4DLab - Critical Thinking  

Can there be different fourth dimensions?

In general, the idea of a fourth dimension seems to precipitate
authors into orgies of occultist mystification,
rather than to lead to clear–sighted
mathematical inquiry.

Rudolf v. B. Rucker²
Geometry, Relativity and the Fourth Dimension

The concept of fourth dimension is an ambiguous one. Most of the time we are meaning some kind of hyperspace, a separate reality from the world we live in. In addition to that, to mathematicians and physicists the term fourth dimension conveys a different meaning from the esoteric, metaphysical, spiritual, and mystic. However, both schools of thought have a common denominator: the abstractness of the idea, and it seems that there is no possibility of simplifying it. In this essay we'll deal with both aspects of the perception of the fourth dimension.

A four–dimensioned word

When we speak of four dimensions is because we also have three previous ones. In that context, no dimension is more important  than the others. In the natural numbers sequence 1, 2, 3, 4, ...., the number four is not more important than the number one, and the same happens with the dimensions of the real world.

What do we understand when we speak of the world of the fourth dimension? Living in all four dimensions  simultaneously? We cannot grasp the meaning of that idea because we live in a world of three physical dimensions. However, our tridimensional world is a sub world, or sub universe, of that tetradimensional universe. The idea of sub worlds was graciously captured and elaborated by Edwin Abbot¹ (1838–1926) in his classic book Flatland. In this book the hyperspace is our three–dimensioned world. For the Flatlanders — the habitants of Flatland — their word was flat, that is, they lived in a plane, in a world of two dimensions. He went further and explored how it would be living in a world of one dimension: a straight line. For Abbot none of these worlds is socially limited, and even in the straight–line world, there was a kingdom. He adopted the classical idea where lines are one–dimensional, planes are two–dimensional and solids are three–dimensional worlds.

In that romance, a strange and transcendental event happened when Sphere —a solid body — visited Flatland. This out–of–their–world event sight caused that Square —the main character of this flat world— was accused of lying and perverting the youth. After many years in jail, without being able to communicate to the external world the truth of his sightings, Square begins to doubt if he really saw a habitant of the third dimension or everything was an illusion of his mind.

From the standpoint of modern physics where for some applications time is accounted as an extra dimension, the habitants of Flatland were really living in a three–dimensional space, and Sphere, the solid sphere that visited to Flatland was a 4–dimension body. This translated to our ordinary lives means that right now we live in a 4–dimensioned universe.

The people of Flatland were free to move, but only in two dimensions. They could move from North to South or from East to West or any combination of them. That was their world; none of them had ever heard of moving in the up direction, it was simply impossible. Thus, for them, the up direction was for them, what is for us the "fourth dimension" in the physical sense. For Flatlanders and for us, the fourth dimension has different meanings: our third physical dimension is their fourth, our fourth dimension is total madness for them. 

The contribution of Charles H. Hinton

Maybe comparing a two-dimension world with a three-dimensional one may sound stretching the comparison too much, and perhaps the original question is still floating in your mind. But the comparison is useful for us to keep in mind that for different worlds, the concept of space may also be different.

Mental images pursuing a unified view of a 4-dimensional world were explored with great success by Charles H. Hinton (1853–1907) in an a series of articles speculating about the fourth dimension. He devised a series a colored cubes to help the readers visualize his ideas and coined the term "tesseract". 

The fourth dimension for some artists

The notion of such a hyperworld and hypercube sparked creative ideas in many artist; among them Salvador Dalí (1904–1989), when he painted Corpus Hypercubus (A Body in a Hypercube). The basic idea behind this painting is that it shows a hypercube unfolded in three dimensions. For comparison, keep in mind that when an everyday 3D-cube is unfolded, it becomes a set of six adjacent 2D-squares. In the case of a 4D-cube is unfolded, it becomes a set of eight adjacent 3D-cubes. 

Another painter, at about the same time, also worked with the mental perception of the fourth dimension. Pablo Picasso (1881–1973) shaped his particular perception of the fourth dimension in the canvas Les Demoiselles d'Avignon(The Young Ladies of Avignon). In this composition he transforms a woman —a tridimensional element— into planar and asymmetric constructions. The painting is about figures seen and painted from different angles. The spectator does not see a classical portrait but a geometrical interpretation of the body. Picasso reduces his figures to bidimensional elements and it is the spectator who reconstruct the object and the scene. In the canvas there are five women each one representing distinct stages of the humanity. The woman that is seated shows the face and her back at the same time. 

In another painting of Picasso —Portrait of Marie-Thérèse— the viewer can see that Marie is shown with her profile as looking to the left while her eyes are looking toward the front. Supposedly, when a being from the fourth dimension looks a three–dimensional object it can see the body from all sides at the same time. With the nails of her left hand happens the same; note in the figure how Picasso deliberately inverted the observers' point of view of the fingers of the left hand, however our mind corrects the distortion, and for us nothing unusual is observed.

In our perception of the fourth dimension we do something similar; since we cannot directly visualize this extra dimension, what we do is to mentally join different scenes to convince ourselves that we have understood that strange and esoteric dimension.

The mathematical fourth dimension

From the mathematical point of view, the fourth dimension has no more significance than the other three, just like the example given above of the first four natural numbers. The 4-dimensional mathematical model offers several properties that cannot be fully understand in the mundane word that surrounds us everywhere.

Suppose we label the four dimensions with the numbers 1, 2, 3, and 4; then there are 4 tridimensional sub spaces of 3-dimensions; namely, the sub space of the labels 1, 2, and 3, the subspace contained in the labels 1, 2 and 4, the sub space contained within the labels 2, 3 and 4, and so forth. It's like asking how many 3-elements subsets we can make out of a 4-element set. Now substitute the labels by the well-known attributes of height, width, length, and time; then we have the same four sub spaces where each solid have dimensions of height, width, length (we see those everywhere all the time), height, width and time (can you think of an example of this type of body?),  width, length, and time (equally impossible to understand, although it can reduced to the preceding example), etc. 

Finally, the point is that if in a 4-dimension world we remove any dimension, then for the remaining three-dimensional world, the removed dimension is the "fourth dimension" not matter what dimension was removed. Therefore, for our physical world of four dimensions, any dimension can be the "fourth" and not necessarily the time dimension.

Unfolding the fourth dimension

Our understanding of the dimensionality of space is tightly related to our concepts of geometry, and geometry is, at the end of road, a planar activity. That is,

we do not do geometry in the space;
we do spatial geometry on a plane.

That was the main reason why the Flatlanders could not understand the "third dimension", because it was out of their geometry, and their geometry was deeply rooted in their minds. Therefore, any concept out of their preconceived "planar space" was merely insane.

Dimensions are supposed to be relatively perpendicular measures one to another. The second dimension is supposed to be perpendicular to the first dimension; its easy to understand that, the width of an object is supposed to be measured perpendicular to the length. That bidimensional world is easy to represent on the plane because both are two–dimensional. Now we add a third dimension, the "height dimension", but now, the height should be perpendicular to the other two. Still easy to understand; all of our surrounding is tridimensional. But the the problem arises when we try to add another physical dimension. That next dimension —whatever it is— should be perpendicular to the other three. If we take time as the fourth dimension, we'll have no problem understanding it, because time surrounds everything like if it were "perpendicular" to all spatial dimensions. However, if we exclude time as a dimension, and think of the fourth dimension as another purely spatial measure, then we'll have have to make strong mental efforts to grasp an idea of it.

One thing that is not questioned is that the mathematical interpretation is the reasoning supposed to be "right" approach.  The mathematical reasoning about the fourth dimension is purely inductive. However, we can bring here an analogy of false inductive reasoning using the sequence of the primes numbers. Leave aside the numbers 1 (for it is not considered prime) and the number 2 (for being the only even prime). The reasoning is as follows: 3 is a prime number, 5 is also a prime , 7 is a prime, so 9 is ... don't say it is prime! The lesson here is that three true "events" does not imply that the next fourth is also true.

This lesson, applied to a possible real fourth dimension, is that there is no guarantee that it will behave as the mathematical fourth dimension implies. We must keep in mind that applied mathematics works with models of the reality. It is not the opposite: reality is not an application of mathematics. So the purely mathematics fourth dimension should always be seen as that: a mathematical object that may have or not a reality equivalent, and in the case it exists, it may not be as expected (is 9 a prime number?).

One approach to understand the mathematical 4-dimensional space is to break it in subspaces of three dimensions each, as done above,  because they are more intuitive for us. When we do this, we then have four subspaces of three dimensions, and each one is completely different to the other four.

Now, our mental task is simply put all four models back together and make a single mental image of this spatial collage like the exercise proposed by Picasso in his painting Les Demoiselles d'Avignon.

So, in the final act, are there as many "fourth dimensions" as people thinking about it?

I call our world Flatland, not because we call it so, but to make 
its nature clearer to you, my happy readers, 
who are privileged to live in Space. 

Edwin A. Abbot
Flatland: A Romance of Many Dimensions.

E. Pérez
jul-10

Flatland is a 2-dimemnsional world. Spaceland ─our world─ is a 3-dimensional world, but there are other coordinate systems for the fourth dimension. To learn more about them see:  Quaternions and transcomplex numbers, and Can there be different fourth dimensions?

Related articles

▀ What is the shape of a wheel in the fourth dimension?

▀ About how A. T. Schofield found the fourth dimension in the Bible

▀ The fourth dimension as seen from Flatland

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