"Gliding" the Cube

What, one may ask, does it mean to "glide" a cube? What is meant by the term "glide"? "Gliding", if you don't already know, is the most-used, perhaps most effective means of introducing the concept of spatial dimensions, and is a procedure you are probably already familiar with. To "glide" a structure means to move a structure at a right angle to itself, and during that transition the structure is assumed to leave behind a "trail". Once the "trail" is complete, it will have formed a structure of the next-higher dimension. We can go even further and attempt to glide that newly-formed structure in the same manner in which the original structure was glided, to form a structure of an even higher dimension, and so on.

Let us advance through the "gliding" process step by step: picture a point - a zero-dimensional structure. Next, imagine moving the point to one side, and as it moves, imagine it to leave behind a "trail" as described above. Once the point has covered a considerable distance, bring it to a halt. Before us lies a segment - a 1-dimensional structure. Next, imagine moving the segment in a direction at a right angle to itself. As it moves, it leaves a "trail" behind as the point did. Once the segment has travelled its designated distance, before us would lie a square - a 2-dimensional structure. We can add further meaning to the concept of "gliding" by moving the square in a direction at a right angle to itself. As before, a "trail" is left behind as the square moves forth. Once the square has covered the desired distance, before us would lie a cube - a 3-dimensional structure. What comes next? The obvious answer as to what comes next is that we glide the cube in a direction at a right angle to itself, to form a 4-dimensional structure. This may seem straightforward at first, but there is a problem that we easily become aware of: HOW DOES ONE GLIDE A CUBE IN A DIRECTION LYING AT RIGHT ANGLE TO ITSELF? This is a question that must be addressed directly, studied, and properly understood, so that we can stretch our visualization capabilities to the very limit, and GLIDE THE CUBE.

Into what directions can a cube glide? Clearly, no 3-dimensional directions remain that allow this to happen. How does one envision a fourth spatial axis into which a cube can glide? This feat has been labeled as non-mathematically impossible - impossible in the sense that it is not spatially visualizable. My own personal belief, however, is that the gliding of the cube can be spatially visualized. What makes the task difficult is that it must be approached in an indirect way. In doing so, one will relate to the third dimension in ways that he never has before. How is this done? The answer is simpler than you might think: one relies upon information obtained from lower dimensions. By examining situations in lower dimensions equivalent in meaning to the corresponding 3-dimensional arrangement, one can, using this knowledge, construct from the bottom up a 3-dimensional model that one would otherwise have no means of bringing forth. Let us begin this process.

A "glided" segment forms a square. A "glided" square, similarly, forms a cube. In this sense we can say that a square is in essence an array of segments, and a cube, an array of squares. Assume that we, along these lines, divide a square into an array of segments, and a cube, into an array of squares. Next, we are to put lower-dimensional creatures on the bottom slice of each structure - the slice where the gliding began. For the sake of the experiment to come, we remove all of the slices above the bottom slice. How would we explain to each creature how his slice "glides" into the next dimension? To perform this feat we must familiarize ourselves with a new concept: the distance mark. As the first step in doing so it is to be emphasized that both creatures are fully unaware of the space above the bottom slice he is on, because his experience of his surroundings is limited totally to that slice. We will use the 'distance mark' here to make these lower-dimensional creatures aware of the space above them - the space into which gliding occurs.

The main reasoning behind the 'distance mark' is based upon what we call perspective: objects that are farther away appear smaller. This could be said to be a property that would apply to a space of any dimension. The farther away an object is, the more 'diminished' it appears. This can be a powerful tool in envisioning the "gliding" of a structure into the next-higher dimension. One need not necessarily be able to comprehend the directions into which the gliding occurs, you see, to understand the nature of the gliding taking place. The lower-dimensional creatures on their 'bottom slices', as you may recall, are aware only of that slice and cannot comprehend the directions into which gliding of their slice would occur. Yet they can comprehend the gliding of their slice as it exists in distance mark form. How would this work? Each creature would first imagine a highlighted version of their slice, equal in size to the slice - a "distance mark" - to occupy the space they perceive their slice to take up. When gliding occurs upward from their slice to form a slice above their slice, each creature imagines the "distance mark" on their slice to become smaller, to a degree equal to the distance the new slice is from that creature's slice. As the gliding extends farther and farther from the creature's slice, the "distance mark" will become smaller and smaller. The logical conception of a square that the creature on the bottom slice of a square being glided, therefore, would be of segments that accumulate on his slice, gradually becoming smaller as the gliding process extends farther away from his slice. To the creature on the bottom slice of a cube being glided, a cube would be, in a way very similar to the previous creature's experience, an outer "distance mark", square in shape, which gets increasingly smaller as the gliding process reaches increasing distances from the creature's slice. How, then, would one apply this to the third dimension?

We can now claim to possess the knowledge to go all the way - to glide the cube. To do so, we simply relate the gliding experiments we've performed to our dimension in the same way that the gliding experiments related to their own dimension. The first, and simplest, conclusion we can come to is that if a square - a 2-dimensional structure - is an array of segments, and a cube - a 3-dimensional structure - is an array of squares, the 4-dimensional structure that is a result of the gliding process must therefore logically be an array of cubes. Could an assumption be simpler? Is there a name, then, for such a structure? Yes - the hypercube. Let us glide the hypercube step by step. We begin on the 'bottom slice', so to speak, of the hypercube to be glided. Think of this 'bottom slice' as a cube located in the spatial center of your room. We are to envision, next, a "distance mark" equal in size to the actual physical size of the cube. To imagine "gliding" this cube along a four-dimensional axis (at a right angle to your room which we are assuming to hold the bottom slice of the hypercube), we conceive of the "distance mark" shrinking inward from its initial 'bottom slice' size. We then mentally visualize the "distance mark" to produce cubical imprintings of itself - 'distance marks' - that gradually get smaller and smaller over time, until the gliding of the hypercube comes to a halt upon its completion. Take note, if you will, that the cube is not actually getting physically smaller - it is, rather, undergoing, so to speak, 4-dimensional perspective. In conclusion, then, our best conception of a "gliding" cube is of a cube that has an initial size and then begins to shrink, continues to shrink, and finally stops shrinking when it has covered the appropriate 4-dimensional distance required to form a hypercube.