I can't understand more than 3 spatial dimensions

[Alan1]

Dear Physics enthusiasts,

I'm just a curious guy, I don't have any fancy credential, and in fact I don't have formal education at all (for 'legal' purposes I'm just "literate"). I'm being honest about that right away so if that's a problem (what seems to be the case for a lot of scientists/scholars) you can just ignore this humble topic without further wasting your time. I must also say that English isn't my native language therefore some terms may be incorrect.

I understand the mathematical concept of a dimension, and I also understand that the use of coordinate systems for physical purposes is just a representation of the reality, 3 dimensions is the absolute minimal quantity of information necessary to represent a volume, in this case in an Euclidean space 3 coordinates would be a point (while obviously 3 numbers could represent a vector, I'm being specific and making assumptions for simplicity, so let's not discuss semantics/concepts please). Everything makes perfect sense so far. For simplification, in analogical examples we tend to drop one axis in our coordinate system, the "flatlanders" explanation on how 3D objects would look on a 2D plane in my opinion is a decent example as long as you keep in mind all the time that it's just that, a representation, an exemplification just like the light cone generally is shown in a 2D space. The mathematics work perfectly, but I can't imagine a physical world like that, especially if it exists within a "3D" world, the first question that come to my mind, from a purely theoretic view is where would the plane/axis be aligned to in this 3D space? Would the 2D space within the 3D space even look like a "plane" (flat surface)? I mean, in any decent coordinate system the axis orientations don't matter: you can have two distinct 'origins' for two separate representations of the space using the same coordinate system, with unaligned axis, and you would be able to convert back and forth a coordinate (point) between these two representations you're using, the maths would work on any of them, that emphasizes the fact that they're just representations of the reality. Another question is on the practical side of things. You have 3 coordinates because things in real world they occupy a 'volume' in space, would these hypothetical objects in 2D spaces have no volume, or would the objects in the 2D space have a volume that's only observable from a 3D world that contains the 2D space? Those are just some 'philosophical' questions that shows me how the whole 2D exemplification of space is only useful when you don't think of it in literal terms, but always as an exemplification for something that's already a pretty rough approximation of nature.

I don't understand extra spatial dimensions because of (among other things) the mathematical implications that emerge from that. Say you have one dimension, a 'line', it's quantisized in integers and is finite, say, from >0 to 10 (for simplicity), so you have a line like a ruler and you can mark 10 positions on it. Now imagine one extra axis, same rules, you now have X and Y and you can mark 10*10 positions. I observe here that every new dimension I add implies that all possible positions are exponentiated to the number of dimensions, in other words it's the same as having 10 lines stacked now, another dimension you have 10³, that means to me that the state of all previously added dimensions is multiplied by 10 every time you add a new dimension, in other words, say at this point you have 'nothing' in this 3D space/volume you're representing, if you add a 4th dimension, you now have 10 of these 'empty 3D spaces/volumes'. That could represent for example a space of 10x10x10 meters during 10 seconds. If you take multiple 'snapshots' of our current universe at different times, you exist in each one of them, in other words, you are 'duplicated' (not exactly) in each one of these snapshots, the information about you is 'repeated' in each of them.

I can perfectly understand objects existing in four dimensions, when I see a "hypercube" I generally think of an object in a 'snapshot' of time that comprises a time 'interval', not a single point in time, for example on some images you can clearly picture a cube translating or scaling 'over time', but time is pictured in a single frame what appears like a time interval that is 'frozen', something like superposing pictures at different times, or a kind of 'motion' blur if you will, I can perfectly understand that, for example this image:
https://en.wikipedia.org/wiki/File:Dimension_levels.svg
Represents to me a cube translating over time.
And I can understand the math behind this representation:
https://en.wikipedia.org/wiki/File:8-cell.gif
of a "4D" cube rotating, but I don't see how that's not totally arbitrary and invalid. I mean, rotation in this example ultimately means changing the coordinates of each 'point' of the object, in case of a cube, if you could rotate a 2D square in a 3D space, the square projection on the 2D space wouldn't keep the same 'area', but the square would have to exist in the 3D space because you can't figure out each points 'height' simply from the projection in the 2D plane, and I see that's the same concept, but apparently the same manifestation won't occur perfectly in this case (as volume), and again, I fail to see how that's not completely arbitrary.

I can understand how we could go with extra dimensions. Applying the few concepts I can grasp of physics and math I can say that an extra dimension (beyond time), could very well look like the so-called multiverse. So to represent an event you would need its spatial and temporal location, and its "multiversial" coordinate(s). Everything works beautifully in my mind when I think like that, what really can't make sense of is thinking about 4 spatial dimensions plus time. I mean, time is the 4th 'spatial' dimension to a sense isn't it? I guess that's pretty much common sense in physics these days.

I'm reading about things like some new unified physical theory proposals and I just can't get a good grasp because each one of them is about multidimensional spatial objects manifesting in our ordinary 'spatial volumetric' universe and I simply can't fathom things like that such as shape shifting objects manifestations and such. So if anybody could explain to me in layman's terms how to go about wrapping your head on the basic concept of that, I would really appreciate. It's not that I'm going to use that knowledge somehow because I'm just an uneducated and by comparison here a possibly quite dumb guy, but I'd really like to know more about what the prodigious minds are thinking of in order to better understand the beauty of our universe, even if only to examine the possible candidates and have an 'intuitive' opinion on them (since the theories are too complex).

Does having an extra spatial dimension not imply another exponential number in this case? I mean, that all the other 3 dimensions they have to exist again for each state of this extra dimension just like the example above? If that's so, in this case the mathematical concepts I know won't work very well for this 'reality' (at least for understanding) and I wonder if there's something that represents the universe better than them.

Thank you very much.

 

[Simon Bridge]

Welcome to PF;

... let's not discuss semantics/concepts please...

... except we will when your use of a word does not match the use within physics or mathematics ... in which case we will correct you in the interests of helping you communicate better with people in the field.
Otherwise fine: we don't want to get bogged down in the finer points of the meanings of words either. To that end, you can count on us to attempt to adress the ideas you are talking about rather than focus on the words you use.

I can't imagine a physical world like that, especially if it exists within a "3D" world, the first question that come to my mind, from a purely theoretic view is where would the plane/axis be aligned to in this 3D space?

Wherever you need it to be. You can cut a loaf of bread on any angle you like and still get a slice. You can also cut each slice as wiggly as you like. What angle you choose is up to you - so choose one that is useful and easy.

Would the 2D space within the 3D space even look like a "plane" (flat surface)?

A 2D surface embedded in a 3D World, viewed from the 3D Worldview, can look like any surface does. It does not have to be a plane - it could be a sphere or anything. Consider how complicated the surface of the Earth is - that is a 2D surface in 3D.

From inside the 2D perspective things are harder to imagine. You have no concept of the 3rd dimension but you can detect curvature by making triangles. The surface you live in is fat if the angles of the triangles all add up to 180deg. Apart from that, everything looks much the same ... they'd notice stuff like how objects may tend to accumulate in the high curvature parts or it may be easier to travel in some directions than others.

Another question is on the practical side of things. You have 3 coordinates because things in real world they occupy a 'volume' in space, would these hypothetical objects in 2D spaces have no volume, or would the objects in the 2D space have a volume that's only observable from a 3D world that contains the 2D space?

2D objects have no volume and 2D people would have no concept of volume (except maybe as an abstract mathematical concept ... a hyper-area say). 3D people have a concept of volume and can see that it does not apply to 2D objects ... 2D objects have lengths, and an area.

Those are just some 'philosophical' questions that shows me how the whole 2D exemplification of space is only useful when you don't think of it in literal terms, but always as an exemplification for something that's already a pretty rough approximation of nature.

Space-time is typically represented as 1D of space and 1D of time ... and only because the surface of our page is 2D. We do commonly plot 2D slices of space (ie. in a CT scan) and a 2D projection from 3D space (ie a photograph) at a short interval of time. These are called snapshots. A series of snapshots can be used to build a space-time picture. These things are usually used in literal terms and that is fine as long as you are careful about what you are being literal about.

But what an axis on a coordinate system represents can be arbitrarily abstract.
Here you want to focus on spatial dimensions.

I don't understand extra spatial dimensions because of (among other things) the mathematical implications that emerge from that.

It is the other way around ... more dimensions for space (in the physics sense of space) is an implication that arises from what we know about 3. Why they count as space dimensions is technical, and has to do with the way they transform, and this allows us to distinguish between space dimensions and, say, time dimensions... or more abstract dimensions.

So if we can have a 3D position dented, say, (x,y,z), in rectangular coords, then we can say that position is $\sqrt{x^2+y^2+z^2}$ far away.
So maybe if we have 4D position (w,x,y,z), then we can say that it is $\sqrt{w^2+x^2+y^2+z^2}$ far away ... this formula follows from our understanding of 3 space dimensions. If you think it is unclear what "how far away" means for a 4D position, you are right. usually "distance" is not the concept used for this sort of thing ... just like if V=XYZ is the 3D volume ... then WXYZ is the hypervolume, not the same sort of concept as a volume.

... but if we have 3 space and 1 time dimension, (x,y,z,t) then we have to say that point is $\sqrt{x^2+y^2+z^2-t^2}$ away (spot the minus sign, and be careful with units).

We do not have any reason to think that physics happens in anything other than 3 space and 1 time dimension ... however, it is often useful to represent things using more dimensions than that. The extra ones are usually quite abstract and I don't think this should be confused with the idea of 3+1D space-time where we treat time as a special kind of space.

Similarly, mathematics uses the word "space" as a technical term in algebra. This also should not be confused with the idea of a physical space.

...If you take multiple 'snapshots' of our current universe at different times, you exist in each one of them, in other words, you are 'duplicated' (not exactly) in each one of these snapshots, the information about you is 'repeated' in each of them.

... there is a distinction to be made between a time axis and a space dimension. Yes, you can take a snapshot of a volume at a particular time, and then run those snapshots along a number line marked out in, say, seconds ... but that isn't the same thing as a physical space dimension.

I can perfectly understand...

... I am going to dispute the term "perfectly" here ... clearly you do not understand perfectly or you would not have these questions. I would argue that perfect understanding is not possible. So it is unclear what you mean when you say you understand something perfectly.
This is not a nitpick over semantics, I am pointing out a place where your language is unclear to your listeners to the extent that there is a hig chance of a misunderstanding. Where you have used this wording, I have just ignored it.

objects existing in four dimensions, when I see a "hypercube" I generally think of an object in a 'snapshot' of time that comprises a time 'interval', not a single point in time, for example on some images you can clearly picture a cube translating or scaling 'over time', but time is pictured in a single frame what appears like a time interval that is 'frozen', something like superposing pictures at different times, or a kind of 'motion' blur if you will, I can perfectly understand that, for example this image:
https://en.wikipedia.org/wiki/File:Dimension_levels.svg
Represents to me a cube translating over time.

From what I can see, it isn't.
W is a dimension of space ... as it is difficult to represent 3D on a sheet of paper, it is even harder to do this for a 4D object.

And I can understand the math behind this representation:
https://en.wikipedia.org/wiki/File:8-cell.gif
of a "4D" cube rotating, but I don't see how that's not totally arbitrary and invalid. I mean, rotation in this example ultimately means changing the coordinates of each 'point' of the object, in case of a cube, if you could rotate a 2D square in a 3D space, the square projection on the 2D space wouldn't keep the same 'area', but the square would have to exist in the 3D space because you can't figure out each points 'height' simply from the projection in the 2D plane, and I see that's the same concept, but apparently the same manifestation won't occur perfectly in this case (as volume), and again, I fail to see how that's not completely arbitrary.

What they are producing is a 2D projection of a 4D object that is moving. You are more used to the 2D projection of a 3D object that is moving - this is called a motion picture and you view them for entertainment at the cinema.
You can easily understand a 3D cube rotating in front of a camera and the resulting images projected on to a screen.
The camera angle and axis of rotation have had to be chosen by someone so the resulting movie is arbitrary in that sense.

Being arbitrary does not make it invalid or useless.

What really can't make sense of is thinking about 4 spatial dimensions plus time. I mean, time is the 4th 'spatial' dimension to a sense isn't it? I guess that's pretty much common sense in physics these days.

The gif of the hypercube rotating is what you get with 4 space dimensions and a time dimension. It is unclear what the experience of a 4D space would be. Without a physical context, it's all just maths. I tried to deal with how time works as a special space dimension above.

I'm reading about things like some new unified physical theory proposals and I just can't get a good grasp because each one of them is about multidimensional spatial objects manifesting in our ordinary 'spatial volumetric' universe and I simply can't fathom things like that such as shape shifting objects manifestations and such.

And yet you just watched a mivie of a 4D object manifesting on a 2D screen in a 3D World ... in some esoteric physical models the experience of the extra dimensions is that we see fundamental particles. The dimensions are curled up really tight compared with the regular space dimensions that our own motion along these dimensions results in nothing detectable. This is supposed to be a bit like how curvature in 3+1 dimensions is experienced as gravity.

These theories are highly speculative and you are best advised to stay away from them until you have grasped 3+1 dimensions.

So if anybody could explain to me in layman's terms how to go about wrapping your head on the basic concept of that, I would really appreciate.

In laymans terms? That nobody can do. If the concepts could be understood in layman's terms we wouldn't need them.

Have a look at:
https://www.quora.com/Why-is-it-tho...ns-of-time-and-space-might-behave-differently
... it is about effects of the number of space and time dimensions, what happens to physics if we try to extrapolate what we know to different dimensionalities(?)
What makes physics distinct from mathematics is that the physicists maths has to be congruent to reality.

Another conceptual exercize comes from Greg Bear - who put a lot of work into 4+0 dimensional space. That is 4 space dimensions and none of time.
http://gregegan.customer.netspace.net.au/ORTHOGONAL/ORTHOGONAL.html
... the bottom line is that developing a reliable grasp of unfamiliar things amounts to living amongst the concepts for a long time.
It helps to take pains to be clear in your language though.

There does seem to be an easy confusion about the number of axes we have on a graph and the number of dimensions in space. I can describe space dimensions as axes on a graph, but I can describe lots of other things as well. For instance:
If I want to describe the state of a bicycle, I need 3 axes for three space dimensions; one for time, I'd also want two for the rotational state of each wheel, one for the gear the bike is in, one for the rotation of the handlebars, and two for the state of the front and back breaks ... I make that 9 dimensions for the description of the bike and I haven't got to the pedals, the luggage rack, or how far the bike leans, yet. but most of these are not dimensions of space ... though they all form part of the state space of the bicycle.

 

[Alan1]

Thank you for your reply and your welcome.

Wherever you need it to be. You can cut a loaf of bread on any angle you like and still get a slice. You can also cut each slice as wiggly as you like. What angle you choose is up to you - so choose one that is useful and easy.

A 2D surface embedded in a 3D World, viewed from the 3D Worldview, can look like any surface does. It does not have to be a plane - it could be a sphere or anything. Consider how complicated the surface of the Earth is - that is a 2D surface in 3D.

From inside the 2D perspective things are harder to imagine. You have no concept of the 3rd dimension but you can detect curvature by making triangles. The surface you live in is fat if the angles of the triangles all add up to 180deg. Apart from that, everything looks much the same ... they'd notice stuff like how objects may tend to accumulate in the high curvature parts or it may be easier to travel in some directions than others.

2D objects have no volume and 2D people would have no concept of volume (except maybe as an abstract mathematical concept ... a hyper-area say). 3D people have a concept of volume and can see that it does not apply to 2D objects ... 2D objects have lengths, and an area.

Space-time is typically represented as 1D of space and 1D of time ... and only because the surface of our page is 2D. We do commonly plot 2D slices of space (ie. in a CT scan) and a 2D projection from 3D space (ie a photograph) at a short interval of time. These are called snapshots. A series of snapshots can be used to build a space-time picture. These things are usually used in literal terms and that is fine as long as you are careful about what you are being literal about.

But what an axis on a coordinate system represents can be arbitrarily abstract.
Here you want to focus on spatial dimensions.

Yes, that's what I tried to imply with the 'conversion' example. These questions were actually just 'reflections' on the "2D on 3D" analogy, which by the way I think start to lose meaning when you try to apply it to the real world without abstraction (more on this below).

It is the other way around ... more dimensions for space (in the physics sense of space) is an implication that arises from what we know about 3. Why they count as space dimensions is technical, and has to do with the way they transform, and this allows us to distinguish between space dimensions and, say, time dimensions... or more abstract dimensions.

So if we can have a 3D position dented, say, (x,y,z), in rectangular coords, then we can say that position is √x2+y2+z2\sqrt{x^2+y^2+z^2} far away.
So maybe if we have 4D position (w,x,y,z), then we can say that it is √w2+x2+y2+z2\sqrt{w^2+x^2+y^2+z^2} far away ... this formula follows from our understanding of 3 space dimensions. If you think it is unclear what "how far away" means for a 4D position, you are right. usually "distance" is not the concept used for this sort of thing ... just like if V=XYZ is the 3D volume ... then WXYZ is the hypervolume, not the same sort of concept as a volume.

... but if we have 3 space and 1 time dimension, (x,y,z,t) then we have to say that point is √x2+y2+z2−t2\sqrt{x^2+y^2+z^2-t^2} away (spot the minus sign, and be careful with units).

We do not have any reason to think that physics happens in anything other than 3 space and 1 time dimension ... however, it is often useful to represent things using more dimensions than that. The extra ones are usually quite abstract and I don't think this should be confused with the idea of 3+1D space-time where we treat time as a special kind of space.

Similarly, mathematics uses the word "space" as a technical term in algebra. This also should not be confused with the idea of a physical space.

Well, isn't a 'hypervolume' the same as, say, getting two positions at different times for an object and 'interpolating' the volume between these positions, and in this case picturing it as an "extruded" figure in 3D like if it left a 'trail' of itself? I find that very meaningful because the hypervolume in this case would be dependent on the speed of the objects, what remarkably seems to be the case, but that's only true when you treat the 4th dimension as time. If there's yet another dimension I don't see how the 'volume' in all of these dimensions wouldn't be an information that also depends on time information, or if it can be represented without taking time into account (what I guess is what 'spatial' implies), then what implications that would have in the representation of time itself.

Regarding the "how far away" question, I think that's a perfect example on how it's hard to get a grasp of the 4D space. If you think about the 4th dimension in terms of time, it makes total sense. It doesn't matter to know where an even occurred if you don't know when. In this case two events may be near in space but far in time, and that makes me wonder about the nature of the math behind it. You can also have two points that are near on X and Y, but are far on Z, however, that concept is totally dependent on your axes alignment, and it's funny to think that generally you don't consider the same for a 3+1D spacetime, you generally don't think of the alignment of the time axis, however the velocity vector can be interpreted in terms of that, so when you're going really fast objects will seem closer/farther in space exactly as if you had objects that look close on a XY plane when you disconsider the Z axis, but since they are far in the Z axis when you 'tilt' your coordinate system a bit, they will appear on different positions on this 2D projection. I know it appears I'm stepping back from my goal, but I fail to see the meaning in the very concept of 'distance' in a 4D space, that again makes me wonder about the validity of an extra dimension for space. 3 dimensions is the very least for a volume, I think we can call it the most efficient way to represent a volume, however, you could represent it in other ways, just as you can represent a vector with a 'point' or with angles and magnitude, and what's really strange is that "an extra dimension" have different implications depending on how you're representing the space, and I guess in the end it all comes down to this in fact. I can see how you could represent a space using another (non-euclidean/cartesian) system, and you could have a system where a 'dimension' doesn't have much meaning, and adding an extra information analogue to a 'dimension' to this system would have very different implications, so, basically, when you say it's the other way around, maybe I'm really too slow but I simply can't fathom that, because what I see here is this: you have a coordinate system to represent the real world, it works nicely so far and happens to be minimal, so you now 'modify' your coordinate system from a purely mathematical standpoint and add extra data to it, and somehow it now should represent what really happens in the real world; but that is fully dependent on the mathematical model you're using, and fully independent from the real world, although you could argue that the math (mostly) works, I can't help but wonder if that is unscientific; to me it's like trying to represent points on the surface of a sphere using separate 2D and 3D coordinates, in the 2D space a point on the sphere surface is 2 coords, in a 3D space it's 3 coords so you could consider it less efficient because you're constrained to the points that are on the surface (most values simply aren't valid), but now think of adding an extra dimension on the 2D representation, an extra dimension on the 2D plane will provide an extra axis that's always perpendicular to the sphere surface when seen from a 3D space, but it's very different from the actual axes of the original 3D space, although you could surely 'convert' any point that is not on the surface of the sphere back and forth between these spaces.

I'd also talking about real world problems, I know you're not talking in literal terms, but you said a 2D space in a 3D world would look like a surface, but that's totally meaningless to me, because for instance it effectively won't 'look' like anything in practice because the hypothetical objects in this 2D world won't be able to interact with the photons to begin with. They won't be able to interact with literally nothing in our universe, so to some extent they're the dragon in Sagan's garage, and so I suspect that 3D objects won't be able to interact with the 2D space too.

Ultimately, even forgetting all the pictures and such, what I see in my mind when I think about 4 spatial dimensions is nothing more than a 'wonky' system to represent space being 'misused', basically I see a somewhat redundant (less efficient) 4D system to represent space where certain constraints have to be observed, like some points obligatory have to 'match' (or have compatible values) with other data, but since that's not being observed you end up with a 'fuzzy' and imprecise representation of volumes in an "atemporal" volumetric space. I fail to see more than that and perhaps the concept is just beyond me.

... there is a distinction to be made between a time axis and a space dimension. Yes, you can take a snapshot of a volume at a particular time, and then run those snapshots along a number line marked out in, say, seconds ... but that isn't the same thing as a physical space dimension.

I guess this is a key point I'm missing.

From what I can see, it isn't.
W is a dimension of space ... as it is difficult to represent 3D on a sheet of paper, it is even harder to do this for a 4D object.

Yes, representing "representations" of a multidimensional world in restricted dimensions is always a problem. I wonder if projections for representations are really just part of our culture and condition, besides the images projected in our retinas, a shadow for example is a projection, so we're used to it, and from a distant light (e.g.: sun) it's almost orthogonal, so in some sense it could be considered somewhat universal, but even though it's dependent on interpretation (unrelated: I always though the pioneer plaques should have projections like shadows instead of outlines. I really doubt aliens will understand an outline while something like a 'shadow' is more universal).

And yet you just watched a mivie of a 4D object manifesting on a 2D screen in a 3D World ... in some esoteric physical models the experience of the extra dimensions is that we see fundamental particles. The dimensions are curled up really tight compared with the regular space dimensions that our own motion along these dimensions results in nothing detectable. This is supposed to be a bit like how curvature in 3+1 dimensions is experienced as gravity.

These theories are highly speculative and you are best advised to stay away from them until you have grasped 3+1 dimensions.

Maybe I have some false confidence but don't see any problem in understanding the 3+1D space(time). I mean, it just makes sense to me both from a mathematical theoretical standpoint and from a practical standpoint too. The math is beautiful and such for n dimensions, but the problems start when I try to relate that to the reality, because although I've used less than 4 dimensions in the examples above, I fail to see how something could even exist without these 4 dimensions, and I guess extrapolation could be applied here to say that physical entities simply can't exist without a currently arbitrary number of dimensions beyond that, albeit far from our understanding. Something in 2 dimensions can't have volume, thus simply won't be able to interact with the real world. It doesn't matter how you represent a 'volume', it's just a property of our physical world, you can use whatever system or convention you want, in the end you're representing the same thing, not a "3D" object because that refers to a mathematical system, but simply a 'volume'. The same can be applied to time. An object simply can't exist without existing in these 4 dimensions. Separating spatial and temporal dimensions seems somewhat bogus to me. I read a discussion about 2 temporal dimensions, just like my previous multiverse example, the extra temporal dimension could represent multiple independent timelines (I'm imagining stacked timelines), but these timelines could also be seen as multiple universes, in other words it's just a new property that 'multiplies' all others, so I don't even see the need to explicitly differentiate spatial and temporal dimensions, since the 4th dimension automatically would mean time for me. The problem is when you have dimensions that violate that basic rule of adding one to the power, what seems to be the case of extra spatial dimensions, but maybe it's just a concept that can't really be understood (at least by me). Although I don't see the need to treat the 'types' of dimensions separately (I disagree with the concept of attributing 'types' to them), I understand that for practical purpose and exemplification it would work. You can perfectly use 3 dimensions to represent a building, or 2 to represent its area on the terrain, but that doesn't change the fact that what you are representing exists in all 'physical dimensions'. It's a requirement for something that 'exists' to exist in all physical dimensions, but I wouldn't use the term 'dimension' for that at all, I would just say the 'physical world'. While it's obvious that something that exists should exist in the physical world, in the way/system we represent that physical world is where dimensions can be considered, not in the world itself from my understanding.

I only see the 'division' between the 4th dimension meaningful because there's a limitation in the propagation of information, what ultimately is linked to the spatial information, which I think should be seen not as a constant but in fact as a proportion between space and time, but again, information can't exist nor propagate without existing in all 'dimensions' (or physical world), and of course time is perfectly compatible with both my mathematical and physical/practical concept of dimension.

If you're referring to the string theory (esp. m variant) yes, that's something I have some hard time understanding, I see the concept of dimension is very different from our basic interpretation, and I have many doubts concerning specific concepts too, for example, I don't see how these dimensions aren't just exactly like my ball example above, in other words, just entities that can be represented in the ordinary dimensions but have a 'convention' to be represented in their own 'local' coordinate system. When I think of new dimensions I don't think of objects in our word.

There does seem to be an easy confusion about the number of axes we have on a graph and the number of dimensions in space. I can describe space dimensions as axes on a graph, but I can describe lots of other things as well. For instance:
If I want to describe the state of a bicycle, I need 3 axes for three space dimensions; one for time, I'd also want two for the rotational state of each wheel, one for the gear the bike is in, one for the rotation of the handlebars, and two for the state of the front and back breaks ... I make that 9 dimensions for the description of the bike and I haven't got to the pedals, the luggage rack, or how far the bike leans, yet. but most of these are not dimensions of space ... though they all form part of the state space of the bicycle.

Exactly, and when you add, say, a 'gear' dimension, that implies that now you have multiplied the total of possible states in the system by the number of gears, in this case if you had, say, the total number of combinations lat*long*time, now the total is lat*long*time*gears. That doesn't seem to be the case for spatial dimension, an extra spatial dimension from what I read doesn't imply the total number of states in the system is going to be multiplied, what is what makes perfect sense when you think of X,Y,Z and time: each one effect the system exactly like your bicycle example.

Thank you for your reply and the links, I'm going to take a look on them.
I want to thank you also for the respectful reply and say that I'm not good with words and this reply is not intended to be rude or disrespectful in any way, so please forgive for any parts where that may seem to be the case.

 

[Simon Bridge]

I have to stick to physics though - philosophy forums exist online but this is not one of them.

Well, isn't a 'hypervolume' the same as, say, getting two positions at different times for an object and 'interpolating' the volume between these positions, and in this case picturing it as an "extruded" figure in 3D like if it left a 'trail' of itself?

Um - no.
For one thing a hypervolume of space would have units of type of L⁴ while an extruded volume, if I understand you correctly, has units of L³.

If there's yet another dimension I don't see how the 'volume' in all of these dimensions wouldn't be an information that also depends on time information,

If there is no time dimension in the definition, then how can there be any time information to use?
Of course you can parameterize the hypercube along one specified line ... that what you mean?

... or if it can be represented without taking time into account (what I guess is what 'spatial' implies), then what implications that would have in the representation of time itself.

There are no implications for time since time is not one of the defined dimensions in the definition.
To animate the ypercube, say we want to rotate it about an axis, then we would need 4+1 dimensions. That is to say 4 space dimensions and one of time. That is how the rotating hypercube projection was obtained.

In mathematics all these dimensions bigger than 3 is just an abstract game of "what if" that need have no objective reality in Nature. Mathematicians are logically extending the mathematics of 3D to see what happens.

Regarding the "how far away" question, I think that's a perfect example on how it's hard to get a grasp of the 4D space. If you think about the 4th dimension in terms of time, it makes total sense. It doesn't matter to know where an even occurred if you don't know when. In this case two events may be near in space but far in time, and that makes me wonder about the nature of the math behind it.

But the example I gave was for space dimensions. If the 4th dimension is time then you get a different transformation to determine "how far".
This has an established mathematics and language to answer your questions, look up "Einstein's relativity".
The pythagoras with all plus signs is called Galilean relativity, where there is one minus sign is called special relativity.
The "how far" thing is called the space-time interval, and a point in space-time is called an event.
In Nature, space-time follows a bunch of rules for geometric perspective called the Lorentz transformation.

I'd also talking about real world problems, I know you're not talking in literal terms, but you said a 2D space in a 3D world would look like a surface, but that's totally meaningless to me, because for instance it effectively won't 'look' like anything in practice because the hypothetical objects in this 2D world won't be able to interact with the photons to begin with.

Can you see the tabletop in front of you or not?
That is what I mean when I am talking about a real physical surface.
This kind of surface can be modeled by a mathematical abstraction known as a 2D surface.

BTW: there are classical models for electrodynamics that allow 2D surfaces of charge (surfaces with area and no thickness) to interact with light so, if such things existed (which is the proposal) then you would be able to see them.

Of course, in Nature, classical ideal surfaces do not exist but that does not mean it is meaningless to talk about them.
And at this point we enter philosophy.

 

[Alan1]

Regarding the second link, I sure can imagine different physical laws, and it's interesting to imagine purely mathematical universes, however, when you have full mathematical freedom, you can go far from anything imaginable as existing, although the math can still be correct. It's not hard for me to imagine a universe where some of the 4 laws don't exist, but not one in which geometry doesn't work.

By no means I want to go into philosophy, It's in fact exactly the opposite, in some way I want to stick as far as possible with physics in the sense of interpreting the reality, in that context it's interesting to think that mathematics is a kind of philosophy.

If there is no time dimension in the definition, then how can there be any time information to use?
Of course you can parameterize the hypercube along one specified line ... that what you mean?

There are no implications for time since time is not one of the defined dimensions in the definition.
To animate the ypercube, say we want to rotate it about an axis, then we would need 4+1 dimensions. That is to say 4 space dimensions and one of time. That is how the rotating hypercube projection was obtained.

In mathematics all these dimensions bigger than 3 is just an abstract game of "what if" that need have no objective reality in Nature. Mathematicians are logically extending the mathematics of 3D to see what happens.

But the example I gave was for space dimensions. If the 4th dimension is time then you get a different transformation to determine "how far".
This has an established mathematics and language to answer your questions, look up "Einstein's relativity".
The pythagoras with all plus signs is called Galilean relativity, where there is one minus sign is called special relativity.
The "how far" thing is called the space-time interval, and a point in space-time is called an event.
In Nature, space-time follows a bunch of rules for geometric perspective called the Lorentz transformation.

Can you see the tabletop in front of you or not?
That is what I mean when I am talking about a real physical surface.
This kind of surface can be modeled by a mathematical abstraction known as a 2D surface.

Of course you can go without time, but you then you simply can't have things that depend on time on that model. Everything that 'exists', even if you would consider only a single point in time, in the stricter sense, 'depends' on 3 dimensions (or any model that allows a volume), although surely you can simplify a lot of things to a 2d space, that's even a much less precise representation of the reality.

When it comes to hypothetical 2D worlds, my feeling is that we are just dictating math into these universes, while physics is exactly the opposite, it's about constraining math to the reality. In other words, while a 2D space can be a good model for some applications, it isn't a good model of reality, each dimension you add gets more precise, including time.

When you add an extra dimension, at least in this case, you're not only adding a 'label' for your system, so you can have more of them (systems), but you're actually using the data in the model, so I understand from a mathematical view that when you have a tesseract that manifests in the 3 dimensions, manipulating the value in the 4th dimension is seen manifesting in the other dimensions. The analogy of using a sphere in a 2D plane explains that. You move the sphere on Z and that manifests as a circle increasing/decreasing in the 2D plane, a 3D line could manifest as a point translating on the 2D plane, so it's clear that spatial manifestations from manipulation of variables from extra dimensions depends on the geometry of the object, but there's an important distinction to be made here, that is intersection with is different than projection, in case of a projection (orthogonal), rotation will manifest as shape changing and translation won't. So far everything's fully consistent with the hypercube example as seen on 3D.

I didn't want to imply in my previous post that space and time aren't somehow separated, just that the relation between the physical time and space, is similar to the purely mathematical relation between dimensions n and n-1. Of course there are physical limitations, but that doesn't change the fact that every point in time represents a different state of space. Would every point in a 4th dimension of a 4D space also represent a different state on the 3D space? Mathematically the answer is obviously yes, but I can't picture that in the real world just as I can't picture flatland, it just isn't 'natural'. I always bear in mind that when you use a 3D space to represent the physical space, that's nothing more than a close representation of it. In fact, what's a 'volume' in the strictly physical sense? In math we know that it's the analogue of a line in 1D, an area in 2D, and you can extrapolate that to any dimensions, mathematically, all of that is the same thing, which is just an 'interval' in the current spatial model you're using. But physically, I don't see how you could put it like that. These hyperobjects are very beautiful both visually and mathematically, but they definitely don't help me understand the problem because I can't project those abstractions in the real world.

We use 3 dimensions to represent space because it's useful, it can be used to describe the physical world with great precision, the number of dimension is constrained by the observation of the reality. When you talk about the 2D space, sure it's a valid model and very useful for many real things too, but it's not a good representation of the reality, at least of our reality. I surely can imagine that in other universes where the physical laws are very different there could be flatlanders using a 2D space with great success, but here on this universe it feels like forcing maths into the reality to me. When we use 2D spaces we're still ultimately representing our "volumetric" universe. It just means that the extra complexity and precision made possible by an extra dimension just isn't important for our application.

A 4th dimension implies that there's an infinite number of "3D" spaces, one for each value of the 4th, but say the 4th isn't infinite and it's quantisized somehow, so now you have a finite number of these 3D spaces possible. Each sample in W you got a different state for the entire 3D space. The fact that the state of the current sample in the 4th dimension is dependent on the previous ones, is a property of the whole spacetime as I understand, what's highlighted by the real world physical limitations (there's a limit on the manipulations that can be made to the variables which are interdependent, this is observed in practice). However, that's exactly the definition of time for me. We're all 'moving' in time but that says nothing about our actual spatial position unless of course that if we weren't moving in time we couldn't be moving in space too.

More importantly though is that if more spatial dimensions exist in nature, what guarantees they are dimensions in the euclidean representation of space? That for me sounds like saying the real world is background dependent on the specific coordinate system you're using, because the mathematical 'dimensions' are nothing more than properties of the coordinate system you're using. I find it too much of a leap of faith to modify a mathematical model and assume it's somehow related to the real world, but I could very well be wrong here. It all boils down to that problem of the existence in the strictly physical sense. We know that for something to exist it must somehow produce 'change' in the state of space over time, that's a good definition of 'existence' I guess, but even the most precise mathematical view of that seems an oversimplification.

I think this thing is just beyond me really. I will try to read on the purely physical definition and implications of a 'spatial dimension', but unless it's a semantic problem and they are presented as something substantially different than its mathematical definition, I don't see any hope for me in understanding the whole thing. I just cannot see the math as a perfect non-abstract and non-arbitrary way to represent the universe.

Regarding something like the string theory, is there any support for it other than the pure math?

Thank you.

 

[Tommyboyblitz]

I have thought about how to represent more than 3 dimensions in a way. Its easy enough to represent 3 axis on a piece of paper. For example if you had a cube you could offset that cube along another axis and the quantity you offfset it would be the size for that dimension. It kind of starts to look like a mess after 4 dimensions but it is possible to try and visualise it.

 

[Alan1]

I have thought about how to represent more than 3 dimensions in a way. Its easy enough to represent 3 axis on a piece of paper. For example if you had a cube you could offset that cube along another axis and the quantity you offfset it would be the size for that dimension. It kind of starts to look like a mess after 4 dimensions but it is possible to try and visualise it.

Mathematically, yes, it's pretty easy to have as many dimensions as you want, in fact back in the days I wrote a program that would create the analogue of a cube in any number of dimension you choose and try to 'project' it on the screen, there's a clear pattern of 'extruding' whatever you have in the new dimensions, so it's possible to have generic code for that, the only problem is with angles on the projection, but as I said, each of these objects represent an 'interval' of space on their mathematical model, so they're all analogue to a 'volume'. It helped me understand the issue immensely. So, sure, if you see only the mathematical side of things you can have these 'hyper objects'. The problem is relating that to the real world.

You can use 2D for a lot of practical purposes. Sometimes that's all you need and it's easy enough for us to visualize things like roads or a façade in 2D. However, it's clearly a poor representation of reality; it may be 100% suitable for your needs but it does not look like the actual real world because there's 'something else' we can see and measure in the real world that's not being represented there (besides time), which is the 'volume' of it. I don't know how to put it, but a 3D space seems to be just a better representation, an 'upgrade' of a bad system. It's not in perfect harmony with the universe. There's a theory the universe is simulated, I honestly don't believe that for a second, but if that was the case, I'm pretty sure it's not using cartesian coordinates on an euclidean space. In my opinion the correlation between time and space is very well represented mathematically, and it's easy to see that both in theory as in practice, but space itself isn't very well represented in comparison in the current model, and to say that altering some property of that model will somehow have any relation to the reality, seems to me like assuming the model is perfectly on sync with the reality, so by making it more complex it'll inevitably become more precise, what I can only see as a leap of faith. It's like just because it works mathematically is the duty of the universe to comply with it. I mean, it could be right, sure, not because we want to but because somehow the model is remarkably realistic, but I don't know of any clue, let alone evidence, that it's that way, besides sounding extremely odd to me.

Also, I mentioned earlier that the math would work, but wouldn't it be incompatible with pretty much all the physical laws (e.g.: squares -> cubes, etc.)? I mean, I can visualize the dimensions on the string theory (although I have many doubts, but I can see how it relates to the reality), mostly because their nature is intrinsically different in from my understanding as 'volumes', the problem is a 4th spatial dimension that mathematically behaves exactly like the other 3. I can't relate that to reality.

Thank you.