Moments of Spaces (Part 2 of higher dimensional spaces)
Previously, I talked about spaces with dimenions greater than 3 such as a 5D Melting Ice Cream (https://medium.com/@indiasoale/5d-melting-ice-cream-and-other-real-world-applications-of-n-dimensional-spaces-ea66d49608a7).
In this second part on higher dimensional spaces, we will be exploring what I refer to as ‘moments.’
A moment is exactly what it sounds like. Whenever we talk about the experiences in our lives, we are talking about moments.
But whenever we talk about these experiences we’re never retelling the entire experience, only parts of those experiences.
We can give the most detailed account possible or take a video with the highest quality camera and audio, but we will never capture the entire experience, only some of that experience.
When we take videos we may think we have caught the entire experience, but other variables such as what wasn’t shown on video or the context of the situation are left out. When we give a detailed account of our experiences, we can only give as much as is humanly possible to give, there are endless details in our accounts which will go omitted such as the weather that day, how many rocks we walked by, etc.
The details may be completely irrelevant to the account we are giving but they show that is it never possible to give absolutely every detail. We can only provide a subset of the whole experience.
We say therefore, that all moments are a subset of the experience.
But suppose it was possible to capture the entire experience, then even the entire experience would be a moment of the experience.
So the number of possible moments must be less than or equal to the whole experience.
A moment of an experience is the same number of variables or less than the number of variables in the experience itself.
The same can be said for spaces.
A moment of a space is the same number of variables or less than the number of variables in the space.
When I talk about variables in the context of spaces, I am talking about dimensions.
Suppose we were to model experiences as spaces, then the moments of those spaces would be the spaces themselves and any variables ommitted.
Suppose the entire experience is a space such as as a cube, and we omit it’s depth, now that cube is a square.
Suppose we then omit the the height of the square, now that square is just a line.
The cube, the square and the line are all the moments of the cube.
Referring back to our earlier example, suppose we are giving an account of an experience but we omit the weather that day or how many rocks we walked by. Then the ‘weather’ or ‘rocks walked by’ are two variables (two dimensions) being omitted from the experience — we are essentially giving our account in two dimensions less. Then we think of all the other variables we missed out and we can see that the number of these variables (dimensions) becomes very large.
If we treat an experience as an n-dimensional space, each time we give a detail, the number of dimensions goes up by 1. We talk about how the weather being hot, 1 dimension, we walk by 5 houses on our way to work, 2 dimensions, we cross two different roads, 3 dimensions and so on. We can go all the way up to n dimensions where n reaches closer and closer to infinity but never gets there.
Consider a bowl of ice cream; not only are the values of its width, height and depth changing but so is its temperature and any other variable relating to it that we can think of. Let us denote the number of all the variables we can think of as n.
Let us suppose we are only concerning ourselves with the width, height and depth of ice cream, but not all of the other variables, then we are analysing a three-dimensional subset of an n-dimensional bowl of ice cream. We shall refer to this subset as a moment of k dimensions d1, d2, … dk, denoted m^k where m^k is a subset of an n-dimensional space, S^n where
If this is confusing so far, then it would be worth reading the article I wrote on n-dimensional spaces.
A moment, m^k, is a subset space of a space S^n. It has the same or fewer parameters as S^n.
Informally, a moment is what a space looked like before a dimension was added. A square, for instance, is a moment of a cube — it has the first two dimensions included in the three-dimensional set of the cube and it is also a cube before depth is added:
We can state that a cube, C³ has three moments; a line, c¹, a square, c² and itself, c³.
We can therefore derive the following subsets of C³ which are moments of the cube, C³ and their respective orders:
If we refer to the 5D Melting Ice Cream Space, I⁵, from the previous article, we noted that it had five dimensions d1, d2, d3, d4 and d5:
The ice cream in three dimensions, before the dimensions of “time elapsed” and “temperature gained” were added, is a moment of the melted ice cream space, I⁵.
I will show in a later article why algorithms are very good examples of moments, since the moments of the algorithm are simply the algorithm with the same or fewer arcs.
All n-dimensional spaces have n moments. The melting ice cream space, I⁵ is such that
I⁵ = {d1, d2, d3, d4, d5}
And it’s moments are:
i¹ = {d1}
i² = {d1, d2}
i³ = {d1, d2, d3}
i⁴ = {d1, d2, d3, d4}
and
i⁵ = {d1, d2, d3, d4, d5} = I⁵
(Not to be confused with the complex number, i).
Moments can be thought of as a series of steps where each successive moment, a new parameter is added.
In the next article, Pancake Theory of Dimensions, we shall explore a flattening approach to n-dimensional spaces which I refer to as unionisation.
