Centre of 4th dimension
‘Picasso was particularly struck by
Poincaré’s advice on how to view
the fourth dimension, which artists considered another spatial
dimension. If you could transport yourself into it, you would see every
perspective of a scene at once, but how to project these perspectives
on to canvas?’
Picasso’s dilemma led, of course, to
Cubism. Those with a yen for
early 20th century scientific theory might want to take a look at award-winning Picasso titles, offering a two-dimensional take on the fourth dimension.
-Maurice Golubov on the “fourth dimension”
The term “fourth dimension”, although technically defined using mathematical computation, generally refers to the infinite relationship between time and space, a primary artistic concern during the early 20th century. Rejecting the traditions of onepoint perspective, artists began incorporating elements of multiple perspectives in their work, essentially departing from their own three-dimensional reality and creating a world of endless possibilities. Cubism, for example, shattered the picture plane,
simultaneously portraying all sides of perspective anchored on a singular plane, “painting not as they saw it, but as they thought it”.
Many artists of the early 20th century began using the fourth dimension as the foundation with which to explore the metaphysical content of geometric symbolism. They sought to create works which allowed the viewer to simultaneously exist in both the third and fourth dimension, without the use of traditional perspective or the familiarity of subjective iconography. While the fourth dimension in all spectrums of visual art stands as an extension of the infinite, there are many different means of representation, such as time, space, or spirituality.
Represent the ideologies of the fourth dimension;
namely Nassos Daphnis,
De Hirsch Margules,
Irene Rice Pereira,
and Charmion von Wiegand.
By the arts and science writer,
Arthur I Miller. It
was published to coincide with the centenary of the death of Henri
Poincaré: French physicist, mathematician, philosopher, and,
according to Miller, the missing link between Picasso and
Miller writes that Poincaré 1902 book, Science
Hypothesis, pushed both the Spanish artist and the father of relativity
towards great breakthroughs. Einstein came across Science and
Hypothesis while working as a patent clerk in Bern, and was struck by
the way Poincaré moved from the precepts of scientific theory towards
a more general understanding of knowledge.
Miller quotes Einstein:
“Poincaré realised the truth [of the relation of everyday experience to
scientific concepts] in his book.” Yet, he was also felt restricted by
Poincaré reliance on lab data, pushed Poincaré’s ideas further, and hit
upon the theory of relativity. Meanwhile, a friend of Picasso was also
taken with the book, and took the trouble to explain some of the ideas
to the artist.
The fourth dimension is generally understood to refer to a hypothetical fourth spatial dimension, added on to the standard three dimensions. It should not be confused with the view of space-time, which adds a fourth dimension of time to the universe. The space in which this dimension exists is referred to as 4-dimensional Euclidean space.
Beginning in the early part of the 19th century, people began to consider the possibilities of a fourth dimension of space. Mobius, for example, understood that, in this dimension, a three dimensional object could be taken and rotated on to its mirror image. The most common form of this, the four dimensional cube or tesseract, is generally used as a visual representation of it. Later in the century, Riemann set out the foundations for true four-dimensional geometry, which later mathematicians would build on.
In the three dimensional world, people can look at all space as existing on three planes. All things can move along three different axes: altitude, latitude, and longitude. Altitude would cover the up and down movements, latitude the north and south or forward and backward movements, and longitude the east and west or left and right movements. Each pair of directions is at a right angle to the others, and therefore is referred to as mutually orthogonal.
In the fourth dimension, these same three axes continue to exist. Added to them, however, is another axis entirely. While the three common axes are generally referred to as the x, y, and z axes, the fourth falls on the w axis. The directions that objects move along in that dimension are generally called ana and kata. These terms were coined by Charles Hinton, a British mathematician and sci-fi author, who was particularly interested in the idea. He also coined the term “tesseract” to describe the four dimensional cube.
Understanding the fourth dimension in practical terms can be rather difficult. After all, if someone is told to move five steps forward, six steps to the left, and two steps up, she would know how to move, and where she would end up in relation to where she began. If, on the other hand, a person was told to also move nine steps ana, or five steps kata, she would have no concrete way to understand that, or to visualize where it would place her.
There is a good tool to understand how to visualize this dimension, however, and that is by first looking at how the third dimension is drawn. After all, a piece of paper is a two-dimension object, roughly, and so cannot truly convey a three dimensional object, like a cube. Nonetheless, drawing a cube, and representing three-dimensional space in two dimensions, turns out to be surprisingly easy. What one does is to simply draw two sets of two-dimensional cubes, or squares, and then connect them with diagonal lines that link the vertices. To draw a tesseract, or hypercube, one can follow a similar procedure, drawing multiple cubes and connecting their vertices as well
more dimensions means more freedom of movement. One of the more mundane effects of that is that in 4 dimensional space there’s an extra direction you can move and/or fall over in. So if you want to build a working bar stool you’d need at least 4 legs instead of just 3. In fact, in D-dimensional space bar stools need at least D legs, or they’ll fall over. Just one of the subtle economic effects of higher dimensional living.
You’d also find that in 4 or more dimensions, you’d be able to do a lot of tricks impossible in 3 dimensions, like creating Klein bottles or (equivalently) taping the edges of two Möbius strips together. Sailing knots could take on stunning complexities. In fact, they’d need too! All of the knots that work in 3 dimensions fall apart immediately in 4.
Most physical laws are already written in a dimension-free form. For example, in Newton’s second law, , and are both vectors, but they can be vectors in any number of dimensions. So you can use for objects on a line (1-D), on a table-top (2-D), in space (3-D), or whatever (whatever-D).
There are some laws that are usually written in a 3-D form, but that’s generally a matter of convenience more than necessity. For example, we talk about the “angular momentum vector”, which is defined to be perpendicular to the plane of rotation. It’s convenient because in three dimensions there’s always exactly one perpendicular direction to a plane, whereas in 4 dimensions there are 2.
This is pretty easy to fix and generalize, it just becomes a little more difficult to work with. All that said, while our physical laws can be generalized to any number of dimensions, the manifestation of those laws are wildly different. So, living in higher dimensions would be pretty alien.
Based on our understanding of gravity (gained from studying this podunk universe), gravitational force should drop by , where D is the dimension and R is the distance between the objects in question. It so happens that because of the nature of orbits, a stable orbit can only exist in 2 or 3 dimensions.
In 4 or more dimensions orbits are always unstable, and in 1 dimension the idea of an orbit doesn’t even make sense.
Most physicists consider light to be native to only 3 dimensions, because light is an EM wave and it’s direction of propagation is perpendicular to both its Electric and Magnetic fields. (Fun fact: the direction that light points is called the “Poynting vector“, after John Henry Poynting. Life’s funny.) In 4 or more dimensions this direction isn’t unique, and in two dimensions there’s no direction at all. However, you can express EM waves just in terms of “E” in any dimension without problem.
Assuming light can exist in higher dimensions, it would behave very strangely. Sound waves too. In odd dimensions other than 1 (3, 5, 7, …) waves behave the way we normally see and hear things: a wave is formed, it moves out, and it keeps going. However, in even dimensions, and 1 as well, (1, 2, 4, 6, …) waves “double back” on themselves. You can see this in ripples on the surface of water (2-D waves). Ripples are more complex than just a ring; the entire circle within the ripples is disturbed.
If you set off a firecracker in 3, 5, 7, etc. dimensions, then you’ll see and hear the explosion for a moment, and that’s it. If you set of a firecracker in 4, 6, 8, etc. dimensions, then you’ll see and hear the explosion intensely for a moment, but will continue to see and hear it for a while. For light the effect would be fairly subtle, except for extremely long-distance effects, like somebody reflecting a bright light off of the moon. You probably wouldn’t notice the effect day-to-day. However, it would ruin the experience of sound. In 4 dimensional space the firecracker, even in open air, would sound like thunder; loud at first, and leading into a drawn out boom. It may not even be possible to understand people when they speak.
All the fundamental particles should still exist, but how they interact would be pretty different. Which elements are stable, and the nature of chemical bonds between them, would be completely rearranged. Some things would stay the same, like electrons would still have two spins (up or down). But atomic orbitals, which are determined by spherical harmonics (which in turn are more complicated in higher dimensions), would generally be able to hold more electrons. As just one example (for our chemistry-nerd readers), you’ll always have 1 S orbital in every energy level, but in 4 dimensions you’ll have 4 P orbitals in each energy level, instead of the paltry 3 that we’re used to. This messes up a lot of things. For example, in 4 dimensions Magnesium would be a noble gas instead of a metal. Every element after helium would adopt weird new properties, and the periodic table would be longer left-right and shorter up-down.
So, while the laws of physics are actually the same, if you lived on a four-dimensional Earth in a four-dimensional universe you’d find that (among other things): your bar stool may need an extra leg, Earth wouldn’t be able to orbit anything, you’d never be able to hear anything crisply, and the periodic table of the elements would be seriously rearranged.