Years ago, I wondered how I might visualize a linear equation in 4 dimensions, or the generalized (though finite) "n" dimensions. When I asked for help, I was told, "You don't."
The truth is, you can. Here's an easy way to do so:
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| Playing surface for two (2) NFL teams |
We all know and love NFL Football. Now, take into consideration the above playing field.
Normally, two teams play at once. At the conclusion of a game, there is a final score. Let's go with Bears 21, Packers 14.
Consider the final score as an "ordered pair." We've all plotted points on a X-Y axis system (Cartesian coordinates) when we were in, say 4th grade.
Just to be sure, we're plotting the ordered pair (21, 14) on an (x, y) axis system.
Now, let's introduce another team to the same game. So, let's say it's a three-way football game. Every team is playing on the same field, and with the same singular football. Of course, this isn't how football is played, but it
could be.
Let's say the result is Bears 21, Packers 14, and Steelers 10.
Now we have 3-D, and our ordered triplet is (21, 14, 10) on an (x, y, z) axis system. What about adding one more team??
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| Playing surface for four (4) NFL teams |
The above image demonstrates the playing field for four NFL teams to play against one another, at the same time. Let's say the final score is like so: Bears 21, Packers 14, Steelers 10, and Jets 7. This would allow for an ordered quadruplet (21,14,10,7) on an
(x
1, x
2, x
3, x
4) axis system.
If you're unfamiliar with "n" dimensions, it works like this: You typically use "n" to denote an axis system that is four dimensions and higher. For example, (x1, x2, x3, x4, ... ,xn). So if there are 32 dimensions, n=32 and the ordered list ends with (x1, x2, x3, x4, ... ,x32).The key concept here for our "4-dimensional" football game is that each team is playing on a singular surface, and they're all playing with the same football. If you increase the size of the playing field (with your imagination, you could conceive a set of 32 uprights, a field suitable for every NFL team playing simulatenously).
But for now, we will inspect a 4-dimensional NFL football game.
Imagine a game being played on the field with 4 sets of uprights, each of the four NFL teams battling for touchdowns and field goals.
The football itself, as is sails around the field from one player to another, and through each set of uprights, is a vessel used to facilitate variable interplay amongst each of the four dimensions (i.e. each of the four NFL teams.
The center of the playing surface (called "half-field" when there are only two teams) can be representative of an origin in the context of coordinate systems that stem from central, unifying points. This "center-field" point is equidistant from each of the four uprights and at the center of the playing surface, and does not move.
More specifically, the origin point is always the same: (0,0), (0,0,0), (0,0,0,0) and beyond.
The takeaway here is that as a spectator, if you observe one of these NFL games with four teams playing at once, on this type of field, you're seeing 4 variable components (i.e. dimensions) interacting with one another within a uniform, or linear environment.
The game's interplay is facilitated through a singular, unifying entity (in this case, an object... specifically the leather football itself). The game's outcome is a function of the leather football's trajectory around the field, as constrained by a separate list of official rules for play.
The playing surface is also finite and contains a singular origin point.
Using this example, our 4-way football game reaches a definite conclusion. We can imagine, after a grueling 60 minutes, the end result is Bears 21, Packers 14, Steelers 10, and Jets 7. Each play-call along the way allows for valuable analysis.
So the next time someone says, "You can't see four dimensions," you can reply with:
"Maybe not with perfect exactitude, but with some imagination we can conceive four-dimensional linear systems."
- Mike
Supporting material:
1.
http://eusebeia.dyndns.org/4d/vis/01-intro“Some believe that it is impossible for us to visualize 4D, since we are confined to 3D and therefore cannot directly experience it. However, it is possible to develop a good idea of what 4D objects look like: the key lies in the fact that to see N dimensions, one only needs an (N-1)-dimensional retina.” ... “What our eye sees is in fact not 3D, but a 2D projection of the 3D world we are looking at.”
2.
https://people.eecs.ku.edu/~jrmiller/Papers/IVP.pdf“based on the metaphor of being in a room with glass walls that one "flies through" the n-dimensional data set.”
