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.
> 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.
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.
> 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.
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.
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.
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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.
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).
"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.
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.
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.
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".
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.
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.
If I take a 1D space, I just need an orthogonal vector to find a 2D space (the second dimension).
If I take a 2D space, same...(3rd dimension).
So if I take a 3D space...? Needs a 4th vector (4th dimension) to find a 4 dimensional space?
?_? (wut?) is the pedantry really accurate here or is it sarcasm or should I go sleep..
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.
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.
[1] https://store.steampowered.com/app/2147950/4D_Golf/
What is the Fourth Dimension? (1884) - https://news.ycombinator.com/item?id=27329211 - May 2021 (45 comments)
He also wrote a book called "The 4th dimension" which explores the concept historically and in various ways.
What dimension is thought in?
I propose that thought is the fourth dimension and we are the shadows of our thoughts.
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.
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