5D Melting Ice Cream and Other Real World Applications of Higher Dimensional Spaces
The picture above displays an ice cream melting in five dimensions. The first three coordinates d1, d2 and d3 represent the positive spatial dimensions of the ice cream (x, y and z), d4 represents the time elapsed, in this case 3 units, and d5 represents temperature gained, in this case 5 units.
So how did we get here?
It begins with my theory on n-dimensional spaces or n-spaces — an extract from my most recent Mathematical paper which I have dedicated this article too.
The theory is such that if you have n different variables relating to a space that you wish to show graphically, then you can represent each of those variables on an individual axis. Therefore, if you wish to illustrate a space in n different variables, then you will need n different axes.
The image at the beginning of this article shows a melting ice cream in five different variables with an axis representing each variable.
To understand the concept of an n-dimensional space on the n-axes we must begin conceptualising dimensions beyond those most will be familiar with.
Mathematicians are familiar with the positive x,y axes on the left and also the positive z-axis on the right.
The key thing to notice about the z-axis is that it is a shift of the x,y axes in z. By treating the x,y axis as flat, we conceptualise the z-axis. The same is true for the y-axis, which we can conceptualise by treating the x-axis as flat.
The same perspective can be used to demonstrate a fourth axis as to represent a fourth variable, and a fifth, and a sixth, and so on:
Fourth Dimension
To conceptualise the fourth dimension ‘w’, we must treat the x,y,z axes as pancake-flat. Picture the x,y,z axes as a pancake on a table and the w-axis striking up through the centre of the pancake, along with the table:
We can draw parallel dotted lines of the x,y,z axes along the w-axis to avoid confusing different axes.
To conceptualise this further, let us draw a four- dimensional space on the x,y,z.w axes such as a tesseract.
First draw a cube in the x,y,z axes, draw the same cube in the x,y,z axes but a distance of ‘w’ away from the x,y,z axes. Finally, we join the edges to complete the 4D cube.
Below, the four-dimensional space of the tesseract is illustrated.
Let us denote ‘x’ as ‘d1’, ‘y’ as ‘d2’, ‘z’ as ‘d3’ and ‘w’ as ‘d4.’
We now have the set of four dimensions S⁴:
Fifth Dimension
For the fifth dimension ‘d5’, we do exactly the same as for ‘d3’ and ‘d4.’
By imagining the ‘d1,d2,d3,d4’ axes as a flat, we conceptualise the fifth dimension:
Sixth Dimension
To show ‘d6’, we model the ‘d1,d2,…,d5’ axes as flat:
nth Dimension
We now shall denote all dimensions from d1 to dn as S with order n:
To draw n axes, we must therefore show ‘dn’, by modelling the ‘d1,d2,…,d(n−1)’ axes as flat, where the vertical dots on the axes below represents the axes between ‘d1’ and ‘d(n−1)’:
On seeing n-dimensional axes, we have demonstrated the existence of n axes, i.e, axes with values on each and every axis from d1 to dn.
There is an endless potential for real-world applications of dimensions beyond the traditional two or three axes we are used to using.
Suppose we want to show the effects of an ice-cream cone melting after a certain length of time has elapsed. We can use d1, d2 and d3 to represent the image of the ice cream, d5 to represent ‘temperature gain’ and d4 to represent ‘time elapsed.’
Just as a flat (two-dimensional) image of an ice cream changes as depth is added, the image of the ice cream also changes in d1, d2, d3 as the temperature rises.
As the temperature rises, time elapses. Even if we suppose d4, ‘time elapsing’ does not have a direct effect on the image of the space, the space can still be modelled to have a value in this axis.
We start by drawing the ice cream cone on the d1,d2,d3 axes at the point that the ‘time elapsed’ and that the ‘temperature gain’ variables are at zero:
Next we can draw the ice cream cone on the d1,d2,d3 axes after 3 units of ‘time elapsed’ and 5 units of ‘temperature gain’ have occurred.
We now have the change in space of the ice cream after the occurrence of these two variables:
We can show both of these moments on one set of axes, by drawing both stages of the space to begin with and labelling the new top point of the ice cream (3, 4, 3, 3, 5):
Finally, we can remove the original space before the two variables (dimensions) were added:
We now have a real-world application of a five-dimensional space which concerns five different variables. We can denote the 5D space of the ice cream as I⁵ where:
I⁵ is a space or function with order 5, with dimensions which have a range of possible values.
As we have seen, just as the dimensions of width, height and depth make up how a space appears in the real world, so does the temperature variable.
Reality, however, is far more complex.
‘Time elapsed’ and ‘Temperature gained’ are not the only variables that exist which may or may not have a direct effect on the space of the ice cream cone.
There are countless other variables we can think of, of which the ice cream cone has values, but which may or may not have an effect on the values of other dimensions which we have ignored, even if they are zero values or very trivial variables.
For instance, the number of people that saw the ice cream, or cars that drove by as the ice cream was consumed may not have any effect on the image of the ice cream space, but the ice cream space can be modelled to have parameters in these variables (dimensions), even if they are zero values.
There appears to be no limit on the variables of which a space can have parameters, regardless of how trivial or non- trivial the variables concerning the space may be. We propose that the ice cream cone has values in n variables, making it an n-dimensional space or n-space.
We model spaces to have the number of dimensions as those we have selected, n, which we can infinitely increase as we identify newer parameters, but can never reach infinity itself.
This becomes clearer in what I refer to as the “Pancake Theory of Dimensions” — the third chapter in my mathematical paper.
We can say the same for the largest known space, U^n, the Universe.
If we model the Universe to have values in n different variables, then we say:
If a space has a zero-value in a particular dimension, it can still be modelled to have a parameter in that dimension with the value zero.
Coordinates of an Outcome
Whenever we consider variables which may or may not affect a space, we are thinking of the dimensions they may have values in. Any and every variable which takes a value can be a dimension or a space, as long as it has at least one parameter.
We can represent all variables which take real values on an axis. Even trivial variables which have the state of true or false can be modelled as dimensions. These are Boolean variables which can be represented on an axis by a ‘1’ if the state of the variable is true or a ‘0’ if the state is false.
Examples of Boolean variables or dimensions include “Turning the TV on”, “Going to work”, “Making lunch”, and so on.
We can display the possible range of values for these types of variables or dimensions as follows:
“Went to work” = W ∈ {0, 1}
“Made lunch” = M ∈ {0, 1}
“Turned the TV on” = T ∈ {0, 1}
Suppose you went to work, made lunch and turned the TV on.
Then W=1, M=1 and T=1.
You could model these outcomes on a three-dimensional axes.
Let d1 = W, d2 = M and d3 = T.
Then we have the following point on a graph:
We notice that there is a three dimensional coordinate (d1,d2,d3) = (W,M,T) = (1, 1, 1).
We shall call this the ‘coordinate of the outcome.’
Variables which take a set of string values can be represented numerically on axes too. Suppose the space “Breakfast” has three parameters “Bacon”, “Eggs” and “Sausages”, then the coordinates of the outcome will have ‘1’s or ‘0’s in the parameters of those which are true or false.
B³ = Breakfast = {“Bacon”, “Eggs”, “Sausages”}
Let M = My breakfast, where M is a point on B³
Let’s suppose the outcome of my breakfast includes bacon and sausages but no egg. Then the point M has the coordinates (1, 0, 1) on B³:
Anything and everything that can manifest in the real world can be modelled as a dimension or a space.
A very large number of variables that we know of that apply to the real world take values which are real, Boolean or strings.
Even a trivial variable such as as “Turned the TV on” can be modelled as a space with a large number of dimensions; it will have dimensions in it’s own variables such as pressure applied to remote or number of people in the room as the TV was turned on and any other variables we can think of.
Another way we can increase the number of parameters in a space is by considering variables in which the space does not have values; then we say that the space has zero-length dimensions in those parameters.
In the next article, we explore ‘moments’ of spaces; subset spaces of n-dimensional spaces which allow us to understand their parameters in greater detail.
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
