Naming the Individual Dimensions

[Eric B]

We often say that since the different dimensions are the same in any rotation, that none are fixed. However, we do often think of the 2nd dimensions as lacking "depth", and the 1st dimension as lacking height or width.
Actually, when you think about it, we look at those lower dimensions, we are looking perpendicular to their dimensions. A person in "flatland" does not see himself as lacking "depth"; remember, he is not looking "out" back at us, bur rather to our left or right along the board he is drawn on, and has no consciousness of any "left" or "right" of his own, to "see" us back. So what we see as "width", he sees as his own "depth". Likewise, in lineland, any being would have only two directions to "look out at", and have nothing but "depth".

So from this, we can identify "depth", not as "the third dimension" as many often think, but rather the first and primary dimension! In any space, one has to look "out", "ahead" into that space, in order to perceive that realm, before he can look any other direction like "left/right", or "up/down".
The next important dimension, is the one in which we are anchored to the surface of solid objects in our space. "Up" and "down" become so fixed to us, because we are always pulled along in the vertical dimension on earth. Even if we were floating around in space, we would be used to thinking in terms of our own "up" and "down". So if you had centers of gravity in flatland, then its inhabitants would see their two dimensions as "up and down", (height) and "back and forth" (depth). In lineland, the single dimension would have to take on both functions. Still, with or without a center of gravity, you would think in terms of "ahead" and "behind" before any "up" and "down". But that one dimension would have to serve as both on any center of gravity. (Of course, "planets" with "gravity" are only hypothetical in lineland and flatland) And, of course, the lower the dimension, the less freedom you have.
Even if we conceive of a flatland rotated to the horizontal, so that it appears to have width and depth, and no height, its inhabitants would see one of those dimensions as height. They would also look "straight ahead" at lineland as having height, but no "depth" (or possibly as having depth, but no height), somewhat the way we look at their space.

It is in the third dimension, where we have our foundational "back and forth" (depth) where each observer looks out into his universe; and also the next fundamental "up and down" (height) where he is anchored to his world. We also add this new, extra dimension, which we call "right and left" (width). It is only needed to give us the additional freedom of 3 dimensions. we do not depend on it to look out into the universe, nor be anchored to a body in the universe.

So from this, we can assign:

1st Dimension-depth
2nd Dimension=height
3rd dimension=width.

so now from here, we can begin to go into the fourth dimension. From what I have seen; its two new directions are being likened to our up and down. They are called things like "ana and kata", and "upsilon and delta".
Despite many notions the average person may have of the fourth dimension as some twisted realm where lines and directions have no meaning (as it was portrayed int he Twilight Zone episode "Little Girl Lost"); all it is is our same 3 dimensions plus one. So they too would have depth (the direction each observer looks out into his space), height (the dimension in which he is anchored), and width; an additional dimension. He would also have another additional dimension added to that one. So it seems the fourth dimension, for practical purposes could best be described as an "additional" left and right!

Matching them up would of course depend on how the imbedding of our space was aligned in theirs. If we were aligned with our "down" being the same as theirs, and they looked "straight at" us; they would see us as having height, width, the dimension we regard as "fourth' which we cannot see, but no "depth" (proving depth is not the third dimension). We could attempt to try to represent it by "superimposing" it on our space. Each point in our space would actually be a line, with ghost objects on it, that lie in the additional dimension. Perhaps one direction we could represent as one color, such as red (which would get deeper the further away you got from our space). the other direction would be blue. Our space, in the middle, would be white (like the color temperature scale). Or, we could use different sizes. This would correspond to the usual "perspective projection" diagram of the tesseract, with the smaller cube representing the fourth dimension, withn the larger cube, which would lie in or at least closest to our space. (The other direction would be presumably be represented by larger objects. I think it would be cool if they one day make a video game like that. How about a Super Mario 4D, or (as much as I hate them) a Street Fighter 4D, which would be a simple start, since you don't move so far in any direction. the main problem would be the controls for those games. Perhaps a pair of joysticks, each controlling two dimensions). In the other representation of the tesseract, (parallel or isometric projection) with the fourth dimension represented as a 45° combination of dimensions (such as the ones that appear on this board next to the names of people who have been here a certain amount of time), it would be the same thing. In any of those cases, "up and down" would still be the same as ours, since the new dimension is added to ours. So the additional dimension ould be an extra one like left and right. An object aligned along it would be seen as "horizontal", just like an object aligned either width wise, or depth wise.

If the imbedding diagrams of stars and black holes apply, these would imply that our gravity actually comes from a higher dimension! Think of all those horizontal gridded sheets of flatland, with the 3 dimensional "impressions" made by the star or black hole or wormhole. In those, gravity lies in the third dimension. If you are on flat regions of two-space; you are not affected by it. It is when you approach an object with gravity, in which the space begins to be "curved" toward the third dimension (even though a flatlander couldn't see this), that you are pulled down the "slope", which he would still consider "down", but is still basically "left or right" to us, though it is gradually curving toward our "down". So in that scenario, the higher dimension is "height", and basically perpendicular to the lower dimensional height.
So likewise, if this model is true for us, then upsilon and delta would truly be the "up and down" in four space. Our up and down would then be rotated into one of the other dimensions, if we could look at our space from the perpendicular higher dimension. If we fell into a black hole, space would be so bent, that eventually, our "down" would agree with theirs, with the singularity at the "bottom". (of course, the relativists generally do not necessarily assume that gravity actually comes from the higher dimension, but the illustrations they use of impressions made in flat surfaces would lead to that conclusion.

So likewise, the addtion of any higher dimensions would be the same. More "lefts and rights", unless their "down" is perpendicular to ours.

[jinydu]

In fact, it doesn't really matter which of our dimensions corresponds to which of the 4D ones. Remember that a 3D "hyperplane" could be oriented in any direction with respect to a 4D room. Usually, we think about a 2D world as being "vertical", like a wall; but it could also be "horizontal", like a floor or ceiling. Of course, in between there are infinitely many "diagonal" orientations it could have. And as you pointed out, all dimensions are indistinguishable under rotation.

Of course, we have a definite sense of "forward and backwards", "up and down", but that is an artifact of the way we live, not of the properties of space. As you are surely aware, "forward" and "backward" is not a universal direction; if you turn, which direction you call "forward" will change and different people will have different "forwards" and "backwards" directions. And as you also know, the directions "up" and "down" aren't universal either; this is only an illusion caused by the Earth being much larger than our bodies. Of course, a person standing on another part of the Earth will have a different "up" and "down". In space, we could think about our head as pointing "upwards" and our feet as pointing "downwards", but this classification is problematic because different people will have a different "up" and "down"; and furthermore, if we are rotation, our "up" and "down" would constantly be changing.

When describing motion, we do label one direction as x, another as y and another as z. However, it should be noted that our choice of which letter to assign to which direction is arbitrary, and that any other choice would be equally valid, so long as we remain consistent in our choice throughout the entire problem.

As for your claim about gravity coming from a higher dimension, you're thinking about General Relativity, a theory that describes gravity as being caused by the curvature of spacetime. In presentations to the general public (i.e. nonscientists), physicists often use the analogy of masses on a trampoline net. According to this analogy, masses placed on the trampoline net cause it to curve, so that a second mass traveling close to this first mass will follow a curved path, due to the curvature of the trampoline net.

While this is a convenient way of thinking about it, you should keep in mind that it is only a helpful analogy and doesn't tell the full story of General Relativity. In order to really understand it, you have to understand the mathematics, something that the general public doesn't usually like to do (which is why those analogies have to be made up in the first place). In the trampoline analogy, the trampoline net is curved into a third dimension (our down). However, in reality, the mathematics of General Relativity is able to describe the curvature of spacetime without reference to a higher dimension; i.e. spacetime can curve without a new dimension to "curve into".

[wendy]

The current standing on dimensions is that only two have any particular natural grain.

1. up/down is a fait of gravity.

2. forward/backward is a fait of our ability to move. Trees would not experience a marked direction perpendicular to gravity.

3. "across" is anything left over: that is, after our orientation and gravity is accounted for.

This has some interesting applications for directions in 4d, since one can not communicate 'turn left at the next junction'. since the across-space freely rotates without affecting gravity or motion. You can't simply name these with perpendicular names, because the names don't carry from place to place.

As for depth, shelves are also deep, meaning that they are of distance from front to back. So a cupboard has both height and depth, which may be different!

W

[Eric B]

In fact, it doesn't really matter which of our dimensions corresponds to which of the 4D ones. Remember that a 3D "hyperplane" could be oriented in any direction with respect to a 4D room. Usually, we think about a 2D world as being "vertical", like a wall; but it could also be "horizontal", like a floor or ceiling. Of course, in between there are infinitely many "diagonal" orientations it could have. And as you pointed out, all dimensions are indistinguishable under rotation.

Of course, we have a definite sense of "forward and backwards", "up and down", but that is an artifact of the way we live, not of the properties of space. As you are surely aware, "forward" and "backward" is not a universal direction; if you turn, which direction you call "forward" will change and different people will have different "forwards" and "backwards" directions. And as you also know, the directions "up" and "down" aren't universal either; this is only an illusion caused by the Earth being much larger than our bodies. Of course, a person standing on another part of the Earth will have a different "up" and "down". In space, we could think about our head as pointing "upwards" and our feet as pointing "downwards", but this classification is problematic because different people will have a different "up" and "down"; and furthermore, if we are rotation, our "up" and "down" would constantly be changing.

When describing motion, we do label one direction as x, another as y and another as z. However, it should be noted that our choice of which letter to assign to which direction is arbitrary, and that any other choice would be equally valid, so long as we remain consistent in our choice throughout the entire problem.

Actually, my whole premise was based on an individual's perspective, and at any one given particular point in time. Every being that can perceive space is an idividual, and it does not matter to him at first what others perceive. Their own perception matters first to them. And he can turn around, in which his sense of directions change. But still; I am thinking in terms of inertial frames of reference, so even if he were spinning around like a gyroscope (relative to the rest of the universe), there would still be three static dimensions before him, with the basic one as he "looks out" being depth, and the two perpendicular to that.

As for your claim about gravity coming from a higher dimension, you're thinking about General Relativity, a theory that describes gravity as being caused by the curvature of spacetime. In presentations to the general public (i.e. nonscientists), physicists often use the analogy of masses on a trampoline net. According to this analogy, masses placed on the trampoline net cause it to curve, so that a second mass traveling close to this first mass will follow a curved path, due to the curvature of the trampoline net.

While this is a convenient way of thinking about it, you should keep in mind that it is only a helpful analogy and doesn't tell the full story of General Relativity. In order to really understand it, you have to understand the mathematics, something that the general public doesn't usually like to do (which is why those analogies have to be made up in the first place). In the trampoline analogy, the trampoline net is curved into a third dimension (our down). However, in reality, the mathematics of General Relativity is able to describe the curvature of spacetime without reference to a higher dimension; i.e. spacetime can curve without a new dimension to "curve into".

I'm aware of that also, but since we here like to think of a large fourth spatial dimension (not just the infinitessimal ones string theorists propose now), then we have to allow hypothetically, the possibility that the curvature could actually be in that dimension. I used that as an illustration of the rotating of dimensions anyway.
The one thing about the General Relativitist's trampoline analogy, is that it explains why the curvature bends the path of moving objects, but not why an object at rest still "falls" toward the center of gravity. Of course, for this, quantum and string theorists have come up with a hypothetical particle called a "graviton".

[houserichichi]

Seems like an endeavor to rid ourselves of semantics. What if we're not working with coordinates (like most modern physics)? How then would you propose we "name" the directions?

[thigle]

hi Eric. please check out my thread named 'notes to all nD-applets makers" (http://tetraspace.alkaline.org/forum/viewtopic.php?t=373), in Q&A forum, i think it's structurally coupled with this thread a lot.

now, to your proposal...

*EricB wrote: 'So from this, we can identify "depth", not as "the third dimension" as many often think, but rather the first and primary dimension! In any space, one has to look "out", "ahead" into that space, in order to perceive that realm, before he can look any other direction like "left/right", or "up/down". ' (italics added)
>well, it depends on which end of timeflow you grab first, and if you grab it at all (if not, we got circular-time approach, and ). either you try to understand in what sequence the perception-structuring dimensionalities arose from as far back as you can get to now/here, or you deconstruct backwards, tracking for the origin. or you can consider the perception as it is.
if you consider current mindset, it's one heavy 3d-perception habit. so we have 3 dimensions already at-hand, that's where we usually stand, with 1 timelike dimension. as you rightly notice, the depth modality comes about last. (one-eyed people whose sight have been restored to binocular vision, have to learn/accustom to perceive depth). so you can have depth-less perception, where up/down and right/left still functions.
then, the left/right modality is less primordial than verticality. verticality is the primal coordination, the FIRST after nothing. for ex., in Gibsonian perceptual theories, the whole e-volution of our visual hard/soft-ware is determined by HORIZON of the GROUND, and our feat of having 2, HORIZONTally aligned eyes. the vertical is the normal vector of the ground. the ordinary wilderness. see Merleau-Ponty for his notions of wild-Being. verticality is the raw experience. if we were evolving in horizonless, groundless place (a paradox ideed), our wiring would grow different.

*EricB wrote: 'It is in the third dimension, where we have our foundational "back and forth" (depth) where each observer looks out into his universe; and also the next fundamental "up and down" (height) where he is anchored to his world. We also add this new, extra dimension, which we call "right and left" (width). ' (italics added)
>from what was said above, i think you err here. going timeflow-wise, vertical is first, horizontal is second and depth comes last. 'where each observer looks out...' tells it all: first there has to be where? then there can be how? and then what?. in other words, there has to be somewhere(0) for someone(1) to percieve(2) something(3) in the first place. you start your analysis in the moment where the perceiver is already here. but the seer comes precisely as the absence (of verticality), as a slip from perfect balance of verticality on ground. [btw, in tibetan buddhism, one of the deeper complementaries is sku/yeshes. sku is a term for dimension, for body. in other words, sku is what is: body-ness, thing-ness of appearances. yeshes is a term signifying gnosis, translated often as ‘primordial wisdom‘ or ‚pristine wisdom‘ – what is meant is ‚living knowledge‘. wisdom is not knowledge (which is considered as structuring-process, as in-formation. learning>pondering upon>experiencing), but wisdom as the primordial awareness which is ever-present inseparably with any sku (being) or dimension, i.e. these (sku/yeshes) are indivisible. so space(0) precedes experiencing (1,2,3). and finally, sku/yeshes are experienced/considered a vertical dyad, happening ‚in‘ the groundless ground (‚gzhi‘) – the openess aspect of enlightenment.]
finally, i would like to propose anew: that epistemogenesis comes in sequence:

...0>vertical>horizontal>depth... and possibly can trace itself back to its own origin, thus becoming sheer perception: -depth(4d as 2d) > -rotation around vetical(5d as 1d)>-veticality(6d as point-sphere)

a more bold claim would be: group of all the views(vectors) from unit 3-sphere onto 3-point at origin is a unit sphere of quaternions. group of all the views (without constant-distance restriction) is a 6-manifold, i suppose.

*Jinidu wrote: ‚In fact, it doesn't really matter which of our dimensions corresponds to which of the 4D ones.‘
>this depends on your frame of reference (or framelessness) (or none or both of these)(or none of the previous 4)(can you get the next step ?). actually: it does matter in certain conditions and it doesn’t matter under others. when it does is different from when it doesn’t. first requires embodied cognition, second is (just) abstract thought. so for someone who thinks ‚with his marrow‘, it makes a difference to understand different aspects one’s own current condition as different dimensionalities, to mark them for distinction->to experience marked dynamics>to reflect upon dynamic relations noticed>to create new, refined, more adaptable and optimal coordination-system, for exemple one, in which the experience of 4-space is co-appearing with experience of 3-space. this requires a precise understanding of our 3d-habituation, of its ‚guts‘ so to speak. in this sense, the correspondences are relevant, for it is necessary to integrate new and old understandings. on the other hand, for someone who thinks that any other dimensions above (or under) 3 are just abstract constructs, not seeing that first 3 are abstractions too, although the currently actual ones, it isn’t even significant to such person, it seems to me, that, just 4-space has most regular polytopes, or that n-sphere has the highest hypersurface-area in 7d, or that 24-cell has no analogy in other dimensions, or... you know well.

*wendy wrote: ‚forward/backward is a fait of our ability to move. Trees would not experience a marked direction perpendicular to gravity.‘
>i disagree on this one. even if you cannot move from-to, you can move aroundhere: just rotate. trees experience rotational scope around their verticality as their second degree of freedom. and then, some trees move 2-4 metres per year. there surely arises the question of the choice of horizontal direction (perpendicular to gravity).

*wendy wrote: ‚"across" is anything left over: that is, after our orientation and gravity is accounted for.‘
>sorry, but in my experience, if gravity(verticality) is first, rotation around(anti-&clock-wise alias left/right) is second, and depth(as binocularity) is third, then, if after this, that what is left is ‚across‘ (or ‚diagonal‘), then i agree.

i have a bit more (there's this houserichichi's questions which needs to be given some thought), but must go now. will continue tommorow. love you folks.

[wendy]

The order of dimensions as we set these, are our choices followed by made choices: ie width / depth / height. If one forgets this relation, one adds new dimensions to the end of the list.

In practice, it is better to arrange the dimensions by increasing choices: ie choices made + our choices. When we add extra dimensions, then these extra choices are appended to the end.

Gravity dictates the up/down choice. That is, regardless of where we are, or what direction we face, we essentially share the same up/down environment. Even things that do not move, or freely drift in the water or air, have an up/down direction. It is absent from organisms for whom gravity is not an issue (such as microbes). Fish and birds still have it, but they may freely move up and down as directions.

Forward is an aspect of free motion. Drifting animals (like jellyfish) do not have a forward sense, and while plants do have a naturally set direction (eg sunlight), it's not a major concern to them. Likewise, we note that spiraling actions that plants have is not a naturally set thing, but a convenience of growth.

The across-space in 3d is left-right. Unlike up and forwards, one does not suffer ill effects for confusing these. Even the most inept person, who confuses left/right as a matter of course, is unlikely to walk upside down, or backwards.

To appreciate the across-space in 4d, consider for example weightless astronaughts. Denied of gravity, they can freely orientate themselves at any position. The nature of biology still sets front/back, but my up no longer has to be your up. It can be any direction of up / left / down / right. I can not communicate this sense of direction in the same manner that i can the front and back of the ship. The relationship requires a lot more learning than it does for a sea-ship (where top-side has a clear meaning).

In a 3d world, the across-space is 1d. It is also parity-driven, so one can easily give directions in terms of left/right. Even those who are confused as to left-right are not going to be confused as to which end of the car is front and which side is top.

In 4d, the across-space is 2d. Unlike the astronauts, we have no pre-defined vectors, and we are presented with a 2d across-space. There is a centre = straight forward, and increasing radius makes for harder turns. But what defines angle?

Wendy

[jinydu]

thigle, if you studied mechanics, you would know that rotation about an axis does not generate a new "dimension". Using the right-hand rule, rotation can be represented using a vector that points in the direction of the axis.

[thigle]

jinydu. dimensions (as you mis.understand them) are not generated, they are preconcieved concepts. dimensional generation is an obscure topic, and just stating that rotational movement doesn't play a role in it, is simply simple-minded. however, rotation does not generate new 'dimension', rotation can happen because there already exists a dimension to rotate into ! but nor does a point moving along a line generate a new 'dimension' ! it can move (and thus exemples happen) because of a dimension already being there, before the movement. actually, the classic exemples of 'move-point,get-line/move-line,get-plane/move-plane...' are not about generating dimensions, they are just about going through dimensions already at-hand. if dimensions are generated, how do you get zeroth in the first place ?

as for these paths through dimensionalities: there are many ways, and rotation is not the only one. check for ex. Keith Critchlow's 'Order In Space'. 3 basic ways are shown. linear, circular, discrete. you seem to limit yourself to linear schema.

in my view, your 'problem' is that you rely on catesian spatial-intuition, so you think flat, local and exclusive. try to move your imagination into projective space (which is natural) and embrace infinity. try following thought experiment:
take a line and its tangent.
focus on their touch in a single common point.
now fix this kissing point while moving the centre of the circle orthogonally away from the tangent.
extending the circle's radius to infinity, the local view of circle's arc touching the tangent flattens and becomes locally congruent with it.
now, rotating the circle around its centre at infinity anti-clockwise, the flat arc perceived as line runs to the right, at vice versa.
changing your perceptual space allows you to understand seemingly different viewpoint.

we are here to try to understand creatively, not to define reductively.
so let me just expand on your note. the vector used for representing rotations about arbitrary axis, can be noted is a described by a quadruple of values and thus it an be considered as a quaternion. check SO(3) & SO(4) (groups of rotaations in 3d & 4d respectivelly), as well as SU(2) which is 3d surface of 4d unit sphere in the quaternions, closed under multiplication...

[Eric B]

the depth modality comes about last. (one-eyed people whose sight have been restored to binocular vision, have to learn/accustom to perceive depth). so you can have depth-less perception, where up/down and right/left still functions.

No; they may not comprehend relative distances of depth, but they are still seeing things ahead of them. When they look out "forward"; they are not seeing left or right; or up or down.

*EricB wrote: 'It is in the third dimension, where we have our foundational "back and forth" (depth) where each observer looks out into his universe; and also the next fundamental "up and down" (height) where he is anchored to his world. We also add this new, extra dimension, which we call "right and left" (width). ' (italics added)
>from what was said above, i think you err here. going timeflow-wise, vertical is first, horizontal is second and depth comes last. 'where each observer looks out...' tells it all: first there has to be where? then there can be how? and then what?. in other words, there has to be somewhere(0) for someone(1) to percieve(2) something(3) in the first place. you start your analysis in the moment where the perceiver is already here. but the seer comes precisely as the absence (of verticality), as a slip from perfect balance of verticality on ground. [btw, in tibetan buddhism, one of the deeper complementaries is sku/yeshes. sku is a term for dimension, for body. in other words, sku is what is: body-ness, thing-ness of appearances. yeshes is a term signifying gnosis, translated often as ‘primordial wisdom‘ or ‚pristine wisdom‘ – what is meant is ‚living knowledge‘. wisdom is not knowledge (which is considered as structuring-process, as in-formation. learning>pondering upon>experiencing), but wisdom as the primordial awareness which is ever-present inseparably with any sku (being) or dimension, i.e. these (sku/yeshes) are indivisible. so space(0) precedes experiencing (1,2,3). and finally, sku/yeshes are experienced/considered a vertical dyad, happening ‚in‘ the groundless ground (‚gzhi‘) – the openess aspect of enlightenment.]
finally, i would like to propose anew: that epistemogenesis comes in sequence:

...0>vertical>horizontal>depth... and possibly can trace itself back to its own origin, thus becoming sheer perception: -depth(4d as 2d) > -rotation around vetical(5d as 1d)>-veticality(6d as point-sphere)

a more bold claim would be: group of all the views(vectors) from unit 3-sphere onto 3-point at origin is a unit sphere of quaternions. group of all the views (without constant-distance restriction) is a 6-manifold, i suppose.

I don't see how all of this makes "vertical" the primary dimension. It is only in the sense that it is the dimension defined by gravity, on a gravitational body, to beings accustomed to living on such a body. But we have to look/(think) out ahead of us first, to see we are being pulled down relative to everything around us. Only in linespace would the functions of left/right, and up/down coincide. (of course!)

[jinydu]

jinydu. dimensions (as you mis.understand them) are not generated, they are preconcieved concepts. dimensional generation is an obscure topic, and just stating that rotational movement doesn't play a role in it, is simply simple-minded. however, rotation does not generate new 'dimension', rotation can happen because there already exists a dimension to rotate into ! but nor does a point moving along a line generate a new 'dimension' ! it can move (and thus exemples happen) because of a dimension already being there, before the movement. actually, the classic exemples of 'move-point,get-line/move-line,get-plane/move-plane...' are not about generating dimensions, they are just about going through dimensions already at-hand. if dimensions are generated, how do you get zeroth in the first place ?

So you don't like the word "generate". How about this: Rotational motion about a line is not independent of motion along any three mutually perpendicular lines. That's why we are able to write down equations like:

r(t) = (a*cos(t), a*sin(t))

without invoking a new coordinate axis.

as for these paths through dimensionalities: there are many ways, and rotation is not the only one. check for ex. Keith Critchlow's 'Order In Space'. 3 basic ways are shown. linear, circular, discrete. you seem to limit yourself to linear schema.

I'm not limiting myself to linear motion. I'm just saying that it's possible to express all motions through any n dimensions using the rectangular coordinate system. To be fair though, it is also possible to do this using a spherical coordinate system (or it's generalizations to any number of dimensions) and many others. What I'm saying, though, is that circular motion is not independent of linear motion. If you are using the x-y plane and you want to describe a particle that moves along the z-axis, you need to "add" a new coordinate to your system. However, if the particle is moving in a circle in the plane, the x-y Cartesian coordinate system suffices. Of course, it's good to be familiar with other coordinate systems besides Cartesian, since they can be more convenient for working out certain problems.

in my view, your 'problem' is that you rely on catesian spatial-intuition, so you think flat, local and exclusive. try to move your imagination into projective space (which is natural) and embrace infinity. try following thought experiment:
take a line and its tangent.
focus on their touch in a single common point.
now fix this kissing point while moving the centre of the circle orthogonally away from the tangent.
extending the circle's radius to infinity, the local view of circle's arc touching the tangent flattens and becomes locally congruent with it.
now, rotating the circle around its centre at infinity anti-clockwise, the flat arc perceived as line runs to the right, at vice versa.
changing your perceptual space allows you to understand seemingly different viewpoint.

Sure, that's a useful and handy mnemonic to "seeing" why a circle "looks a lot like it's tangent line, when you're near the point of tangency". In fact, I often use a modified version of that when thinking about problems intuitively. But of course, I'm aware that it is not truly rigorous; when actually writing out the solution to a problem, I should be prepared to use a more precise argument (limits, Taylor series, etc.)

[bsaucer]

If we can break out of our physical 3-space and into the theoretical, I say a higher-dimensional space can have any number of "horizontal" dimensions and any number of "vertical" dimensions. It all depends on how you label them. A tensor can have a number of "subscripts" and "superscripts" (called a "mixed tensor"). The subscripts could represent "horizontal" dimensions, while the superscripts could represent "vertical" dimensions. Who says there needs to be only one "vertical" dimension?

The space-time of Special Relativity has three "space-like" dimensions and one "time-like" dimension. Again, that's arbitrary. You could have as many "space-like" and "time-like" dimensions as you want. All that does is cause the "distance formula" (metric) to have some plus-squared terms and some minus-squared terms under the radical.

Such a space-time continuum would have some kind of "light-cone" which separates the space-like dimensions from the time-like dimensions. The light cone is a set of points having zero "distance" from a given point of origin. Other points have real or imaginary "distance" from the origin.

[thigle]

what points have imaginary distance ?

[bsaucer]

This only applies in spaces that have both space-like and light-light dimensions. An "interval" between two events can be "space-like", "time-like", or "light-like" A space-like "distance" is real, while a time-like "distance" can be imaginary (or vice versa). A light-like "distance" is always zero.

This is true in 4D Minkowskian "space-time", but not in 4D Euclidean space, since Euclidean space has no time-like dimensions.

[otheronenorehto]

I get what you are saying here it is back at you from a different perspective.

The fourth dimensions "left and right" are past and future. Because our bodies exist in 3D we can not see all of the future or all of the past as an observable dimension. The question is how can we have memory or semi acurate predictions beyond a very short time span?

An entity in a line would only be able to observe the entity directly ahead or behid it. It could not see around that other entity because there is no around. Unless the line was a peano like curve then even though the 1 D entity in question only had direct contact with it could possibly be aware (use this term very loosely) of or effected by other segments of the line that curved close to the entity in question.

The same follows for 2 dimensions. In this case though movement is for the first time possible. Not only would it be possible to se arround something but it is possible to move something to the side to get beyond it. That is assuming that movement was possible, which I don't believe it would be until the adition of the 4th dimension. Curving the 2 dimensional plane (made of a 1 dimensional peano like curve) in a peano like fashion allows the 1 dimensional entity to be surrounded/effected by other parts of the line segment.

A peano curve relies on self similarity to extent into the next dimension up from itself. Therefor it is easy to imagine that the logical extension of the peano curve into the fourth dimension consist of self similar 3D groupings. Isn't that why time sometimes seems like a sequence of states or why we are able to represent time in a film by a sequence of selfsimilar images? If time were strictly linear though we would only be aware of the state preceding and upcoming from us.

Why can we remember things in the distant past and predict things in the distant future? I belive it is because time is a complex peano like curve that curves into the 5th dimension. 5 dimensions is why we have the ability to imagine states of the universe that never happened or will happen. it is our 4 dimensional universe being effectedareas of the 5th dimension that are in a close proximity to us. The universes that are most like ours are the ones closest in proximity to us.

I am going to stop talking about my perspective for now though. It boils down to 4D = all the states the universe has or wil occupy 5D = all the posiible states the universe could have.

[Katsushiro_Myoshi]

on the moon, or anywhere else with no gravity (or less than Earth) we can move up, so would that count as a slightly higher dimension or just more than our limitations on Earth where we can only move backwards, frontwards, and sideways?