation to other points. Two points really only give finite definition to each other, so we make distance the finite element.

 

Generically, there is no notion of distance between two points. Although, on a topological space, which I assume we are taking about there is a notion of points being close.

 

Distance between (near by) points requires extra structure. On a (smooth or differentiable) manifold this structure is known as a Riemannian metric. I'll stress this again, it is an extra structure that needs to be specified.

 

 

 

I think the main thing is, a single lone point is not relative to anything else, and any two points are identical, a point can not used to measure anything else, so we can only invent definitions, depending on context.

 

As an aside, I tend to work in the category of smooth supermanifolds. These "spaces" do not have points. (Well, not without some extra work, i.e. the functor of points.) You can do a lot of geometry without points.

[kleinwolf]

Why do we represent nucleons as small sphere when they are build of 3 quarks....this describes a plane (2-dimensions) ?

[ajb]

Why do we represent nucleons as small sphere when they are build of 3 quarks....this describes a plane (2-dimensions) ?

 

Building nucleons from 3 quarks is very nieve.

 

Anyway, it would be wrong to think of a nucleon to be defined by three points. Quantum mechanics will smear this description.

[boxhead]

Time does slow down as you approach the speed of light. For something that actually travels at c, in this case a photon, you will not experience time at all. Photons do not experience time.

 

Photons do experience time as it passes but they can not express it (in mathematical manipulations) because they leave time behind before they can express it in a way that we can understand mathematically.

 

 

[[ Mod note: Link to an email deleted. ]]

Edited February 17, 2009 by mooeypoo

[Sovereign]

A photon has zero dimensions.

 

Wait there are things that exist that have no length, no width, and no height? That's one of the funniest things I've ever heard.

[Klaynos]

Photons do experience time as it passes but they can not express it (in mathematical manipulations) because they leave time behind before they can express it in a way that we can understand mathematically.

 

Would you care to show that mathematically?

[throng]

Wait there are things that exist that have no length, no width, and no height? That's one of the funniest things I've ever heard.

 

Of course massless particles have no length width or height, and do not occupy volume. Is that right? I thought is was a given, but I've since seen debate about it.

 

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In the sense that they are the minimum number of coordinates needed to describe a point.

 

That is quite a concept, since 0D doesn't define itself, it is a concept requiring relative points if it is to be a point.

 

 

 

Generically, there is no notion of distance between two points. Although, on a topological space, which I assume we are taking about there is a notion of points being close.

 

Distance between (near by) points requires extra structure. On a (smooth or differentiable) manifold this structure is known as a Riemannian metric. I'll stress this again, it is an extra structure that needs to be specified.

 

I think I understand, and frankly, I'm glad you said so, because I have alot of trouble expressing this notion.

 

 

 

 

 

As an aside, I tend to work in the category of smooth supermanifolds. These "spaces" do not have points. (Well, not without some extra work, i.e. the functor of points.) You can do a lot of geometry without points.

 

And I can't believe you say 'without points'. I thought I was in my own private Idaho, because I conceive of geometric fundamentals that don't require points, but I have no expression, so I am very interested in that indeed, thanks.

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I'll have to study that tonight, it'll take a while to asorb,

Edited February 18, 2009 by throng
Consecutive posts merged.

[ajb]

You don't need any other points to define a point. Think about [math]\mathbb{R}^{0}[/math]. What is it and how many points does it have (as a set).

 

If you read almost any book on modern geometry (= theory of ( usually differentiable or smooth) manifolds) then the notion of a point is presented as the most important thing. However, you can formulate just about all of this with out talking about points by using sheaf theory which used open sets and not points.

 

For example, smooth manifolds can be defined as locally Euclidean, Hausdorff and second countable topological spaces together with a sheaf of smooth functions. (also known as the structure sheaf)

 

 

This may seem a bit of "overkill" but it becomes the right way to think for the more general geometries that arise in algebraic geometry and noncommutative geometry. (Both of which the notion of a point is far from trivial).

 

The modern way of thinking of a geometry is really as an associative (need not be commutative) algebra which we treat as the structure sheaf of some "space".

 

In doing so we do lose some of our initial geometric feeling. However, if we can formulate some "classical" differential geometry in terms of the structure sheaf, then we stand a chance of that notion passing over to more general geometries.

 

For example, vector fields on a smooth manifold can be defined as derivations of sections of the structure sheaf, i.e. derivatives! This then passes over to the noncommutative case directly without talking about points.

 

So, we can push points to a lower standing in differential geometry via the structure sheaf (though points are needed to define the underlying topological space, i.e. a set + a topology). In doing so we learn how to deal with "spaces without points".

 

Now, supermanifolds are a very special and very nice class of noncommutative or pointless geometries. They arise as a modification of the structure sheaf over a smooth manifold and are a kind of ringed space. They look more like a scheme than a manifold. So, there is the notion of a point as a point on the underlying smooth manifold. However, that is not enough to describe supermanifolds. There is simply "more going on" than just the underlying points. That is why I say they are "non-set theoretical" or "pointless".

 

Supermanifolds admit local coordinates, much like manifolds do. One way of carrying over the notions of geometry to supermanifolds is to exploit the local coordinates. I like this method as it is very direct and clear.

 

This can then be rewritten, if desired in terms of the structure sheaf. Again, we make no mention of points.

[throng]

R^0 = 1, but isn't R refering to a radius or length and therefore the point is actually defined by it's relationship with a line?

 

From these axioms, one can prove that there is exactly one identity morphism for every object.

 

 

The morphisms from x to x are the elements of the monoid

 

'Trivial' also refers to solutions to an equation that have a very simple structure, but for the sake of completeness cannot be omitted.

 

y' = y

 

where y = f(x) is a function whose derivative is y′. The trivial solution is

 

y = 0, the zero function

 

Is there a way of using this geometry to explain that there isn't actual distance between two zero dimensional points, or is there a another way of communicating the concept?

 

I am finding it difficult to grasp all this, I think my math is very primative really.

 

 

I am very pleased that a point is considered to be important, it means I'm not making stuff up, and I thought I was developing a deluded obsession.

Edited February 19, 2009 by throng

[ajb]

[math]\mathbb{R}^{0} = \{p\}[/math], i.e. the set containing one element.

 

Really, by the distance between two (near by) points you are assigning a length to the vector that joins them. This requires a metric as we have discussed.

[throng]

[math]\mathbb{R}^{0} = \{p\}[/math], i.e. the set containing one element.

 

Really, by the distance between two (near by) points you are assigning a length to the vector that joins them. This requires a metric as we have discussed.

 

Oh thanks for that, I'm just reading back to that now.

 

What I am trying to say is two zero dimensional points represent two possible locations, so the distance is zero, because there is no location between the two. Is this what is called 'near by'?

 

x = y. So,

 

d(x,y)=d(y,x)=0

 

Is that the basic jist of it?

 

Three points in a line z = x = y and d(x,z) = 0

 

!!x+y!!=!!x!!+!!y!!

Edited February 19, 2009 by throng

[ajb]

Near by is really a statement in topology. Think of infinitesimally close. Otherwise a vector could not join them and I need a curve. The distance would then be the length of the curve. To simplify things it makes sense to consider very close points, in which the curve is just a straight line, i.e. a vector.

[throng]

Near by is really a statement in topology. Think of infinitesimally close. Otherwise a vector could not join them and I need a curve. The distance would then be the length of the curve. To simplify things it makes sense to consider very close points, in which the curve is just a straight line, i.e. a vector.

 

Thank you very much for all the assistance you have provided me. I find it diffucult to understand this geometry. I don't have familiarality it takes for real understanding of it. I have never been interested in math/physics before, but suddenly I developed an obsession with dots (points), I guess 0 point locations in physics were not accepted as actual phenomena until recently, and now they are considered possible.

 

I don't want to express anything spectacular by learning this kind of mathematical 'language'. And any idea I might have has probably been done before.

 

My main problem is: If only two locations exist, without a location between them, movement between the points can't be measured in time, hence distance, so if I think of an infinitismal distance, that would require a third infinitismal point location.

 

Can two seperate points be described without a vector? I need something that is not a line or distance, and the 0D points are identical of course.

 

I need a line that is two points long, just the two ends of distance, without the middle bit. I'm going to try to grasp length contraction at c to find ideas.

 

I don't know if I understood, but I thought if x=y then d(x,y)=0. Does the d represent distance?

 

Is there a way?

 

Thanks again, I really appreciate your time.

 

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Thief here...

For a moment or two...think like an artist.

On a flat surface...any number of points, any number of angles, any number of contours, let your imagination go as far as you can.

All of the geometry is "plane".

The circle is unusual, as it can be designated in size by the length of one line from center point to boundary (radius), or one line through center (diameter).

Angles are a study almost to themselves (trigonometry), but without that point above or below the plane, all that you see is flat 2d.

With the addition of any one point above or below the plane, 3d takes hold.

Circles becomes cones...or spheres. Circles can also become cylinders.

 

Movement is noted when the position of a figure, is not congruent to the position of another figure. That is to say, the measure between one position is not fixed to the position of a reference, and periodic measuring will note increase or decrease of distance.(time)(4d)

High velocity equations indicate distortion to simple geometry.(physically)

 

 

I think a plane is defined by a minimum of three points, though a point of observation in the third dimension is required to realise a plane.

 

Also if the three points are 'touching' in an equalateral triangle formation, there is no area because there is no distance between points, so perhaps four points in two dimensions are required, because a RELATIVE distance would be inevitable. Otherwise area is not required for 2D.

 

Is this a valid concept do you think?

Edited February 21, 2009 by throng
Consecutive posts merged.

[thief]

Thief here...

Correct to say three points define a plane.

Three points also define the boundaries (area) of a triangle.

Four points for rectangular items,five for pentagrams,etc,etc,

These are plane geometries...still 2d.

 

A point, above or below the plane, is needed for 3d.

 

Observation of 2d plane geometry would be difficult, within the confines of the plane.

Edited February 27, 2009 by thief
typo

[throng]

Thief here...

Correct to say three points define a plane.

Three points also define the boundaries (area) of a triangle.

Four points for rectangular items,five for pentagrams,etc,etc,

These are plane geometries...still 2d.

 

A point, above or below the plane, is needed for 3d.

 

Observation of 2d plane geometry would be difficult, within the confines of the plane.

 

That is what becomes remarkable, because 3D is easily observable from with a volume, and also the number of points is relevent because the universe is a 'relative' thing, and in reality space need only exist if distance is relative, and distance is requisite of locations (in this case 0D points).

 

So if there are three 0D locations equalaterally spaced all the distance/angles are Identical, not relative, however, if a fourth point is added to the plane it is inevitable that a relative distance would occur and a 2D space would be required.

 

Just geometrically musing, so don't shoot me.

 

[thief]

Thief here...I've been busy, Throng....not sure of your last post.

 

Just a quick look and run...

Some artists, (like myself) can make a drawing look quite good...like a photo.

But for all the effort it takes...the drawing is considered 2d.

Looking directly over the edge...you cannot be sure what you see.

Above the plane...the image appears to take shape....be not yet 3d.

Drawings are optical illusions.

 

Translate the drawing into a carved object....3d.

 

You can have any number of points in a plane...show any shape...but without a point above or below the plane....it's all flat....2d....width and length...

no height.