I meant no offense in correcting you today in front of my fellow cadets. As per your request, I’m writing up my argument for submission tomorrow morning. Excuse the typographical errors and sloppy diagrams, I’m having to put this together under my bunk as we spent all afternoon in the simulators and it’s now Lights Out.
M301: Linear Algebra
The Visualization of Extra Dimensions
The difficulty we have in visualizing extra dimensions lies not with our familiarity of the three dimensions that make up R3 space. Rather, it lies with our our false sense of familiarity. The mistakes we make when picturing a point, a line, a plane, an infinite cube get in the way of understanding movement within higher dimensions.
When thinking of a line, it is necessary to remember that points have no physical dimension. They have no edge. It is impossible that they even “touch” one another. Pick any two “points” and I will show you that an infinite number of other points exist between these two. P1 might lie at .01 and P2 at .02 and it is readily apparent that there’s countless points between that have higher precision: P3 at .015 P4 at .0105, and so on.
Therefore, it is impossible to conceive of a line as a solid string of points. It is actually a dotted path, with the dots existing at whatever spacing dictated by the settled-upon limits of precision. A line is not: ———— It is: ………………
Movement along the line involves the theoretical “teleportation” from one point to another. We say that an object was once at .1 and is now at .2. It is impossible to describe its journey through the infinite numbers between, a paradox well-known to the ancients from Zeno (for which I have a nifty solution).
For the same reason, a plane should not be considered a flat “sheet” of connected material. No matter what resolution you choose in which to visualize it, an infinite number of points will exist between the others. And an infinite number of lines between any two lines.
These zero-dimensional constructs are representations of places in space. They are meant to convey location, not substance. For this reason, they should not be considered to “make up” lines and planes and cubes, they merely “divide” them. A point just slices a line into two pieces, telling you how long each piece is. A line slices a plane in the same manner. The physical lies to either side.
So, a plane looks like:
And movement is our figurative teleportation between any two adjacent points. We can zoom in or out and change the number of these points, but the decision is arbitrary. As long as the precision suits our needs, there is no wrong answer.
R3 (the infinite cube we see around us) Is a stack of planes, each made up of rows of lines, each of which is just a string of points. Seeing how each higher dimension is built from the last, it is trivial to continue to R4. But let’s review what we’ve done. We took points and laid them side-by-side, resulting in a line (R1). We laid lines side-by-side, and got planes (R2). We laid planes side-by-side and got a cube (R3).
Take that cube of points in your mind, and create a string of them, laid out side-by-side. That’s R4. Now take more R4’s (just “lines” of dotted cubes) and create a “plane” of them. That’s R5. We can layer these and visualize R6. With a little practice, we can string these layers out in a dense mesh and “see” R7. I tend to fall asleep with a tenuous grasp on the ephemeral R13, but computational display devices could portray dimensions much higher.
Visualizing movement between these new dimensions is easy when we remember that we teleport along simple lines all the time. Let’s take a line segment and a point at the center, or the origin. Our precision is two decimal places. The object can go from 0.00 to 0.01 or to -0.01. Those are it’s only options. It must go to one of these two points, side to side, before it continues anywhere else.
In a plane, it could also go to the adjacent points in two other directions, front and back. In R3, it would have the option of going up or down in addition. We do not have standard names for the next options, but I use in and out for movement along R4. If we go back to our string of dotted cubes, it just means a point at the “center” of one cube is adjacent to the centers of the cubes to either side of it. The point can move to these centers, or even (diagonally) to any point “adjacent” to these centers.
In R5, it can do the same for the “string of dotted cubes” that lies to either side. And then there’s the “plane of dotted cubes” above and below. The fascinating result is obvious: every point is “adjacent” to an infinite number of other points. This isn’t as remarkable as it sounds when we consider: even the smallest segment of a line contains an infinite number of points, which is the same number of points found in an infinitely-sized sphere. Understanding this, and the nature of figurative teleportation within standard dimensions, unlocks the mind to much higher ones.