To simply put, your expression after reading the question is puzzled, right? How can there be fractional dimensions, let alone living in one? Before heading to answer something that tricky, shall we get to ground zero and understand what a dimension is in the first place? Quoting Wikipedia: “In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it”. Did you note the word “informally”? Obviously, there exist some rigid definition, what do you think mathematicians are for :). In fact there are multitudes of definitions in mathematics but will come to that later on. For now let’s just discuss some amazing thoughts based on the wiki answer which will really be a pleasurable experience. The most important thing in science is to ask the right questions so let’s start by a few interesting questions.
How many dimensions do our eyes see? The answer can be derived from this counter question: How many dimensions can a camera capture in an image? Well, that’s simple it is 2 and because our eye is an organic camera it can see only 2 dimensions. But why 2? Hmm, that’s a decent question. That’s because we live in three-dimensional space and one of the dimension(depth) in the image capture setup will be for light to bounce off the object and form image on the retina(photo sensors of our eye). Also, did you know because we have 2 eyes very close to each other we have stereoscopic vision and thanks to that we can visualize depth too. Okay that was more than asked, but how can you claim that the space we live in has just 3 dimensions? Because every point in space can be associated with three numbers in x, y and z direction. But how can you be sure that there are only 3 dimensions? Because we perceive only 3 dimensions. Isn’t that answer more anthropological rather than scientific? Okay, I will end the soliloquy here but see I made my point. Which one, science has anthropology as its basis or soliloquies suck :). No idiot, the point is that there is no concrete proof of space being three-dimensional. By the way, the picture below shows how a Tesseract, a 4-dimensional “cube” will look to us when it is rotating, so Yeah.
Okay, you might be thinking where I am going with this. As I said earlier before moving on to a tricky answer let’s get comfortable with dimensions and related concepts. In the nineteenth century a lot of work was done in Mathematics by some of the most brilliant people such as Bernhard Riemann on higher dimensional geometry. These were later used in physics starting with the Special theory of relativity. In fact it was Hermann Minkowski (Yes, he was a Mathematician and Einstein was his student) not Einstein who gave the 4-dimensional space-time model. Then came along Quantum Mechanics which was completely probabilistic and too random to make sense, although philosophers loved this and created all sorts of crappy theories using the uncertainty of the new science (as is always the case). The uncertainty of the new subject was obviously taunting to many physicists but in the recent decades the dimension picture is attempting to explain why quantum mechanics may look probabilistic because of our 3-dimensional spatial perception. The amazing video below talks about this new approach, now very well-known as M-theory (apparently M stands for Minkowski) and also shows you how to visualize 11 dimensions.
The above picture shows a Cantor set also known as Cantor comb. You get this by taking a line and opening up the middle third of the line. As you can see the second line has an open middle which is 1/3 the length of the line and the third line is got by applying the same rule on the second line. What is so special about this? Actually, everything about this set is special and the most amazing thing, before I stop this first part of my answer (I know, I was cheeky for not giving the answer in this part :)), is that the Cantor set which intuitively looks to have a dimension of 1 (as it is a line), actually has a dimension of 0.630929… Yes, I will explain this in the second part and also answer the question in the title. So, stay tuned
……………….to be continued………………….