a personal pet peeve (brace yourselves, it's pedantry):
Technically speaking, we're talking about four dimensional space. It doesn't really make sense to call such a space "The Fourth Dimension", any more than real life space is "The Third Dimension", or a tabletop is "The Second Dimension". This sometimes trips people up into arguing over whether The Fourth Dimension "is" time, or whatever. For that matter, maybe the first dimension is time, and the second, third, and fourth dimensions are space. These things aren't ordered, and in fact you can't really distinguish between the three familiar spatial dimensions: imagine trying to point along dimension one, whatever that means.
The familiar three-dimensional space as we know it is three dimensional because you can put three straight lines to meet at right angles to each other, and no more. And you can label those lines x, y, and z if you like and pick their orientation. Four dimensional space allows you to cram another in. Two dimensional space only allows two, and one dimensional space is just a single line.
> space as we know it is three dimensional because you can put three straight lines to meet at right angles to each other, and no more.
To continue in the spirit of your pedantry, it’s not really important that the lines meet at right angles; to say that space is three dimensional is just to say that a point in space can be uniquely expressed by specifying three numbers: the ‘coordinates’ of the point. What you say is true, but dimensionality in general doesn’t require the notion of ‘angle’ to make sense.
In vector space/linear algebra terms it’s to say that we live in three dimensional space because 3 is the size of a maximal linearly independent set of vectors, or alternatively the size of a minimal spanning set or vectors. They need not be orthogonal, but orthogonality does imply linear independence (it’s stronger).
Can you give an example of any n-dimensional system where you can't make a basis of n perpendicular vectors?
"Right angle" isn't any kind of angle, it's the angle that means "perpendicular" . You don't necessarily need any other angles besides 0 (parallel) and right (perpendicular) in a system.
> Can you give an example of any n-dimensional system where you can't make a basis of n perpendicular vectors?
Well, yes. In a totally general vector space (in the mathematical sense of a vector space over a field; it’s the same notion as used in physics but perhaps more general) there’s no such thing as ‘perpendicular’ (aka ‘orthogonal’) — you need an inner product space for that. Of course, (at least in finite dimensions) you can always define such an inner product and you can certainly create orthogonal bases (see: Gram-Schmidt process), but my point was that the notion of dimension doesn’t require it to make sense. Indeed it’s not part of the usual definition; we only need to be able to talk about sets of vectors and their spans.
> It doesn't really make sense to call such a space "The Fourth Dimension", any more than real life space is "The Third Dimension",
It makes a lot more sense to refer to time as "the fourth dimension" than to refer to all three spatial dimensions together as "the third dimension." Time is indeed one of the four dimensions that we're discussing here!
Of course you can quibble that the four dimensions are not in some fixed order, but in this context we're clearly referring to time as the fourth dimension because it's the one being introduced as an addition to the other three.
Its not that simple. Time it seems IS the first (0th?) dimension.
A point is space denotes existence in time of the object & observation by the subject. In other words, rate of change of existence is observed as time. Rate of change of a point is observed as a line. A moving point accepts line as its track, moving interval accepts square as its track and moving square accepts cube as its track. 2-eyed observer has 3D vision & 1-eyed observer as 2D vision (try touching your fingers with one-eyes closed exercise) has some role about role of observation as well.
Further, dimensions being relative vs absolute makes more sense. In absolute sense, time is its own dimension T & point line cube are L, L^2 & L^3. A 3D object, a cube, has 2D object, plane, as its boundary & 1D object, lines, as its dimensional denotion. A square has 1D object as its boundary & n-2=0D objects, points, as its dimensions, relatively speaking. This is important because of the number of eyes? So basically, those 2D hypothetical characters in your physics are 1-eyed creatures, lol.
You can point to a dimension; you just have to pin down a coordinate system and then point to an axis.
It does make sense to speak about a dimension. Say that some creatures in 2D space are ganging up from all sides on a creature which has access to 3D. As they close in, that creature can evade them by "disappearing into the third dimension". What it means is that it moves in such a way that its motion has a component that is orthogonal to the 2D plane, and only that component is relevant to the success of its escape. In the coordinate system in which that 2D plane makes up the first two dimensions, the orthogonal axis is "the third dimension".
Meh. If a circle was able to escape Flatland by rising into the third dimension, how would you prefer to describe it?
Similarly, if we were able to escape 3-space by moving into a 4th spatial dimension, what would you call it? If this hypothetical 4-space is Euclidean, then it contains exactly one dimension that is perpendicular to our familiar 3-space, so we would be perfectly justified in calling it The Fourth Dimension.
I’m just an amateur but it seems like the argument above yours is pretty airtight, just based on the difference between the mathematic definition of Dimension (“…is informally defined as the minimum number of coordinates needed to specify any point within [a space]”) and the colloquial definition (“a space”) which your last sentence seems to rely on.
If I’m reading the first paragraph of Wikipedia right (surely an airtight source for a pedantic argument about advanced mathematics!) “dimensionality” is an adjective describing a space (a set?), so saying that we moved to “the fourth dimension” is about as meaningful as saying we moved to “The Euclidean” or “the big”. Rather than “a euclidean space” or “a big space”.
That said you’re colloquially absolutely correct, of course. If I was giving advice to fiction writers or journalists I’d definitely endorse your common-sense argument.
Fair, but two perpendicular Flatlands embedded in the same 3D space won't be able to agree on which is the fourth dimension. It's fine as a shorthand when Flatlanders talk to each other, but "the fourth dimension" still won't be an unambiguous direction, for Flatland A it's actually one of Flatland B's two dimensions, and vice versa. For us any dimensions perpendicular to the entire universe will be "special", but only for us. Native four dimensional critters won't see what's so different about the ana/kata axis, unless our universe happens to be their tabletop RPG.
> Fair, but two perpendicular Flatlands embedded in the same 3D space won't be able to agree on which is the fourth dimension.
A lot of religious/spiritual people will say that God or the metaphysical realm (some people agree, some disagree...just like certain ideas in physics) is where many higher dimensions can be found. Say what you want, but people's incorrect models of reality having more influence than "reality itself" isn't nothing (if it kills people, it's at least something[1]). Besides, almost everyone complains about it, they just don't consider it (thus it is not) dimensional, it's "just reality", kind of like how a lot of phenomena now understood due to science were(!) formerly "just reality".
Well that's where I think the Flatland analogy is helpful, there's nothing inherently mystical about a fourth spacial dimension any more than Flatlanders ought to be worshiping us because we have one more dimension than they do.
> there's nothing inherently mystical about a fourth spacial dimension
Weirdly enough, there are big distinctions between three and four dimensions when it comes to geometry and topology. For example: “Four is the only dimension n for which R^n can have an exotic smooth structure. R^4 has an uncountable number of exotic smooth structures; see exotic R^4.” [0]
It turns out that having four dimensions really is magically different from having any other finite number.
Does this mean anything physical for 4-D spacetime? I'd guess yes because "manifold" is a general turn that includes the asymmetric curvature of spacetime.
I guess, but there may be (is, in my opinion) something mystical about higher non-spatial, metaphysical dimensions, and I think the difficulty one experiences conceptualizing a 4th spatial dimension can be useful in attempting to conceptualize the possibility that difficulty also exists in conceptualizing these non-spatial dimensions.
Or in other words: People don't think it be like it is, but it do.
I think you’re making my point. The Fourth Dimension is still uniquely defined for our 3-space (which is the entire universe, as far as we know). I think that’s pretty solid justification for giving it a distinct name.
Hypothetically, if there are other (infinite) 3-spaces embedded in our 4D metaverse, then we either intersect them (which would cause all sorts of problems), or they are parallel and agree with our definition of The Fourth Dimension.
Our universe might be a much more complicated manifold than a plane in the space it's embedded in. Of course, we experience it as a flattish 3D space, but maybe gravity makes it four dimensionally lumpy. Or something else, who knows. Maybe it's actually closed and is the surface of a very, very large 4D sphere.
This would make the ana/kata vectors pointing outwards away from the universe where I am and the ana/kata vectors where you are not line up, and there wouldn't be any way to decide which ana/kata vector is the special one, even between different points inside our universe.
But really, my objection is mostly that "The Fourth Dimension" makes it sound like a "Dimension" is a kind of place, which is confusing.
This is one of the biggest hurdles of learning modern physics. We know, from experiment, that the universe is flat at large distances, and that there is not a fourth dimension into which it is folded or bent.
However, we also talk about "curvature" of space-time, and people's intuitive first understanding of that is that space-time somehow gets bent from the perspective of a higher number of dimensions.
Curvature is very badly explained in physics, starting from the highly misleading rubber sheet analogy.
If you want more pet peeves for your collection, try every explanation of gravity that shows rubber sheets, and (for advanced peeving) rubber sheet diagrams of Schwarzschild black holes that put the event horizon partway down the well, instead of where the coördinate singularity actually places it (which is infinitely far down at a finite radius).
Isn't that just a matter of visualization, though?
You can visualize (a static snapshot of) a 1-dimensional compression wave as the flattening of a 2 dimensional sine wave.
More generally, any n-dimensional space with varying "density" can be viewed an n+1 dimensional space, where the extra dimension is the "density" dimension.
Maybe a "stretch".
The article is extremely clear in explaining the fourth dimension is a direction, not a place. You appear to be projecting your past experiences onto the article.
Quite likely I am! And of course the article was fine. But I will note there was already a comment on it about how "the fourth dimension is time, actually." So this particular phrasing does seem to keep confusing people.
If two flatlands are embedded in 3d space, there's only three dimensions, and one dimension shared between the two flatlands, or one common 3rd dimension if the flatlands are parallel.
What they may disagree on is which the third dimension is. But, so what? subjectivity isn't wrong. You can make the same argument to say that "right" and "left" don't exist because they are subjective.
No, one couldn't; because roywiggins isn't arguing that the directions don't exist. Given that the ideas of there being no privileged frame of reference and the universe being isotropic are fairly basic concepts of modern physics, it is quite surprising how many people don't grasp the quite simple argument that "the" is the wrong article to use.
It was even wrong by 1884 standards, although one could excuse a treatment of it at the novice level from explaining that not all subspaces of 4-dimensional spaces are necessarily parallel.
if we were able to escape 3-space by moving into a 4th spatial dimension
We are perpetually suspended in this '4th dimension' given that we are orbiting a galaxy and star. Find a way out of the observable universe which doesn't move and you might escape this fourth dimension.
The point is "a" fourth dimension vs "the" fourth dimension.
Our 3D space uses all 3 dimensions interchangeably. It makes sense to talk about "the" 3 dimensions of space.
If you look at 3D + time, time acts differently than the other 3D.
- we only move through time in 1 direction, at a rate of pretty much 1 second per second.
- things break more easily than they go back together (ex. shattering a glass)
So time doesn't fit in with the other dimensions. Or does it? Einstein crammed it in there, and proved that time would be the same except we're being constantly pulled in the timewards direction by a really heavy object that we can't see. Or something like that.
> These things aren't ordered, and in fact you can't really distinguish between the three familiar spatial dimensions: imagine trying to point along dimension one, whatever that means.
I actually don't know what this means. You can certainly distinguish between the three spatial dimensions. Pointing along dimension one is irrelevant. Once you point in a direction, you can define the others given an orientation, and there are only two unique orientations on a three dimensional manifold.
What they are saying is that you can't make a non-arbitrary identification of each dimension. You can call them 1,2,3, and I can call them 2,3,1, and that changes nothing except your arbitrary labels.
A third party has no way to know whose choice of "1" is better for any reason related to the space itself, only by reference to some preferred object in the space, like the direction my pencil is pointing right now. Space itself has no such preferred object or direction.
> What they are saying is that you can't make a non-arbitrary identification of each dimension. You can call them 1,2,3, and I can call them 2,3,1, and that changes nothing except your arbitrary labels.
I don't understand the point. What is the supposed consequence of that? As far as I can tell, it's an empty statement.
And as I said, there are only two orientations in three dimensions. For an orientation, which is an equivalence class of ordered bases, the order actually does matter. You can call them whatever you want, but once you've called them something, their order matters, which was my original retort.
I highly recommend a book called "Spaceland" by Rudy Rucker. It's like a modern take on "Flatland". In it a Silicon Valley hotshot gets visited by a 4th dimensional entity called Momo.
He also wrote a book called "The 4th dimension" which explores the concept historically and in various ways.
Weird synchronicity, I just had a long conversation about this book a couple days ago, because the subject of jungle gyms came up. I was also wondering wondering whether Hinton had any freemason connections, since I learned about him from From Hell, and a lot of that book seems to draw on masonic references.
I love reading this for the eloquent, antiquated style of the prose. The part where he writes about experimentally detecting four dimensional space by the behavior of matter was interesting - I’ve never read anything like that before.
3D objects cast shadows, but not all 2D objects are shadows. A flat piece of paper is (an approximation of) a 2D object, but it's not a shadow of anything 3D.
Shadows behave really differently to real objects. They disappear into nothing, they can move faster than light, they can fully overlap each other and then separate again.
Think of a sunset just before the sun appears to fall below the horizon as the length of the shadows approaches infinity.
Not only shadows can move faster than light. Any projection can. Take a laser pointer and aim for the moon, then flick your wrist back and forth. The point appears to move faster than light across the surface of the moon. The photons still travel in a straight line, at the speed of light; but you are sending new photons in a different direction. There really is no single photon actually moving across the surface of the moon; it is merely an image.
Yes, this is what I meant. There is no single photon (or any other single object) moving across the surface of the moon. No individual thing is moving faster than the speed of light in this scenario.
Projection. If your shadow is further away than your object then the dimensions of the shadow will be exaggerated and so will it's motion. The light itself travels at light speed (duh?), but the image moves faster. Think how your monitor makes moving images without moving pixels.
If you build a arbitrarily large Dyson sphere, the shadows cast by planets orbiting the sun inside the Dyson sphere onto the inner surface of the Dyson sphere will appear to "move" arbitrarily fast.
If a planet orbits in, say, a day, its shadow will make a full circuit of the Dyson sphere in a day. Make the circumference of the Dyson sphere larger than a light-day, and now the shadow is "moving" faster than light.
Yes, but if the Dyson sphere is 2 light hours away from the planet, the planet will cross the actual geometric line between the Sun and a given point on the Dyson sphere two hours before the shadow is visible.
What you're saying is like saying that vision is faster than light, because I can look at Sirius, and then look at Aldebaran a second later.
Well yeah, that's why I said it appears to move. Shadows aren't objects at all so they don't really move, but colloquially we think of them as moving as much as we think of a laser dot moving (which also can "move" faster than light).
I think this - thought/'inner life' being the fourth dimension. But the problem is that there are several senses of the term dimension - there is the mathematical/topological sense and the philosophical sense, and possibly others. Confusing contexts doesn't help in talking about this stuff.
I imagine there would still be a relationship with `c` in a fourth spatial dimension, but it might be useful for taking shortcuts around 3d space without having to violate causality.
Technically speaking, we're talking about four dimensional space. It doesn't really make sense to call such a space "The Fourth Dimension", any more than real life space is "The Third Dimension", or a tabletop is "The Second Dimension". This sometimes trips people up into arguing over whether The Fourth Dimension "is" time, or whatever. For that matter, maybe the first dimension is time, and the second, third, and fourth dimensions are space. These things aren't ordered, and in fact you can't really distinguish between the three familiar spatial dimensions: imagine trying to point along dimension one, whatever that means.
The familiar three-dimensional space as we know it is three dimensional because you can put three straight lines to meet at right angles to each other, and no more. And you can label those lines x, y, and z if you like and pick their orientation. Four dimensional space allows you to cram another in. Two dimensional space only allows two, and one dimensional space is just a single line.