Introducing Spatial Dimensions Beyond Three: The Theory of Infinite Dimensions

For centuries we have become accustomed to what I refer to as The 3D Generalisation — the idea that everything we see or everything around us is limited to three dimensions, and that anything beyond three is beyond us, which evidently below, is not the case.

A point is said to have zero dimensions and a line is said to have one dimension, as the line connects two zero-dimensional points. A square is said to have two dimensions, as its second dimension (height) connects two one-dimensional lines (width). A cube is said to have three dimensions, as its third dimension (depth) connects two two-dimensional squares.

It should logically follow therefore, that an object or space which connects two three-dimensional objects or spaces by a length , is a four-dimensional object or space. Or more generally, a space is said to have n dimensions if it connects two n − 1 dimensional spaces. As shown above, the distance between the two ‘3D’ drink tables is not inside the third dimension, but is a fourth spatial dimension, which connects two three-dimensional tables to form one four-dimensional space. As one will discover later on, even ‘3D’ spaces like the drink tables individually below are not merely ‘3D.’

The 4D shape on the top right, which consists of the two drink tables and the space between them, can be displayed on a 4D set of axes, where width, height, depth and the space between the tables are represented by the variables x, y, z and w respectively.

In both the photos above, the lines of the same colour are all parallel to each other in reality, but because of lens distortion, they appear as if they do not look parallel.

If one were there in the room for their self, the drink tables would both appear to be parallel to each other in the fourth dimension, w, and the ‘3’ dimensions (x, y and z) of one drink table would be parallel to the ‘3’ dimensions (x, y and z) of the other drink table. The same applies for any scenario involving two ‘3D’ objects being placed on a relatively flat surface.

The Theory of Infinite Dimensions is in simple terms the concept that the third spatial dimension is a generalisation for all spatial dimensions between 2 and infinity. Reality is modelled as three-dimensional when it is in fact infinitely-dimensional. The Theory of Infinite Dimensions is the first instalment of The Theory of Space-Matter , a collection of three theories which can be found here.

Visualising a fifth dimension is the same process as visualising a fourth dimension — the space between two 4D objects is not inside the fourth dimension but is a fifth spatial dimension. The picture on the following page shows five spatial dimensions in the real world. All four objects exist within their own individual ‘3D’ spaces.

In the image below, the pepper, the cork and the space between them make a 4D space. The salt and the pen lid also make a 4D space together. Both of these 4D spaces are separated by a fifth spatial dimension, denoted v.

Equally, salt and pepper both make a 4D space in the v-axis, as does the pen lid and the cork. From that perspective, the fifth spatial dimension would be in the w-axis and the fourth dimension would be in the v-axis.

If we observe the origin point, O, as if it were a much larger point containing all ‘’three’ spatial dimensions of pepper— that is to say that if the respective x,y,z axes encapsulating pepper were inside the origin point itself, then cork is a distance w away from pepper in the w-axis, whereas pepper is at zero in both the w-axis and v-axis because pepper’s entire dimensions are at the origin of those axes. At the same time, salt is a distance v away from pepper in the v-axis.

Here is an example of dimensions beyond three which may be much easier for most to conceptualise. The image below illustrates six spatial dimensions in the real world.

Contained within the origin point are the ‘three’ dimensions of a box — that is, the width, height and depth of the box. Part of this spherical origin point would obviously be underground and the part of the sphere encapsulating the box would be above ground. Alternatively, we can ignore the sphere altogether and simply treat the white box as the origin point. However, visualising the origin point as a sphere which contains the box is a more accurate approach for further on in this article.

One will notice that outside of this origin point there are another ‘three’ dimensions, x, y and z.

In this case, the x and z axes are on the ground, whereas y is above and below the ground.

If the box were to expand outside of the origin point which contains all of its dimensions, it would gain dimensions in the x, y and z axes. It would then not only have the ‘three’ dimensions of the origin point or any point of the same size, but an additional three dimensions in x, y and z.

If it were not expanded, but simply moved away from the origin point in the x-direction, it would still have its ‘three’ dimensions, but it would have been moved into a fourth spatial dimension, x. The same would occur if it were only moved in either the y or z directions.

If it were moved in both x and y, it would have been moved inside a fourth and fifth dimension. If it were moved in the x, y and z directions, then it would be moved inside a fourth, fifth and sixth dimension.

Another way of looking at the image above, is that the box has relatively no dimensions at all to the dimensions outside of it, as it has no lengths in x, y or z. Nor has the box moved in any of these axes. Relative to the box, however, it already has its three dimensions and will gain dimensions in x, y or z if a person were to approach the box and move it away from the origin.

Now model all six dimensions as the new origin point and draw another set of x, y, z axes outside of that origin point. We would then have nine spatial dimensions as shown below.

In the images below, we have the dimensions of the box and the dimensions outside of the box contained within a larger origin point. This larger origin point is denoted O2. By basing the ‘3’ dimensions of the box and the ‘3’ dimensions outside of the box as contained within the new origin point, O2 , we can then conceptualise another ‘3’ set of axes branching out of O2 , which are denoted x2, y2 and z2.

If the box is moved outside of the new origin point, it will start to move in the dimensions x2, y2 and z2. The box can only fill an O-sized point. Therefore, at any fixed speed, it will take longer for the box to travel an equivalent distance of, say 2 metres, in x2, y2 or z2 than the equivalent distance of 2 metres in x, y or z. This is because the dimensions of

x2, y2 and z2 are larger than x, y and z. Unless the box is moved outside of the new origin point, O2, then it will not

move in x2, y2 and z2 at all. If the box grew outside of the new origin point, it would not only have the six dimensions illustrated in the previous example, but dimensions in x2, y2 and z2.

If the box fit the size of the new origin point, O2 , it would be equivalent to a point in the x2, y2 and z2 axes. An equivalent distance of 2 metres travelled in x2, y2 or z2 would translate to more metres travelled in x, y or z. This is analogous to the idea that if you measure any distance in metres, you would measure more metres than kilometres — what may be a large number in metres is a small number in kilometres.

The same process can be repeated to show 12 spatial dimensions. By modelling the 9 dimensions as the new origin point and drawing another set of x, y, z axes outside of that, denoted x3, y3 and z3, we increase the number of spatial dimensions to 12.

Every time the white box is to move in x3, y3 or z3 it has to travel another point the size of the O3.

The same process used to show 12 spatial dimensions, can be used to show 15 spatial dimensions, and so on. So far I have shown instances of increasing dimensions with a new set of x, y, z axes for each new origin point, but the increase in dimensions with each larger space is not limited to just increments of 3. Let us suppose that every possible line extending outside of an origin point is a dimension or axis.

If we had infinite lines extending outside of a point or sphere, what shape would we get? A sphere. A ‘3D’ sphere in the real world has infinite dimensions.

As shown in the bottom left image, one can conceptualise this by imagining a sphere encapsulating the white box or any other object and denoting that sphere as the ‘origin point’, O. One then imagines that sphere within another sphere, denoted ‘origin point 2’ or ‘O2’. In this case, O is the centre of the O2 sphere. Now imagine every possible line joining O to O2. There are infinite possible lines joining O to O2 .And each and every one of them is a new axis or dimension. Every radius in the O2 sphere is an axis or dimension. To begin with, let us imagine the x, y, z axes all acting as radiuses of the sphere, O2 as shown in the bottom left image. Part of the O2 sphere like O would obviously be underground. Now imagine, the other two axes we explored on page 12, w and v, also acting as radiuses of the sphere, O2 as shown in the bottom right image.

Every axis extending from O to O2 is a line connecting O and the surface of O2 together. All of the axes here behave as radiuses within the sphere.

Now imagine all of the other possible lines, or radiuses connecting the centre of O2 to O2 itself. Eventually, we will fill the inside of the O2 sphere up with axes.

All of the space between O and O2 is filled by infinite red lines where each and every red line is a dimension, and so one can see that O2 has infinite dimensions.

The sphere of space formed by the infinite axes is what I refer to as the ‘dimensional density’ or ‘dimensity.’

In the image above, the space between O2 and its centre, O is occupied by infinite red lines or radiuses. Therefore, there are infinite dimensions in the space between O and O2, and therefore there is infinite dimensity in that space.

If a point or sphere is the centre of another sphere, then the dimensity will be infinite, even if the larger sphere is only marginally larger than the sphere it encapsulates.

As shown below, if O2 were only marginally larger than O, there would still be infinite dimensity, but the axes or radiuses connecting O to O2 would be smaller, as we saw earlier.

As shown in the right image above, if we make the sphere O2 about half as big, the original axes which were drawn start to get crammed together. And if one makes O2 even smaller as shown in the left image below, the axes will become so crammed that they will start to merge together.

It does not matter how thin we draw the red axis lines. As O_2 continues to shrink smaller and smaller until it is almost equal to O, the axes will always eventually be too big and start to merge together.

At the same time, thinner or smaller red axis lines can be drawn to get past this and then the dimensity will appear just as infinite.

So what is occurring? The dimensity is always infinite unless O2 is exactly equal to O, but the number of dimensions still falls, which leads one to the conclusion that the infinite number of dimensions in smaller spheres are smaller than the infinite dimensions of larger spheres.

In Mathematics, there are already instances where infinite quantities are larger than other infinite quantities.

For instance, the set of positive rational numbers is larger than the set of positive integers. Both are infinite, yet the set of positive rational numbers includes the set of positive integers, as all integers are divisible by 1. Therefore, the infinite quantity of rational numbers in the set of positive rational numbers is larger than the infinite quantity of integers in the set of positive integers. In a similar fashion, the infinite quantity of dimensions displacing O and O2 varies based on how large O2 is relative to O.

If O exists within a significantly larger O2, then O shall be said to exist inside of a high dimensity relative to O2.

If, on the other hand, O2, is only marginally larger than O then O shall be said to exist inside of a low dimensity relative to O2.

A car, for instance, is a low dimensity for a human, but a high dimensity for an ant.

Earth exists in a high dimensity relative to our Solar System as it is small relative to the rest of the Solar System. The Sun, however, accounts for a very large proportion of the Solar System. Therefore, the Sun exists in a low dimensity relative to the Solar System. The Milky Way, however, is huge relative to our Sun, so the Sun exists in a huge dimensity relative to the Milky Way.

High dimensities provide small spaces or objects more dimensions to move in.

If a space or object is only marginally smaller than the space or object it occupies (i.e a low dimensity) then there will be little space for that object or space or object to move in.

To a human, a telephone box would be a low dimensity, providing very little room for the human to move in or explore.

To a small spider, however, that same telephone box would be a massive place to move and explore, because it provides more dimensions relative to the spider than to the human.

Let us suppose that both a small spider and a human are standing inside of a telephone box, and the human is tall enough that their head touches the roof of the telephone box. We suppose that the spider is to crawl from the floor of the telephone box to the roof of the telephone box. No matter how fast the spider moves, the human is already there at both the starting line and the finishing line. So in a race from the bottom of the telephone box to the top of the telephone box, the human in this example will always win. The human, unlike the spider, does not need to increase or decrease their speed to win the race.

Larger objects or spaces have what I refer to as an engulfing effect on smaller objects or spaces.

The engulfing effect is such that the dimensions of a large object’s trajectory engulf the dimensions of smaller objects.

Imagine a game of chess where there is only a white king and a black king. And each column, row and diagonal of the chessboard relative to each piece is a different dimension. In this case, the number of possible dimensions each king can move in is limited to eight — a king can move one space in four diagonal directions, two vertical directions and two horizontal directions.

In the left image below we have all of the possible trajectories of the black king piece and all of the possible trajectories of the white king piece. Each possible trajectory per piece is in a different dimension. Each piece therefore can move in eight different dimensions. The outcome of this game cannot be decided in a single move as both pieces are of equal size, both are displaced by two squares, and a king can only move one square at a time.

However, if the black king were 3×3 squares large as shown in the right image above, then the equivalent of one move to the black king now takes up 9 squares. Therefore, the black king will take the white king in a single turn irrespective of whose turn it is.

If it is the black’s turn, then the white king will be taken in a single move. If it is the white’s turn, then the white king piece can only be moved one space relative to its own size and will be taken by the black king in the following turn.

Because the black king piece is now larger, its trajectory now encapsulates all of the possible trajectories of the white king.

A single dimension relative to the black king swallows up the dimensions of the white king, as well as the eight possible dimensions for the white king to move in. This is an example of the engulfing effect — a single dimension outside of the black king is equivalent to all eight dimensions outside of the white king and the dimensions of the white king itself.

To not be taken by the black king, the white king in this instance would now have to travel more than one square in a single turn. This idea is explored in greater in the Theory of Relative Displacement .

The engulfing effect also applies to the O2 sphere from earlier. O2 encapsulates all of the possible trajectories of O. O2 therefore has an engulfing effect on O as shown below.

As we saw previously, if an object or space exists within a much larger object or space, then the smaller object or space is said to exist inside of a high dimensity.

If, on the other hand, the larger object or space is only marginally larger than the object or space contained within it, then the smaller object or space shall be said to exist inside of a low dimensity.

So far we have observed dimensions in terms of what is outside an object or space.

But what about the object or space itself? As I mentioned earlier, even ‘3D’ objects are not merely 3D. This is because any ‘3D’ object such as the white box from earlier can fit inside of a sphere proportional to its size. We established earlier that spheres have infinite dimensions. It must therefore follow that any ‘3D’ object which can fit inside a sphere must also have infinite dimensions.

Imagine a ‘3D’ box inside a sphere such that the centre of the sphere is inside the box. The centre of the sphere is the origin point which all axes extend out of.

As we saw earlier, there are infinite radiuses in a sphere where each and every radius is a dimension from the centre of the sphere to the boundary of that sphere. A sphere must have infinite radiuses as it has infinite points on its surface. Therefore, there must be infinite lines or radiuses connecting the centre of the sphere to each point on the sphere’s surface.

As all of the radiuses (dimensions) from the origin point at the centre of the sphere must pass through the box to get to the surface of the sphere, it must follow that the box also has infinite dimensions.

As all physical objects or spaces can fit inside a sphere proportional to their own size, they must therefore all have infinite dimensions.

The Theory of Infinite Dimensions is the first instalment of The Theory of Space-Matter, a collection of three theories which can be found here.