Mathematical Theory of Infinite Dimensions
According to my theory, all dimensions contain an astronomically large number of sub-dimensions. Equally, any number of dimensions, no matter how large, amount to one dimension.
To understand why this is the truth of the reality we live in, that there are astronomically more dimensions than 3, 4 or 10 (as is proposed in String Theory), we have to first of all consider what a dimension is.
A dimension is merely a number-line for the values a variable can take, and these are not limited to the width, length and depth of a space, but all other variables we can think of.
This article is part 3 of my series on higher dimensional spaces.
In Part 1 we showed that an ice cream can be modelled to have 5 dimensions. Below is a 5D melting ice cream with values in width, height, depth, time elapsed and temperature gained.
We established that any variable concerning any space, no matter how trivial can be modelled as a dimension. I decided to model breakfast as a 3D space with bacon, eggs and sausages as it’s three dimensions. In this example my breakfast included bacon and eggs but no sausages, giving us the coordinate (1,0,1) on the d1,d2,d3 axes.
We demonstrated visually that any space can have up to n dimensions (where n can be astronomically large).
In Part 2, we explored the observation that all spaces which we observe can be thought of as snippets of higher-dimensional spaces (a square can be thought of as a snippet of a cube, a line can be thought of as a snippet of a square, etc).
We think of our world in three dimensions, but this is merely an oversimplification of a far more complex reality.
The more variables we observe concerning a space, the greater the number of dimensions. If we observe 30 different variables concerning a space, then the number of dimensions of that space we observe is 30 dimensions.
However, these are only the dimensions we have chosen to observe. The number of dimensions of a space is independent of our observation and is always greater than the number of dimensions we observe as there is no limit on how many variables we can think of that concern a space.
The variables are endless, so the dimensions representing each and every one of the endless number of variables must also be endless.
The number of dimensions we observe expands or contracts depending on our observation.
If we observe a cube from one angle, the cube appears two-dimensional. However, if the cube starts rotating, the change of the shape in our 2D perception allows us to perceive it’s third dimension.
Suppose the cube is made of ice cream and is melting before our eyes, it’s shape is also changing in a fourth dimension, temperature.
However, it didn’t have to be temperature, suppose someone dropped a tennis ball in the cube. This too would be a dimension as the appearance of the shape changes in response to a new variable.
It wouldn’t even have to be a tennis ball, it could be dust falling on the cube or any other change we can think of.
As our perception picks up these dimensions, the number of dimensions in our perception grows. This is an expansion process which I refer to as ‘deunionisation.’
Deunionisation is seeing a space and picking apart it’s dimensions.
This makes it an additive process — as we perceive more dimensions in a space, the number of dimensions in our perception increases.
The reverse of increasing the number of dimensions in a space (deunionisation) is decreasing the number of dimensions in a space (unionisation).
Instead of perceiving a cube as three-dimensional we negate it’s third dimension, perceiving it as two-dimensional.
A cube from any angle looks like a 2D shape. This 2D perception is contractive as we are reducing the number of dimensions of the cube from 3 to 2. We’re essentially unifying the third dimension in 2D, which is why I refer to the process as ‘unionisation.’
Suppose I’m only interested in the three dimensions of a cube and none of the other variables I can think of, then I am unifying all the variables in three dimensions.
We can also perceive things as one-dimensional.
Place a pen on the table, walk further and further away from it, and it will eventually appear like a one-dimensional line from the distance. Be far enough from a road filled with cars and it will appear as a line where each car is a point on that line.
You don’t even have to walk a distance away. Simply place two 3D apples on a table. Then these two 3D apples are just two one-dimensional points on a one-dimensional line of 3D apples.
In both of these one-dimensional examples we’re unifying all dimensions as one dimension.
In all cases where we unify dimensions, we decrease the overall number, so unifying dimensions is a deductive process.
We say therefore, that the minimum number of dimensions is 1 and the maximum is n. So the range of dimensions of a space are between 1 and n where n can be astronomically large.
The theory of expanding the number of dimensions (deunionisation) and decreasing the number of dimensions (unionisation) is what I refer to as the ‘Pancake Theory of Dimensions’.
We will often subconsciously treat objects as one-dimensional.
Let’s suppose I have 5 3D apples and have to report this observation to the manager of a grocery store.
I don’t care about the width, height and depth of the apples as the store manager is only concerned with the number of apples there are.
I simply negate the height, width and depth variables and create a one-dimensional number line ‘number of apples’ with the value 5.
Rather than treating the apples as three-dimensional, I decided to treat them as one-dimensional, reporting only a single variable, ‘the number of apples.’
This process is unionisation as we have treated all dimensions as one.
Let’s suppose the grocery store manager now wants to know what the apples look like, we would then extend our model not only to include the number of apples, but what the apples actually look like (the width of the apple, the height of the apple, the colour of the apple, how old the apple is, etc).
The more variables we explore, the more the number of dimensions increases.
This process is deunionisation as we are now increasing the number of dimensions we are perceiving by inspecting more.
Now let’s apply this visually to all spaces generally.
Unionisation
The Pancake Theory of Dimensions is such that all dimensions (d1, d2, d3, …, dn) can be modelled together as flat to appear as a one dimensional line.
Consider one dimension as a pancake on its side. Then that pancake includes all the sub-dimensions of that dimension within its flat pancake surface while appearing as a line from its side.
For instance, if the x,y axes are modelled as a pancake, then it includes all coordinates of x,y while appearing as a single line from the side of the pancake. We shall call this one dimensional line a unionisation of x and y which is denoted as x ∪ y.
The bottom left picture displays the surface of the pancake whereas the picture on the right displays its side.
Think of the bottom left picture as the pancake you’re about to eat, it’s surface is x, y which displays whatever toppings (be it lemon, chocolate, etc) that you put on your pancake. If you rotate the pancake to it’s side, so that the side of the pancake is facing you, it now looks like a one-dimensional line. Even though the pancake now appears one-dimensional, it still includes everything that you put on the pancake.
You’ve turned your pancake on it’s side in the right picture above. The surface of your pancake is the x and y axes, so the space around your pancake is the z axis.
Now we have modelled x and y as one dimension, the z-axis is now taking on the role of the new x-axis.
Label x as d1, y as d2 and z as d3.
Then x ∪ y = d1 ∪ d2
We let p2 = d1 ∪ d2
Where p2 is the unification of two dimensions.
Then p3 is the unification of three dimensions and so on.
The same process can be done for a fourth dimension.
Imagine my flat pancake surface now not only includes d1 and d2, but now d3.
My pancake is now a sphere but I continue to model it as flat as this allows me to conceptualise a fourth dimension.
Because if the d1,d2,d3 axes are modelled as a 3-dimensional pancake, then d4 is the space around it:
For further help on visualising a fourth dimension, fifth dimension and so on please read Part 1 of my series on higher dimensional spaces.
If the d1,d2,d3,d4 axes are modelled as a 4-dimensional pancake, then d5, the fifth dimension is the space around that pancake.
And so on.
If the d1,d2,…,dn axes are modelled as an n-dimensional pancake, then we have the pancake pn (p_n, not p times n) where pn = d1 ∪ d2 ∪ … ∪ dn
Deunionisation
In everyday life we are constantly unionising dimensions as three-dimensional. When we are walking out of our house, one of the dimensions we are likely to be concerned with is the space between us and the outside of the house.
But there are other dimensions (variables) which we unionise with the dimension of space. For instance, the time that will pass in the moment, the temperature when we leave the house, even trivial variables such as the number of shoes we put on and the angle at which the door must be opened to walk through.
If we look close enough, the number of variables which we unionise with space are very large indeed.
As we pick apart these variables we ‘split’ the dimension of space into a larger number of variables (sub-dimensions) — we shall call this technique ‘deunionisation.’
‘Deunionisation’ is the reverse of ‘unionisation’ — instead of modelling multiple dimensions as one, we split them into sub-dimensions.
The right picture below is a complete unionisation of the dimensions of an n-dimensional space.
For the deunionisation of an n-dimensional space we unpack the dimensions.
If we expand (unpack) the pancakes out, then we have n pancakes, where all the little ‘…’ on the graph represents the pancakes between p4 and p(n−1) and the big three dots represent the dimensions (axes) between d4 and dn:
A clearer illustration of this is the computerised version I made:
Our lives mostly concern the space that surrounds us, so it is easy to unionise all dimensions, but humanly impossible to pick them all apart (i.e, completely deunionise).
At best we can partially deunionise the fully unionised dimension of space to the activities which concern us in the moments of our lives.
Some deunionisations, such as going about our daily activities, may be straightforward while others, such as being the best at a sport, becoming famous or other very specific desired outcomes for instance, may be far more complex as they may require knowledge of a higher number of variables than activities we perform which are more straightforward.
Irrespective of whether or not different outcomes can be objectively proven to be simple or complex relative to others, we can say that certain resulting outcomes of spaces will require knowledge of a larger number of parameters.
If we could know all variables and move in all parameters, no outcome would be complex and every coordinate of outcome would be simple to achieve. For instance, reading an encrypted letter would be no different to reading a letter written in plain English.
For more on spatial dimensions beyond three check out my new article where spatial dimensions beyond three are observed in the physical world: https://medium.com/@indiasoale/introducing-spatial-dimensions-beyond-three-the-theory-of-infinite-dimensions-253292bbd16f
