What characterizes a dimension? That is, how do I know a dimension when I see one? String theory says there might be up to eleven of those suckers, I want to know what one looks like if I happen to step in one.
One of the obvious characteristics is duality (left/right, up/down, back/forth, forward in time/backward in time).
There also tends to be a property of movement within the dimension. Non-living, non-energized matter cannot move voluntarily within dimensions, living matter can move voluntarily within dimensions, and energized non-living matter also seems to be able to move voluntarily.
It seems that explaining any one dimension on it’s own without utilizing other dimensions for comparison is very difficult.
Shadows exist between dimensions. That is, one dimension can generate an effect in another dimension through a difference in energy. That’s probably a debatable definition, but it seems accurate to me.
More advanced entities are able to voluntarily travel in (and perhaps otherwise experience) more dimensions. This too is debatable, but given the solid fact mentioned in #2, it seems to be true.
Is there anything else that might characterize a dimension?
Well, in essense, it’s something that allows a degree of freedom of motion. I’m not sure if a better definition of a dimension exists. All the other stuff seems to be consequences of the existence of one or many dimensions.
Dimensions are the measurements required to indication the location of a point relative to another point.
Thus, in a one-dimensional space (a line), you need one number to locate another point on the line (e.g., 2 inches, assuming the minus sign shows direction).
In two-dimensional space, you need two number to locate. They are usually length and width, but not necessarily (you could locate a point by using the angle and distance of a line given that angle).
We live in four-dimensional space; you need four number to accurately map the location. However, in most instances, time is assumed to be “now.” But if you were mapping the location of a fly in a room, the time dimension is important.
Or, I suppose, a coordinate needed to describe location. We’re limiting this to space-time dimensions, right? Otherwise, almost any quality (direction, volume, mass, intensity) tacked onto a scalar could be thought of as a “dimension”.
AFAIK, no matter how many dimensions the latest trendy theory says there are, we will only ever be able to perceive three (four if you include time, which you must admit is not quite the same as the others).
The number of dimensions is the number of variables needed to unambiguously describe a position relative to another position in a particular model – be it space, space-time, or a 107-dimensional model that a mathematician dreamed up. It doesn’t really make sense talking about “stepping into another dimension”. But if you “step” into, say, a five-dimensional universe (and your brain and body magically transform to cope with this), you will need five numbers to describe a position (relative to another position).
I’d also like to point out that non-living things cannot do anything voluntarily. Hopefully someone will be along shortly to explain that motion is all relative anyway – I’m too tired to put it into words.
Dimension is the cardinality of a basis for a vector space. In a manifold it’s the dimension of the tangent space (in this sense) at any point.
More generally, it’s the dimension over the field of all F-valued functions modulo the maximal ideal at any point of the vector space given by that maximal ideal modulo its square, where F is the base-field of your manifold.
There’s also a notion of Hausdorff dimension, but that only really applies to subspaces.
So this brings up an interesting question for me. Is MATH a dimension? In order to make any one dimension (either the three spatial or the one temporal) make sense, it seems to require math.
What I mean is, if you read a description of what it’s like to exist in a one or two spatial dimension world, it’s never quite right. The 3D world always tends to sneak in and taint the description. So it’s very difficult to explain without another dimension to compare it to. It occurred to me that math is the same way. You cannot describe another dimension without math coming into play. Just like RealityChuck’s example above. And there is a degree of difficulty describe math, even if it’s just numbers, without using something from one or more of the other dimensions. Doesn’t that at least hint at the idea that math too is a dimension?
If we bounce that idea off the list of characteristcs of the other dimensions:
There is a duality in negative and positive numbers.
You can “move” within mathematics. Although it’s more of conceptual movement that real movement.
Explaining math has been discussed.
Math can exhibit a shadow of sorts. It can describe things in higher dimensions, but it lacks the substance of those higher dimensions.
An entity must be more advanced to experience math.
We didn’t even realize time WAS a dimension until early in the 20th century. Is there no room for the possibilty that we might be identify another dimension, or ten, or 206? And if so, why not?
As for pointing out that non-living things cannot do anything voluntarily, there was no need for that. I brought it up only because I was sure that someone on these boards would feel the need to point out that non-living things DO in fact move through dimensions, but only because they happen to be on a spinning rock.
No. I’m not quite sure what you’re getting at. Math is an entire system of thought. A dimension can be expressed as a cardinal number in a coordinate system. You might need math to accurately understand and map out a 4-dimentional coordinate, but in and of itself it’s not a dimension.
How do describe the position of an object? There’s an x-coordinate, a y-coordinate, and a z-coordinate.
x-coordinate: 54 degrees N. Lots of things exist at 54 degrees N. We need to narrow it down to describe the position of our object.
y-coordinate: 17 degrees W. Now we have two dimensions: 54N 17W. But our object can be on the ground, in the air, or way under ground.
z-coordinate: 34,000 ft above sea level. OK, we’re getting somewhere. 54N 17W, 34,000 ft up. Can another object also exist in this exact same space? No, not at any one time. Ah ha! Time! So to accurately describe our object, we also need a time coordinate. After all at 54N 17W 34000ft, there were different objects there in 1890 and on April 15, 2004, 9:31:04 p.m.
time: 54N 17W 34,000ft, 4/15/04 9:31:04 pm. We can now accurately describe the position of our airplane in space-time. We have no need for a fifth dimension in this case, unless we can somehow find a way that two objects can exist independently in the same place in space-time.
Another thing that I was thinking about is that it seems to me that mathematics is something that has existed since the beginning of the universe. As soon as time ticked one nanosecond on the clock, and as soon as energy and matter exploded a Planck’s length from the theoretical singularity that existed prior to the Big Bang, math had to exist to provide a framwork for its existence. Math was born as soon as time and space was. It could be argued that math is a construct of the human mind, but it seems to me that one plus one equals two and it doesn’t matter if there’s a human mind present to conceive of it or not.
The idea that there are four dimensions between space and time isn’t so much that time is like the spatial dimensions. It’s that the more important underlying truth is that there are four dimensions that all behave similarly, that each have some space and a little time in them.
If you move in a certain way, you perceive three spatial dimensions and one time dimension. But if someone else moves in a different way, the three spatial dimensions they perceive and the time they perceive don’t quite coincide with yours.
A good example is that two events can’t just happen at the same time if they are at two different locations. For some observers one will happen first, and for other observers the other will happen first. And this isn’t just because of how long it takes them to see the events because of how far the light must travel - of course we’re already taking all that into consideration.
Any perception of three spatial and one time dimension is a 4 dimensional coordinate system. Someone else moving differently has a different 4 dimensional coordinate system that is a rotated version of the first one. So the underlying truth is more nearly that there is some kind of 4 dimensional mathematical space made up of 4 spacetime coordinates in which all 3 dimensional spatial coordinate systems and times are just specifically oriented instances of coordinate systems. At least, this is what is left of some courses in relativity I took light years ago. Ah -Heh.
Thanks for the replies. The “is math a discovery or invention” thread was good stuff.
What led me to this was the question of what “concepts” (which is a word that seems incomplete) were born at the beginning of the universe, and in what order? Obviously, this gets very complicated in mere nanoseconds after the Bang starts, but I imagined that this would happen in a manner similar to dendritic scaling; a few concepts giving birth to others, and they in turn bearing their own child concepts, and on and on.
I would expect that the concept of Creation would be first. That would begat matter and energy, which would require spatial dimensions, time, and mathematics, which in turn would require laws to govern their behavior, etc…
You see what I’m getting at? Is there a progression of “concepts” that can be chronologically listed in order of creation, after Creation? It doesn’t seem to me to be logical that they all are created in one fell swoop, so there would seem to be a set of “base concepts” that spawned all others.
Well, no. Concept of a Creator would have to preceed the concept of Creation. If you don’t abide by that concept, you cannot have creation. The Big Bang then must be explained by other means.
BTW, we will never know if there are other dimensions because we are stuck in the 3 or 4 we are in. If somehow you can walk through a wall, then you will have discovered another dimension.
