Complexity, Simplicity, Dimensionality

I want you to go on to picture the enlightenment or ignorance of our human condition somewhat as follows. Imagine an underground chamber like a cave, with a long entrance open to the daylight and as wide as the cave. In this chamber are men who have been prisoners there since they were children, their legs and necks being so fastened that they can only look straight ahead of them and cannot turn their heads. Some way off, behind and higher up, a fire is burning, and between the fire and the prisoners and above them runs a road, in front of which a curtain-wall has been built, like the screen at puppet shows between the operators and their audience, above which they show their puppets. (Plato, The Republic)

And even as we, who are now in Space, look down on Flatland and see the insides of all things, so of a certainty there is yet above us some higher, purer region, whither thou dost surely purpose to lead me — O Thou Whom I shall always call, everywhere and in all Dimensions, my Priest, Philosopher, and Friend — some yet more spacious Space, some more dimensionable Dimensionality, from the vantage-ground of which we shall look down together upon the revealed insides of Solid things, and where thine own intestines, and those of thy kindred Spheres, will lie exposed to the view of the poor wandering exile from Flatland, to whom so much has already been vouchsafed. (Abbott, Flatland)

Don't run the movie yet! Just look at the first frame when it loads. What do you see? Eight dots inside a box? I hope so. Is there anything unusual about the arrangement of the dots? Some pattern, perhaps? Maybe yes, maybe no.



OK, then, run the movie. Do you detect any kind of order or coherence that you did not see before? Would it help if I told you that the dots mark the eight vertices of a rotating cube? (The cube is rotating at a different rate about each of its three axes.) There are a few observations that I think are interesting:

• What might have initially looked like a random collection of dots in 3-space was in fact far from random, being a projection of a very simple and coherent structure, the cube.


• Even before you recognized the rotating cube for what it was, you probably became sensitive to the fact that there was some structure or organization to the motion.


• The rotating cube is not always easily recognizable as a cube. As is the case in the opening frame, there are many static views of the cube's vertices from which it is very difficult to infer the underlying structure (most of these views look like semi-random dot patterns in 2D). But I think you will agree that there are also sequences of the movie during which the cube seems to disintegrate into just a collection of moving dots. This also seems to happen if you close your eyes for about 5 seconds during the movie, and then reopen them. For the first second or so you may not see the cube, but only the dots.

Go through the same exercise with the second and third movies. First assess whether you detect any structure or coherence in the pattern of dots on the initial frame. Then run the movie and reassess the situation. It's unlikely you will deduce the actual structure underlying the motion of the dots (as you may have been able to do for the first movie), although you may still observe the motion to be highly organized, perhaps more so at some points than at others. In fact, the actual structures determining the motion of the dots in second and third movies are very similar to that in the first movie. In the second movie, a 6-D hypercube spins at a different rate along each of its axes. The dots again represent the vertices of the hypercube, of which there are 64, projected into 3-space. In the third movie, it is a 9D hypercube that rotates, the dots marking the 512 vertices.







What is the point, you ask?

Well, besides the point of "keeping the blog alive" while I work on preparing my epochal post on the Unknowability of God (coming soon to an epoch near you!), there is also the following point:

What people usually intend when they invoke Plato's Allegory of the Cave (or Abbot's Flatland) is that information is lost when a high-dimensional structure is projected into a lower-dimensional space. (Obviously, the "spaces" in question need not be physical spaces. The argument goes through for vector spaces in general). Thus an entity or system which exists in 10 dimensions — and whose dynamics are described by relationships among 10 variables — will probably not be completely observable when viewed through a reduced subset of those dimensions. ("Probably" is the correct term to use there.)

The intuition is that what you will get when you observe, say, only three of the phenomenon's 10 dimensions is some diminished and highly oversimplified view of that phenomenon — in a sense only seeing 30% of the actual phenomenon, and thus utterly missing the other 70%.

But that is not typically the case, as the movies show. What you get on projecting a N-dimensional structure into lower set of M dimensions is a compression of the N dimensions into M dimensions, and while there is certainly a loss of information, there may also be a dramatic increase in apparent complexity. Thus, a phenomenon that is simple and compact to describe in 10 dimensions, may prove difficult and verbose to describe in three dimensions. In the extreme, a high-dimensional structure may appear essentially random when projected into lower dimension.

The Cave Allegory is a favorite of theological dabblers. Most people, when they first hear of it, feel that they have been hit by a bolt of pure wisdom from God. And maybe they have, although most probably do not realize that dimensional compression is the bread-and-butter of many specialties in science and engineering — e.g., pattern recognition, optimum signal processing, relational databases, etc. — and that there is an entire area of statistics called "projection pursuit." In any case, people immediately latch onto the idea (like Plato, perhaps) that this physical world is just a simplified projection of some transcendent realm — a pale shadow of some grander and more real reality that actually exists in much higher dimensions.

The prospect that things might appear simpler in some other dimensions is very attractive, and it provides the motivation for the development of most scientific models (although the preference is usually to find lower-dimensional representations rather than higher-dimensional representations). However, the appeal of this idea should be tempered by the fact that structures projected from higher dimensions may appear considerably more complex or even random in lower dimensions.

As a result of projection, diverse and unrelated dimensions in the higher dimensional space might be collapsed together or "confounded." From studying the resulting projection in lower dimensions, how then do you determine what the actual dimensions of the original space are? Perhaps, as a result of projection, what we observe in our reduced-dimension space as a seemingly unitary variable "X" is actually the combined effect of variables A,B,C,D,E,F,G,H,I,J which exist in 10-dimensional space. It is therefore not the case that the "real" 10 dimensional reality simply adds nine more variables to the observed variable "X"; It is rather that there is no variable "X" in the real reality — there is no spoon!

So when the amateur theologians ask us to "imagine that our world is a projection from a higher-dimensional space," what exactly is it that we are supposed to imagine? It is not fair to simply imagine a reality like this one, but with a lot more properties. Even if we knew the exact nature of the projection in question, we would still be left with inductive ambiguity due to the resulting compression, but the fact of the matter is that no theologian has offered us even the slightest idea about the nature of the proposed projection. In the absence of this, it is possible to imagine almost any situation obtaining in the higher-dimensional world while yet being compatible with (and in some sense identical with) the observed events of our own shadowy world.

In summary, the Allegory of the Cave may provide the welcome insight that there is something beyond the world of our experience, but it cannot provide any genuine theological comfort, because in the absence of constraints we have no idea how the world of our experience corresponds to that "greater world," the real reality, the more spacious Space, the more dimensionable Dimensionality. We are still lost.