I use the word “dimension” in my writing quite often. I did not get a chance to clarify what I mean by it. The word “dimension” has multiple meanings in various contexts and it is worth spending few minutes to explain the differences.
Object dimensions refer to the extent of an object in external space. Objects in external space have 3 dimensions: height, width and depth. All objects occupy a 3-dimensional space. Even the elementary particles occupy a 3-dimensional space.
In the so-called “Standard Model” of physics, the elementary particles are modeled as having no extension in space. This is because with our most advanced microscopes (particle colliders like Tevatron and LHC) we cannot detect any size for them but it does not mean they have no extension.
The latest physics theories such as the “String/M Theory” claim that the elementary particles should have some extension in external space. I agree with the String/M theory. The elementary particles are extremely tiny but they still occupy space. The “Standard Model” assumption of point particle is problematic, it leads to nonsensical infinities in computations and many other theoretical problems. Physicists have found ways to sweep these nonsensical infinities under the rug but the solution is clear, elementary particles must have an extension in external space.
You may have noticed that I keep saying “external space” rather than simply “space”. It is because there is also “internal space” where elementary particle properties such as “spin” and “charge” belong. I will write about “internal space” later.
Space dimensions and object dimensions are the same. External space and objects embedded in external space cannot be separated in our minds. The 3-dimensions of objects mean the 3 dimensions of the external space. The reason I decided to discuss object dimensions and space dimensions separately is because we also have “empty space”.
What is the dimensionality of the empty space? This is a loaded question. One can digress into many deeper philosophical subjects but let’s stay focused.
If I had a small object (even a point particle) somewhere in external space how would I describe the position of that point? How would I describe the coordinates? The answer is: with 3 numbers. The coordinates can be described with 3 numbers. This is why the external space is 3-dimensional.
Space dimension is a geometrical concept. There is a branch of mathematics known as topology which deals with the dimensionality of n-dimensional spaces. Mathematics does not have to be tied to the physical reality so mathematicians freely explore the higher dimensional abstract spaces.
Compact Space Dimensions
Lisa Randall describes dimensions as “passages”. If an elementary particle can move along those “passages” then we can call those “passages” as dimensions. This definition applies to large space dimensions as well as the so-called compact (rolled-up) dimensions. Lisa Randall’s book titled “Warped Passages” and Brian Green’s books “Elegant Universe” and “The Fabric of the Cosmos” are excellent resources to learn about these rolled-up space dimensions. These compact space dimensions have not been proven with experiments yet but these are very compelling theories.
Dimensions as Degrees of Freedom
An object in external space can move up/down, left/right and forward/backward. These 3 degrees of freedom constitute the dimensionality of the external space. The “up/down” is one degree of freedom, the “left/right” is another degree of freedom and the “forward/backward” is another.
If you consider a tiny object such as a fundamental string/membrane, it has the “up/down”, “left/right”, “forward/backward” degrees of freedom but it can also move along the compact (rolled-up) space dimensions. So the degrees of freedom for these tiny objects are greater.
In my opinion, the definition of “dimension” as “degree of freedom” is the most general one.
Einstein’s theory of relativity (which was proved by experiments many times in the last century and is proved everyday in particle accelerators) describes a connection between the external space, time and the object embedded in the external space. This complicated relationship is described using the language of geometry. In this theory “time” is referred to as a “dimension” because the theory is based on geometrical concepts.
Fractals are mathematical objects, they are abstract but there are many physical objects that exhibit fractal properties. One of the essential characteristics of fractals is “self-similarity.” In a self-similar structure the whole has the same shape as the parts. Many popularized science articles give the shape of a coastline as an example for fractal but I will skip that and tell you about two self-similar mathematical objects (fractals) known as Sierpinski triangle and Cantor set. Physicists will discover the physical significance of these mathematical objects in the future.
Sierpinski triangle is a union of 3 copies of itself. Each copy is shrunk by a factor of 1/2.
Cantor set is a union of two copies of itself. Each copy is shrunk by a factor of 1/3.
The Sierpinski triangle (many iterations are shown here)
The Cantor set (first 6 iterations)
As you may have noticed the concepts of “iteration” and “recursion” are the other characteristics of fractals, “self-similarity” being the essential one.
What is the dimensionality of a fractal? Can we say Sierpinski triangle is a 2-dimensional shape? It has width and height but no depth. But obviously there are holes in the Sierpinkski triangle. Some areas within the original triangle are blank. If we define “dimension” as a degree of freedom then the Sierpinski triangle must have a dimensionality less than 2. The dimensionality of a Sierpinski triangle must be between 1 and 2. It cannot be less than 1 because it surely looks like a 2-dimensional shape but it not exactly 2-dimensional because there are holes (blank areas).
There is a way to measure this kind of dimensionality. It is known as the Hausdorff dimension.
Hausdoff dimension of the Sierpinki triangle is log(3)/log(2) = 1.5849 which is less then 2.
Hausdoff dimension of the Cantor set is log(2)/log(3) = 0.6309.
I cannot resist the temptation to mention that 1.5849 is very close to the golden ratio 1.618. There is something to explore here.
Statistical Factors as Dimensions
In finance, economics, politics, sociology and many other fields statistical modeling is very important. In fact, many decisions impacting the lives of billions of people are made based on these statistical models. I should mention climatology, the models of global warming explicitly and the field of pattern recognition or artificial intelligence. Statistical models are used everywhere. Even in web advertising.
In the statistical models we have multiple explanatory factors. We assume that those explanatory factors are independent of each other. Modelers (scientists) try very hard to come up with the statistically independent explanatory factors. Then the actual model itself is a formula that assigns weights to those explanatory factors. Those weights are called coefficients which are not precise numbers. Each coefficient has an uncertainty associated with it. The prediction number which is what the model formula produces has an error associated with it as well.
The explanatory factors can be described as dimensions (statistical dimensions or statistical coordinates). When the explanatory factors are not statistically independent then the statistical dimensionality is reduced. This situation is similar to the dimensionality of a fractal. The 2-dimensional looking fractal shape (Sierpinski triangle for example) has a a dimension less than 2. You may have 5 explanatory factors in your statistical model but the actual statistical dimensionality may be 4.5 because two of your explanatory factors may in fact be correlated with each other.