In order to describe my definition of a Fractal Manifold we must first begin with some very basic mathematical ingredients, the first being a manifold itself. For those of you how do not know what a manifold is you can either read my description of it here (or wiki it here) or just read on.
Let’s begin at the very start. Suppose all you had was a collection of some point-like objects, and that these points had little regions surrounding each of them. Looking at this picture you can start asking some very simple questions such as how many points are there? Another question might be what is the area surrounding each point? Yet another might be how close are two points? These sorts of questions asked of this very simplistic mathematical setting is really how Set Theory was born. See “sets” are just collections of points and “subsets” are just other, sometimes smaller, sets contained in bigger sets. Like a hierarchy of collecting and categorising points in space.
Once you press the “I believe” button on Set Theory things start getting a bit more interesting. A special sort of set becomes apparent, the empty set. Further, you get for free two very fundamental operations that you can perform – both of which are very similar to counting points. These are the union and the intersection. Basically the union combines sets together to form bigger sets without violating any logical laws, and intersection splits sets up into smaller sets, again without violating any logical laws.
The astute reader probably knows at this point where I’m going with all of this. Yes, in fact what we have already in just the first couple of paragraphs of this article is enough to define a topology. A topology is just a prescription or a rule that lets mathematicians talk about things that are open or closed. Why is it important to want to know if something is open (or closed) you ask? Because it let’s you figure out how close something is to something else and one of the fundamental questions we want answered of our simplistic mathematical construction is how close are two points?
How close two things are together is a fundamental question that has many physical and philosophical implications. Evolutionary speaking, we as humans probably evolved with a constant natural desire to know how close two things together – I’m thinking here about how close that hungry tiger is to me and how far away my cave is. One might argue that it was this very question which began physics. Philosophy aside, closeness is important for another reason, a mathematical one. Two things can be classified as being mathematically different if there is some non-zero separation between them. Being different, mathematically, is very interesting (the complement, that is everything being exactly the same, is very boring) and long ago we constructed a tool to help us determine this separation – it’s called a metric.
A metric is a tool and what we call a piece of structure that a mathematician can add to his canvas to enrich his masterpiece. But what exactly is a metric? A metric is a function – which means that it eats something and spits something out (or in other words you put something in and it produces something back for you – quite like a primitive sort of machine). A metric is actually a special kind of function called a functional, yes just add that ‘al‘ at the end and you change its meaning. A functional is basically a function which spits out a single number. So, you could feed a functional a complicated (and I mean really complicated) thing, a functional will go ahead and chew on that and spit out a single lousy number. Well wait a second, that makes sense! If we want to know the distance between two objects then the thing that tells us that distance better be able to consider those two objects and spit out a single number representing that distance.
However, can we define distance without having to equip ourselves with a metric? The answer is yes we can. A purely topological space (one with just points and neighbourhoods of points) gets a distance measuring tool for free: the open set. An open set can be defined without a metric and with logic alone. Two points are topologically distinguishable if about one point there exists some open set not containing the other. If the second point was within the open set of the first then both points are inside an open set and thus are identical so far as the topology is concerned. So how does one define the open set about a point? There are three rules for constructing an open set:
- The whole collection of all points and the empty collection is an open set,
- Any union of any number of open sets are open sets,
- Any finite intersection of open sets are open sets.
This blue print for open sets is a bit self-fulfilling, but you can just start with the whole collection and the empty set and start creating unions and intersections of varying combinations of the two and create all the open sets possible.
A collection of points and a topology (a collection of open sets) defines what is called a topological space. It’s pretty generic and bland but it doesn’t demand much structure or rules, so it is quite a basic mathematical object and a good starting point for creating a Fractal Manifold. Furthermore, many mathematical frameworks are special examples of topological spaces so we say that a topological space generalises many other types of spaces, one of them, as we will see is a Fractal Manifold.
If you equip a topological space with a metric then you get a metric space. A Fractal Manifold is a metric space but the functional is a little more complicated than the standard Euclidean one – so we leave that definition for later. Instead we next define what a manifold is.
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