“A dimension is a set of values from which an object may only possess one.”
An object can have a temperature and a width simultaneously, so those things must be different dimensions. An object can’t be both 4" wide and also 5" wide. So those things must belong to the same dimension, and that dimension is labeled “width”. Similarly, you can be red-orange or red or orange. You can’t be more than one of those options, so that makes “color” a dimension. If it’s possible to have more than one value for that argument, then you haven’t defined the dimensions of that particular…scenario…well enough.
Am I making sense? Is that definition sufficiently broad but also specific enough?
Separately, I don’t understand the definition in the OP. That sounds like it’s defining dimensions, or more technically, a set of coordinates for a specific point. I mean, it’s defining {2, 5, 9, hot, blue} but that’s not a dimension. At best, it’s five dimensions, but I wouldn’t even call it that.
To strip things down to the more pragmatic (or the everyday dimensions we live with all the time):
3D = A model globe of the earth, resting on your desk. (length, width, height… a sphere)
2D = The circular shadow it casts on the desktop. (length, width… a plane)
1D = The circular shadow’s diameter. (length… a line)
0D = The dimensionless point the globe is in contact with the desk. (a point)
These are the spatial dimensions that we know of.
Other than that, you can go any which way, the most common being time as the 4th dimension, but this is not the same as a 4th spatial dimension; rather it is a temporal dimension.
As to trans-3D spacial dimensions, you’ll have to dive into manifold physics and mathematics, as these extra dimensions are next to impossible to fully imagine.
This first time I saw this, I wept. IANAWhatever, but after the explanation of the 3rd dimension, it quickly veers off into unsubstantiated illustrations that appear to be shoe-horned into the “point, line, fold” model by shifting it into “meta-dimensions”.
And if you had a bunch of dimensions, you could call the set of values coordinates!
The problem with this definition is that it doesn’t specify that the coordinates are independent. The surface of a sphere in three-dimensional space is two-dimensional because knowing any two of the (x, y, z) coordinates allows you to determine the third, but your definition doesn’t let us conclude that.
But the general notion of “dimensionality” can have as many dimensions as you would like. The OP didn’t restrict the definition to the physical universe but to the dictionary definition, which is a bit more general.
Obviously the term ‘dimension’ has a number of rigourous definitions, so any general definition that tries to encompass all these definitions is going to be prone to be imprecise and worse easy to find exceptions too.
However I think the best general definition I have read is the minimum number of coorindates needed to continuously paramaterize a space.
The use of minimum ensures that they are independent (i.e. we’re not using more coordinates than we need to.
coorindates are just a set of numbers assigned to every point. They needn’t necessarily be real numbers.
The use of continuous is sort of suggested by the use of cooridnates and it’s there to make sure we’re not using too few cooridinates and the actual cooridnates hold useful informaion about the point they describe. So for example whilst maps from R[sup]2[/sup]->R certainly exist, they do not describe the kind of ‘paramterization’ that is useful for use in this context. That said the dimensionality of a vector space for example does not rely directly on notions of continuity.
Parametrizejust describes the act of assigning cooridnates to every point.
Space is a bit of a fudge just having the vague meaning of a set with some additional structure.
Yes it’s simalir to the OP, but the key word IMO the definition in the OP is missing is the word ‘continuously’. As I mentioned you could assign every point in R[sup]2[/sup] a unique real number and call it a cooridinate.
It’s due to Professor Bernard Schutz of the Max Planck institute who wrote a very good introductry book on general relativity (though he may’ve been paraphrasing someone else).
For the purposes of physics where there’s nearly always a notion of continuity this admittedly vague defintion is quite useful.
Yes I’m aware of the definition of dimension in vector spaces as a specifically mentioned them:confused:
I’m not sure what it means, to explain why something is nonsense. The YouTube video is nonsense because he says he’s talking about dimensions, but the things he’s saying have nothing to do with dimensions whatsoever.
It means (loosely) that some points in the space are “close” to each other, and that if they are, their coordinates should be “close”. A formal definition would involve epsilons and deltas.
If you relax this, you can fill a 2D space with a 1D Peano Curve.
Once the video starts talking about dimensions higher than four it becomes completely disconnected from physics. The extra dimensions of string theory are not the extra dimensions described in the video. The extra dimensions described in the video are just an unsupported riff on the many-worlds interpretation of quantum mechanics.
