Just what IS a dimension?

A direct link to the above video can be found at http://www.youtube.com/watch?v=RmQqDJwnmAw

The above video is about a meme that has risen in popularity this year: "it's incorrect to talk about imagining the fourth dimension, one should instead talk about 'imagining four dimensions' because these dimensions are all intertwined". My response is that's at best a semantics discussion, and at worse leads to the faulty conclusion that there is no difference between the degrees of freedom available to a 1D line versus a 2D plane and so on. Would I be saying different things if my project were called Imagining Ten Dimensions instead of Imagining the Tenth Dimension? No. Both are different ways of talking about the same idea. So, while it's all well and good to say "let's imagine ten dimensions", perhaps there's an even more basic question to ask before we begin:

"Just what IS a dimension?"! 

Let's take a look.

Very generally, if you wanted to describe all the possible states for a certain quality of something, that would be a dimension. We could create a database of temperature for a given location, and all possible temperatures for that location would be a dimension. But we would need to add a dimension if we also wanted to plot windspeed. And so, if this is the approach we are using to define the word "dimension", then there is no reason to assume there are only ten: couldn't there potentially be an infinite number of ways to describe something, and therefore an infinite number of dimensions?

Theorists have said our reality comes from ten spatial (or "space-like") dimensions. How can we imagine them? Well, the point-line-plane postulate is the accepted way to visualize any number of spatial dimensions. Please note, though, those two important words: "any number". This means that we can easily say that this postulate results in an infinite number of spatial dimensions, because there's no reason to stop at any particular number.

But with this project, I do indeed say that we can stop at a number, and that there are really only ten dimensions... or eleven if you count the "zeroth" dimension, the point that we start from. Wow, isn't that quite a jump, from the mind-boggling realm of infinity down to a measly ten dimensions? And yet with this project, I insist that because we are assigning meaning to each of these dimensions rather than just abstractly adding one upon another, we really have reached an infinite "everything" by the time we get to ten.

As I've said elsewhere, the point-line-plane postulate and the line/branch/fold visualization that Imagining the Tenth Dimension uses are very related concepts, two different ways of describing the same idea. Both say there is a repeating logical structure we can use to extend from our intuitive knowledge of the first three spatial dimensions into the extra spatial dimensions that lie beyond.

One of the words used to describe spatial dimensions is that each is orthogonal to the next. "Orthogonal", as defined in the Mirriam-Webster Online Dictionary, means "intersecting or lying at right angles". "Perpendicular" has much the same definition, and the two words are often used interchangeably.

Let's go back to our windspeed/temperature example for one way to think about dimensions. What if we were to add an additional dimension which plotted the elevation above sea level? Now we have three dimensions for our given location, two of which are constantly changing, one which stays the same. In this way we can see how with multiple dimensions, some can be "pinned in place" so to speak, while others change. With my approach to visualizing the dimensions, I suggest that our universe is "pinned in place" at a position within the seventh dimension and above, with the sixth dimension and below allowing for the phase space of all possible states for a unique universe such as ours to be expressed. (String theorists have said our universe is embedded within a seven-dimensional "brane", or membrane, which could be another way of expressing the same idea).

Are windspeed and temperature "orthogonal" to one another? Only in one sense of the word. If you read through the entire Mirriam-Webster definition for orthogonal, the last interpretation listed is "statistically independent". Does temperature have to go up when windspeed goes down, or do both have to go up and down in lockstep? No. In other words, they are statistically independent. One could even say that windspeed, temperature, and elevation above sea level are at right angles to each other, in that you could plot these values on a three dimensional graph, with each axis at right angles to the others.

But how can we visualize a four-dimensional, a five-dimensional graph (and so on), where each axis is at right angles to the others? This is very hard for our monkey brains to envision, and that's the beauty of the point-line-postulate: it gives us a way to keep building the idea of spatial dimensions one upon another in our minds. 

So, we've established what dimensions are, but have we established what spatial dimensions are yet? Here's how I would define the difference between elevation/wind speed/temperature as a set of three dimensions, as opposed to three spatial dimensions such as length, width, and height:

1. wind speed does not require temperature to exist, or vice versa: so those are not spatial dimensions. The third dimension that you and I are within, on the other hand, can't exist without the first and second, because they are all spatial dimensions.
2. The third dimension, like any of the spatial dimensions, is really a set of dimensions that are intertwined. Because of this interdependence, it doesn't matter what label you put on the third dimension: so while height or depth are different ways of thinking about what gets added by the third dimension, any term you use is dependent upon your reference frame. As I've often said, changing labels doesn't change what we're talking about, and "a rose is still a rose by any other name". On the other hand, you can't take your values for wind speed and say they are now temperature: the labels are not interchangeable, so those are not spatial dimensions.

That's it in a nutshell. There are many dimensional systems which can lay claim to an unlimited number, and that includes spatial dimensions if you're speaking in abstract terms. But with this project, we discuss the quality that gets added with each spatial dimension, and that's how we end up with the bold statement that there are really only ten spatial dimensions.

Enjoy the journey!

Rob Bryanton