MOVEMENT AND DIMENSIONALITY
Dimensionality is not always a clean cut. It is true that a coordinate system is either two dimensional (or whatever dimension for that matter) or it is not; there's no sliding scale in this case. However, when one thinks about dimensional beings living in dimensional worlds, focusing especially on the dimensionality of that beings existence, it becomes less obvious whether something is "this" or "that" dimensional.
Let's begin with us. For arguments sake assume that our construction is limited to only three dimensions; in that respect we are three dimensional beings. What then is the dimensionality of our movement? We can surely jump up into the air, fall down a flight of stairs, move our index finger right-left, forward-back, and up-down; we can fly to Mexico (or to Mars for that matter), spelunk 1000 feet into a cavern . . . (as an exercise, think of some more examples on your own!). Many times our third dimension of movement is related to being on a particular floor of an office building in a city. The middle-earth correlation would be a multi-layered maze found in a complex system of mines where staircases and ramps and descending slopes abound. But aren't these three dimensional movements unique in a way? Where is the third dimension coming into play?
How about animals that can fly? Their degree of freedom is obviously augmented compared to ours (they don't need stairs), but how much difference really exists? Sea dwelling creatures can move very similar to birds, but some move quite differently. Where's the difference come from? What do land animals, birds, and sea dwelling creatures all have in common that would have great significance on their respective degrees of freedom (or perception of movement)? We will return shortly to this part of the discussion.
In Flatland Abbott creates a two dimensional world in which, unlike our world, every creature has a full two degrees of freedom. Unlike humans, who are constrained by a strong gravitational pull to the two dimensional surface we live on, the gravitational force in Flatland does not have so great an effect on its inhabitants. Thus Abbott's creatures move freely about their entire two dimensional evironment with little regard to orientation. Knowing this we realize that without the slight effectiveness of their gravity and the subsequent fall of rain from the north, any set orientation in Flatland would be a convention and have nothing to do with their physics or biology.
Abbot's world in many ways was created by projecting aspects of life on earth, Victorian England especially, onto a two dimensional plane. A different projection of our world into the plane is outlined in "The Planiverse". Here the dimensionality of movement is maintained from our world to theirs so that the living creatures in this two dimensional world live constrained (by gravity) onto a single dimension (a line) yet still posses and utilize the ability to move into their other dimension.
What is the difference between these two dimensional worlds? In one, gravity's force is much less constraining / effective than it is in the other. Thus we are able to think of Flatlanders as living in a realm analogous to our outer space where gravity does not constrain / effect us as much, permitting more freedom of movement. This idea promotes the image of Flatlanders, and everything else in Flatland, simply floating about however it pleases. Another way of looking at Flatland's movement is instead of decreasing the strength of the gravitational force, increase the strength of an opposing force that could counter-balance the force of gravity and provide more freedom of movement. In fact, this is very similar, on earth, to the case present under water where the force opposing gravity is called buoyancy (which is directly proportional to volume, while gravitational force is directly proportional to mass). Here sea dwelling creatures like the porpoise and sea turtle possess a higher degree of freedom than the rest of earth's animals. In this scenario the random drift of objects that was present in the zero-gravity scenario is accounted for by the viscosity of the fluid they live in. In the same manner, water is not the only medium they could live in: the same effect would be possible in a gel like matter, or in a dense gaseous atmosphere (if Flatlanders were paper thin). Each scenario is very interesting and necessitates much more fine detail analysis that won't be done here.
So now we can discuss our own dimensionality of movement with respect to Flatland's (going from 2D to 3D rather than the other way around). Unlike Flatlanders, we can not flip ourselves upside down and then move into the air and lower ourselves head first through our friend's sun-roof; our movement is two dimensional. Even when we do move into the third dimension we still perceive it in terms of two dimensions ("we'll take the elevator to the third floor"; "go up the steps to the plane that is eight feet above the one you are on right now"). Most likely only those in the Air Force or NASA ever experience anything close to true three degrees of freedom. But what if we did have three degrees of freedom? The possibilities are very interesting, all of which are loaded with assumptions: volumetric cities (radius of New York cubed); three dimensional network of highways (and wrecks); etc.
What parts of us would be wrong or inadequate (on the macroscopic level) living with three degrees of freedom? Our eyes for instance? As it is our eyes, designed for two dimensions, cover probably 35-40 percent of the horizontal plane, 15-20 percent of the vertical plane. We don't worry about the vertical plane as much as the horizontal, so we are pretty much suited to our environment (unless someone sneaks up from behind you, or we're casually cruising in the jungle, or we're diving in the ocean). Having this type of eyesight is the common bond between land, air, and sea dwelling creatures in terms of their dimensionality perspective. But in three dimensions our visual spectrum would be quite inadequate, "tunneled" in fact, covering (I believe) less than 10 percent of the new three dimensional spectrum. To function reasonably well in three dimensions we would most likely need another eye (and a rearranged brain), just like by moving down one dimension we could lose an eye (A Square has only one eye and can still see probably 40 percent of his visual world at any given time, a sufficient amount).
We could go on for a long time like this because I've come to realize that when you move something to other (higher especially) dimensions, the amount of stuff that changes is quite drastic (think about the amount of different shapes between 1D -> 2D -> 3D -> 4D). To reiterate my main points (the ones that sparked the whole idea in my head): We think in a restricted two dimensional way. Birds less so, but their world is still a stack of two dimensional planes because they have to orient their wings so they are counteracting the force of gravity. Sea animals are the "least two dimensional" beings (besides Jeremy Kahn's bacteria in a drop of water scenario), using their buoyancy to balance gravity's strength, providing their movement with a higher degree of freedom.