I recently made a purchase of a hand-blown Klein bottle. For those not familiar with the concept, a Klein bottle is an unorientable surface that was constructed by sewing two Möbius strips together. These surfaces are interesting in that we have a three-dimensional structure that appears to have two surfaces. However, closer inspection reveals that these two “sides” are both the same surface. This is thus a projection of a lower number of dimensions onto a higher order. If you are interested in these, I recommend a beautiful little short story by A.J. Deutsch, “A Subway Named Mobius.”

Another projection that may interest you is known as a hypercube, or tesseract. This is not the same tesseract from Madeleine L’Engel’s A Wrinkle in Time, but parallels could be drawn. A hypercube is a four-dimensional object projected onto three dimensions. Within the hypercube, one should see eight cubical cells. Look closely at the projection on the link above. There is a large cube, a small cube, and six distorted “cubes” connecting them. This distortion is a byproduct of the projection onto a lower number of dimensions. To better illustrate this distortion, consider a three-dimensional cube projected as a wireframe onto two dimensions. As opposed to searching for eight cubical cells, we can see six “square” cells. There are two squares, one in the front, and one in the back. These are then connected by four additional “squares.” This projection of a three-dimensional cube onto a two-dimensional surface follows the same concepts of the four-dimensional hypercube projected into three-dimensional space.

However, we cannot visualize four spatial dimensions. This makes the concepts of additional dimensions quite confusing. Should we believe that such dimensions exist? Another interesting story on this topic is that of a world known as Flatland. The story, written in the 19th century, describes a world where only two dimensions exist. Males are placed into social classes by the number of sides in their structure, where circles are the highest order of priests. Females are line segments and, as you can imagine, are quite dangerous if approached from the “front.” The novella delves into the natural laws of this world, the communities, the buildings, and the social norms of this world. The story then focused on a Square, who is visited by a Sphere in his dreams. The Sphere describes the third dimension to the Square (Spaceland), but he cannot understand it. Only by introducing the Square to Lineland and Pointland can he begin to believe in a place called Spaceland. It is a wonderfully-entertaining pamphlet, and I highly recommend reading it.

Let us assume, however, that in another iteration of Flatland, one that follows all the same natural laws of our three-dimensional Spaceland, the Square is not visited by the Sphere. For some reason, the Square is deluded into the heresy that another dimension exists. Without knowledge from some higher-order Sphere, how can he, the Square, demonstrate the existence of a third dimension? Is it even possible?

We need to make two assumptions. First, this version of Flatland follows all the rules of our world. Second, Flatland is a sheet within our world, meaning that there is space above and below Flatland, but the inhabitants of Flatland are unaware of “up” and “down.” Taking these into account, we can then answer this question quite simply. The Square can perform a fairly simple experiment. I must state, however, that this experiment will only provide evidence of a third dimension, and other models of the Flatland Universe could reach the same conclusion. That being said, bear with me.

In our world, at certain spatial dimensions (not very small, and not very large), forces exerted by two objects from the forces of gravity or electromagnetism propagate in three-dimensional space. This results in a reduction of the forces exerted by the objects upon one another as the radius between them increases.  The law they follow is an inverse-square law, where the force exerted is proportional to 1/R^2. However, when we are in a universe limited to only two dimensions, assuming isotropy, there would be no additional spreading in a third dimension, leading the force to follow a simple inverse law, where force is proportional to 1/R. If the Square took two magnets at a reasonable size and distance and measured the forces acting upon them as the radius was changed, he could make a plot of force versus radius. The relationship would presumably follow an inverse-square law, and the Square would have evidence that a third dimension exists! Again, this would be met with scrutiny from the Circles.

Though we cannot always visualize additional dimensions or scales, we can perform experiments to not only demonstrate their existence, but to observe phenomena at an otherwise unobservable scale. This is an aspect of experimentation that I find fascinating. I hope my introduction to dimensional projections, if nothing else, will bring a new perspective on observations around you.

1. - June 4, 2013

[…] foundations for graph theory and topology. For a related post in topology, I recommend my post, Diving through Dimensions. Some of you may be aware of this problem, and I hope I do it justice. Let us begin by traveling to […]