new theory

[Jragon]

the "zero" dimension is a segment, no height or length
1st dimension is a line going a single direction
2nd dimension it is possible to travel any direction along the line, but not up or down.
3rd dimension you can follow the line, or go up or down and leave the line, you can go any direction you want.
5th dimension is time, you can go through time, things change in time, but once time happens, you cannot reverse it and you cannot live it again.
the 4th dimension, opposite of the zero dimension, in zero there is no measurments, in the 4th all measurments have no end. the only 4th dimension is the universe as a whole, it never ends, and never begins and there is no exit or way out. it has always been and always will be. in the 2nd dimension you cannot exit it if the line is connected at both ends to eachother, you will travel around the path. in the 3rd you can exit by going up or down until you leave, but the 4th prevents leaving using the directions involved with the 3rd.

any comments would be great, and if anything is not understood i will try to explain.

[wendy]

Making time into a place or axis implies fatalism. This is because, like a movie, the past, present and future always exist and are fixed.

The escape from this fatalism is to posit the existance of parallel universes. However, the bifracation on this is hideous, unless one believes in parallel choices, too: ie X chooses x implies Y chooses y.

The fault in this comes from representing space-time, not as a graph, but as something that exists. That is, in space-time, there is no notion of continuing time. Instead, one has just lines in space.

The other thing, is that some of us have been out to six, eight, ten dimensions, and we do not see time-as-dimension.

[Jragon]

interesting

but time, like dimension 1 has only one direction.. maybe time isn't a dimension, at this moment i cannot think of anything similar or opposite of time but perhaps there is. much like the oppoiste of zero is 5 in my theory.

[timespace]

the 4th dimension, opposite of the zero dimension

your problem is that you're thinking of a new theory in relativity to older theories.

5th dimension is time, you can go through time, things change in time, but once time happens, you cannot reverse it and you cannot live it again.

If time were a dimension, couldnt you move backwards and forwards through it?

just a few thoughts, its just a theory youre giving, after all

[Jragon]

well, in time you can only travel one direction, in the 1st dimension you can only move one way as well.

and i figure, the dimensions are set up like this:
0, 1, 2, 3, 5, 4
a, b, c, d, b, a
(matching letter means works the same, but infact opposite).
though i have not came up with one to match the 2nd dimension, and the 3rd dimension i pit in the centre, and as the centre there is no opposite/similar.
and again, the 4th could be considered time and 5th can be considered the endless dimension.

thanks for the replies so far.

[jinydu]

Sigh, this is not a theory (in science, a precise, logical framework, built up from a set of well-defined postulates, well supported by experimental evidence). It is only a collection of ideas, without much evidence or logic behind it.

At the most basic level, the dimension of a geometric figure is the number of mutually independent directions in which a particle, when placed on the figure, can travel.

the "zero" dimension is a segment, no height or length

According to the basic geometric definition from high school level math, a zero-dimensional object is a point. A point has no dimensions at all, which is partly why we call it zero-dimensional (not some mystical "0th dimension").

1st dimension is a line going a single direction

What you have in mind there is called a ray; part of a line that starts at a point and continues indefinitely in one direction.

In general, any curve could be considered a one-dimensional object; however, it is generally accepted that the most basic one dimensional object is a straight line. Again, we talk about "one-dimensional figures", not "the first dimension", as if it were some kind of mystical place.

2nd dimension it is possible to travel any direction along the line, but not up or down.

No, that would be one dimensional. Note that the two directions you're probably thinking of ("forward" and "backward", which themselves you haven't properly defined) are not mutually independent, since travelling in the "forward" direction causes you to undergo a negative displacement with respect to the "backwards" direction.

In general, a two-dimensional object is called a surface. Examples are the surface of a sphere and the plane (generally considered the simplest two-dimensional object). Furthermore, even if you choose your plane so that the two mutually independent directions are "left-right" and "up-down" (in your frame of reference), the choice of which of the two to call "the" second dimension is purely arbitrary.

3rd dimension you can follow the line, or go up or down and leave the line,

If the line was vertical (in your frame of reference), the particle will still be moving through one-dimensional space. If, on the other hand, the line were perpendicular to the "up-down" direction (again, in your frame of reference), the particle would be in 2 dimensional space, not three dimensions.

you can go any direction you want.

Taken literally, that refers to an infinite-dimensional space. If you mean that with respect to some particular space, then the dimension is equal to the space's dimension.

5th dimension is time, you can go through time, things change in time, but once time happens, you cannot reverse it and you cannot live it again.

By introducing time, you've now broken away from (pure) geometry. Furthermore, irreversibility has nothing to do with dimensions.

the 4th dimension, opposite of the zero dimension, in zero there is no measurments, in the 4th all measurments have no end.

It seems you're just throwing out more and more assumptions with no chain of logic linking them, let alone evidence to support any of them. First of all, the choice of which dimension (in a four-dimensional space) to call "the" fourth is arbitrary. Furthermore, of course there is an end to the measurements in a (finite) four-dimensional space. And of course, you haven't even properly defined what you mean by "all measurements have no end".

the only 4th dimension is the universe as a whole,

First of all; as far as we know, the universe has 3 spatial dimensions. However, in the Theory of Relativity, events can conveniently be represented using a 4-dimensional spacetime vector; although that's besides the point in this case.

Here's how to construct another four-dimensional space. Choose a point. Draw a line through it. Draw a second line that's perpendicular to the first. Draw a third line that's perpendicular to the first two. Draw a fourth line that's perpendicular to the first three. Choose a distance, and call it the unit distance. Now you have a four dimensional space.

it never ends, and never begins and there is no exit or way out. it has always been and always will be.

It seems you've now digressed into cosmology, a different subject altogether. You're claiming that the universe is infinite in space and time, a claim that 1) Is not relevant to the subject of dimensions and 2) Must be decided at least in part by experimental evidence.

in the 2nd dimension you cannot exit it if the line is connected at both ends to eachother, you will travel around the path.

Now, you seem to be talking about not a ray, or a line, but a closed curve (such as a circle, or ellipse). And closed curves, being curves, are one-dimensional.

in the 3rd you can exit by going up or down until you leave,

That can only happen 1) After you've defined which direction represents "up-down" and 2) If the 3-dimensional space is finite in the up-down direction, in which case the statement is trivially true.

but the 4th prevents leaving using the directions involved with the 3rd.

Only if the four-dimensional space is finite in those directions. And please remember that in a four-dimensional space, which direction you choose to call "the fourth" is purely arbitrary.

any comments would be great, and if anything is not understood i will try to explain.

I suggest you do some reading on dimensions before you try to make "theories" like this. At the very least, you should understand what a dimension is before make "theories" about them. A good place to start would be alkaline's website.

[Jragon]

thanks jinydu, i enjoyed reading your comments

[Keiji]

<insert long post here>

Amen to that! If only this forum had a rewards system...

Anyway, now I will try to interpret this topic myself.

the "zero" dimension is a segment, no height or length

What you are thinking of is a point, not a segment.

1st dimension is a line going a single direction

correct.

2nd dimension it is possible to travel any direction along the line, but not up or down.

No, in the second dimension you can go in four cardinal directions: up, down, left and right...

3rd dimension you can follow the line, or go up or down and leave the line, you can go any direction you want.

Like jinydu said, to go in "any" direction you have to be in an infinite-dimensional space.

5th dimension is time, you can go through time, things change in time, but once time happens, you cannot reverse it and you cannot live it again.

You don't move in time, time just passes. Which is why it isn't a dimension; if it was you would be able to go backwards, or go faster or slower, or stop.

the 4th dimension, opposite of the zero dimension, in zero there is no measurments, in the 4th all measurments have no end.

Measurements don't have ends in any dimensions.

the only 4th dimension is the universe as a whole, it never ends, and never begins and there is no exit or way out. it has always been and always will be.

Everything must start somewhere...

in the 2nd dimension you cannot exit it if the line is connected at both ends to eachother, you will travel around the path.

True, but this:

in the 3rd you can exit by going up or down until you leave

is wrong. No matter what the dimension, you can't exit it...

but the 4th prevents leaving using the directions involved with the 3rd.

How exactly does that work? To stop yourself escaping the 3rd dimension into higher dimensions, you have to restrict yourself to three dimensions, not four.

[Jragon]

hmm i understand more now. what do you mean exactly by infinite-dimensional space is the only way to travel any direction?

[houserichichi]

It's getting technical with the semantics.

When someone (in regular conversation) says they can go in "any direction" they mean they can move up, down, left, right, forward, and backward. Mathematically though when one says "any" direction without first referring to an embedding space, another CAN infer that you require an infinite number of dimensions to go "anywhere". You clearly weren't talking about an infinite dimensional embedding space so I'd disregard those arguments. That's just my impression of what you were trying to say, anyway...though I'm not saying either side was flat out wrong.

[Jragon]

i definatly did not try to say move anyway in mathematical terms. but i understand what it is now.

[RMac2k5]

Jinydu,
If I may be so bold as to comment against what you say in the slightest of manners, while holding onto the fact that you have studied this a great deal more than I have, and conceding the fact that I required a dictionary to fully understand what you were conveying, I think I may in at least some way be enlightening, or at least put a new chess piece on the table for you to take.

You stated,

According to the basic geometric definition from high school level math, a zero-dimensional object is a point. A point has no dimensions at all, which is partly why we call it zero-dimensional (not some mystical "0th dimension").

I disagree a zero-dimensional object has no dimensions.

Now if you fast forward 3 years to the basics of Calculus. You learn in Volume by slicing that you can "stack" 2 dimensional objects face to face to create a 3 dimensional object. Now with this being said. If a plane had no depth, then why when you stack several together bound by given equations do you create a volume?

In a similar case when finding area between curves. Again using integrals, you stack lines side by side to creat an area. If a line has no width how can it create an area?

Again to create the length of a line. You stack points next to eachother to create a line. If a point has no length how can it create a line?

So if points create a line, lines can create a plane, planes can create an object that takes up space, then is not a three dimensional object created out of points? How can that be if it has no height width or depth?

How I see things, a point has infinitely small existence in all dimesnions as do lines, planes, and objects in the third dimension.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

I think it a viable argument that should be considered, that all things exist in all dimensions. Even if only in an infinitely small way.

Oh and something that I found interesting. Not saying anything against what you said, because this isn't even something I've researched. I read once that Einstein said, "As an object approaches infinity, it then approaches zero." So is a ray theoreticaly a universal circle? That's just a question I have. Please respond.

[jinydu]

I think it a viable argument that should be considered, that all things exist in all dimensions. Even if only in an infinitely small way.

You bring up a good point which I did not mention, although your statements may confuse a pre-calculus student.

At a basic level of math, we are taught that (for instance), 2 and 3 dimensional objects are necessarily distinct, that a square must have zero volume and that no amount of stacking will turn a bunch of squares into a cube. This works just fine until...

...we meet calculus (to be honest, actually, I saw this idea in Physics class instead of Math class). Suddenly, squares really do have an (infinitessimal) thickness, and by summing together infinitely many of them (via a definite integral), it is possible to form a cube. And yes, you're right, it is possible to construct any object (whether it has 1, 2, 3 or a million dimensions) by adding together points.

But now we have seem to have a paradox. On the one hand, the ghost of our old middle school geometry insists that we stick with the "squares have zero volume" position (What do you mean you're adding together infinitely many things! Infinity is not a number!). On the other hand, our calculus teacher tells us that assigning an infinitessimal thickness to a square is not only a valid move, but allows us to do grand things we've never done before.

So who should we believe? Isn't mathematics supposed to be consistent (i.e. it contains no contradictions)? For a long time, this question gave mathematicians endless headaches and embarassment. Someone by the name of "Bishop Berkeley" once said: "And what are these same evanescent increments? They are neither finite quantities nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities." He then went on to claim that mathematics was based on an illogical foundation and shouldn't be taken seriously.

Stung by criticisms like these (and wanting to reexamine the foundations of mathematics anyway), mathematicians in the 19th century sought to find a logically precise and unambiguous definition of the "limit", the basic concept in calculus. It was finally Karl Weierstrass who came up with the answer (you can read about it here: http://mathworld.wolfram.com/Limit.html, you only need the first sentence). Mathematicians then went on to show that all the major known theorems of calculus could be recovered from this definition (via the infamous epsilon-delta proofs that most college students fear).

In the case of the definite integral (which is used to form higher-dimensional objects from lower-dimensional ones), the precise definition is given by the Riemann Sum (http://mathworld.wolfram.com/RiemannSum.html). Among other things, it is possible to prove that the actual value of a definite integral "behave as if" 3D objects really could be built out of 2D objects with an "infinitessimal" thickness.

This then, is the resolution of the (apparent) paradox: Your old middle school geometry teacher is (technically) right; squares really do have no thickness. However, treating squares as if they did have an infinitessimal thickness is a useful and intuitive way of dealing with problems, and always gives you the correct answer in the end (so long as you build up the higher-dimensional object in a 'valid' way). Still, it's important to keep in mind that this is just a useful shorthand, and not really the way things are. To see how things really are, you have to look at the precise definition. However, the precise definition (as you will see if you look at the link) is complicated, hard to visualize and difficult to use, which is why people prefer to use the "infinitessimal shortcut".

But keep up the good work! It seems you've been paying attention in class!

Oh and something that I found interesting. Not saying anything against what you said, because this isn't even something I've researched. I read once that Einstein said, "As an object approaches infinity, it then approaches zero." So is a ray theoreticaly a universal circle? That's just a question I have. Please respond.

Like almost all other human beings, Einstein liked to make jokes and witty remarks every once in a while, so you shouldn't take every Einstein quote seriously :wink: . As for your question, I think what you're really trying to ask is:

Is a line segment just a finite part of an infinitely large circle?

The answer is: For the purposes of calculation and intuition yes. I'm sure that in school, you've probably done projectile motion problems. In those problems, the Earth is treated as if it were flat, even though we know that the Earth is in fact round. Why is that? Because the Earth is so large that this curvatue is negligible for the purposes of the problem. To put it more geometrically "on scales much smaller than the circumference of the circle, a circle is indistinguishable from it's tangent line".

If you want the precise mathematical formulation of this idea, study Taylor Series.

[houserichichi]

As an addendum, there's actually a very intuitive and logically consistent way of doing calculus via infinitesimals. We're taught in high school (or wherever you learn your first bouts of calculus nowadays) that the dx at the end of an integral represents an infinitely small limit of a point. In fact this was never shown to be logically consistent (through all calculus) until the 1950s, surprisingly enough. Though it makes calculations much easier and intuitive to grasp, sometimes what we're taught in elementary school, high school, and even early university isn't the whole truth. (For instance, a fraction is nothing like what we intuitively grasp it to be. We assume that 1/2 and 2/4 are the same thing but they're not. That's counterintuitive.)

Another note...the circle is not homeomorphic to its tangent line. They are mathematically distinguishable by that fact alone. For the purpose of intuition I suppose I'll agree with what you wrote (but then, intuition tends to be wrong when we start playing with extremes), but for calculation we'd definitely need to be working in another space.

On that note, what's the general consensus nowadays on the curvature of space? Positive, zero, or negative? Depending on curvature I'd alter my statement above...(thank god for stickler mathematicians, eh?)

[jinydu]

I have in fact heard that some mathematicians have attempted to define infinitessimals rigorously and use them as a foundation for calculus. In fact, they defined a new number line with real numbers, infinitessimal numbers (defined to have an absolute value smaller than any real numbers, except zero) and infinitely large numbers (larger than any real numbers). However, I think we've gotten advanced enough, especially since epsilon-delta is what most students learn.

As for your question, do you mean "the curvature of empty space in our Universe"? That is a question that must be determined experimentally; and I don't see how it's relevant to pure mathematics.

[houserichichi]

Well my logic behind the curvature of the universe would be that the universe is itself the embedding space and if its curvature is positive then a line in space could in fact meet itself (assuming curvature is constant). If that's the case then a projection of the line from the embedding space (universe) down to E^2 would be homeomorphic to a circle. Otherwise this is not the case in general.

By the same logic (I'm just typing as I'm thinking here, bear with), since the universe isn't just some abstract notion but, in fact, the reality in which we exist, physically the line wouldn't be homeomorphic to the circle ever in "real life" because on large scales the universe wouldn't have the structure of E^2, so I suppose that was a futile thought argument anyway.

When I was learning calculus early on I found it much easier to grasp limits, integrals, and derivatives via operations on what I considered infinitesimals. It wasn't until university and classes in analysis and differential geometry that the dim bulb brightened...I suppose in general practice, however, that ignorance is bliss and not to be trifled with...unless you want to be a stickler mathematician too :wink:

Oh, and if I haven't posted this already (or stole it from here), take a look at the file below - it's a free, online textbook on elementary calculus written via infinitesimals. I thought it a gem for those who've taken a first year in the subject and wanted an alternative viewpoint (and perhaps the more intuitive one at that? I'll leave it up for debate)

http://www.math.wisc.edu/~keisler/keislercalc1.pdf
(Large file: 24 megs. PDF. Right click and Save As)

[Eric B]

You don't move in time, time just passes. Which is why it isn't a dimension; if it was you would be able to go backwards, or go faster or slower, or stop.

Actually; I have seen relativists claim that time was a dimension that we are automatically moving in one direction at light speed in when we are sitting still in the other 3 dimensions. When we move in one of those other dimensions toward the speed of light, then we slow down in time, and that space dimension we are moving in becomes like time. (It also supposedly shrinks down to zero, but I don't rememner how they explained that).

[jinydu]

You don't move in time, time just passes. Which is why it isn't a dimension; if it was you would be able to go backwards, or go faster or slower, or stop.

Actually; I have seen relativists claim that time was a dimension that we are automatically moving in one direction at light speed in when we are sitting still in the other 3 dimensions. When we move in one of those other dimensions toward the speed of light, then we slow down in time, and that space dimension we are moving in becomes like time. (It also supposedly shrinks down to zero, but I don't rememner how they explained that).

You're thinking about time dilation, part of the Theory of Special Relativity. Yes, physicists who work with Special Relativity often treat time as a dimension. Why? Because in Special Relativity, events are reprsented using 4-dimensional spacetime vectors. However, in the context of this forum, "dimension' is generally taken to mean "perpendicular direction in space".