For a number of very practical reasons, human beings tend to proclaim their successes and conceal their failures. Failure, therefore, is often mysterious, a secret kept in a box opened only when we are alone. Nevertheless, I would like to discuss one of my recent failures. Perhaps by sharing this apparently insurmountable problem, I can at least get a grasp on it and force it back into its box. As it is, the problem is loose and gives me no rest.
As some of you know, I have been studying a mathematical structure which I call the StarMaze for over ten years now. I am also a student of the I Ching, an ancient Chinese book of divinations. Over the years I have gradually become aware that these two topics are deeply related.
I have already prepared several installments of Mr. Wizard which deal with the Starmaze; I hope to share these with you in the next several issues of Archipelago. For now let me just say that the starmaze consists of 512 rooms and 2304 passageways. Each room has 9 doors, some open, some closed. Each passageway is unidirectional, that is, you can only travel forward through each passage, never retracing your steps. The maze has many mysterious and surprising properties.
The I Ching, or Book of Changes, consists of 64 patterns called hexagrams. The patterns have evocative names like "The Caldron" or "Darkening of the Light." Each pattern is associated with a series of poems and philosophical commentary. These patterns are supposed to represent archetypal states or moments in a universe in a constant state of flux. Thus each pattern has the potential of changing into any other pattern. The patterns are drawn with strong or weak lines which indicate the current direction of change. Therefore there are 64x64=4096 different possible situations.
As it turns out, this is a number of some relevance to the Starmaze. As I mentioned above, each of the 512 rooms in the maze has 9 doors. These 9 doors fall into 3 categories: there are 4 "yin" doors, 4 "yang" doors, and 1 "center" door. Thus the total number of "yin and yang" doors in the maze is, you guessed it, 4096.
Now then, here is what I set out to accomplish (about 2 years ago). I want to assign a I Ching hexagram to every room in the Starmaze in such a way that each one of the 4096 possible transformations occurs once and only once. Moreover, I want this assignment to be consistent. Let me give you an example.
Suppose that you are standing in room 372. The room is shaped like an octagon with a door in each of the eight walls. (There is also a trap door in the center of the room, but we don't care about that.) Imagine a huge hexagram painted on the ceiling of this room, and imagine further that each of the 8 doors has a different hexagram painted on it. You decide to pass through door number 3. The hexagram on the ceiling is, say, number 56, "The Wanderer," and the hexagram on door number 3 is, say, number 61, "Inner Truth." Thus choosing door number 3 in this case would correspond to the situation of hexagram 56 changing into hexagram 61.
Got all that? Now then, here's what I mean by consistency. Door number 3 in room 372 happens to lead to room 504. When we arrive in room 504, I would expect to find hexagram 61, "Inner Truth," PAINTED ON THE CEILING. That is, whenever a hexagram is painted on a door, that door should lead to a room devoted to that hexagram. This is what I mean by consistency.
Since each of the 512 rooms is devoted to a single hexagram (the one painted on its ceiling), in order to complete each room I will need 8 copies of each hexagram (8x64=512). The problem, then, is how to assign 512 hexagrams to the 512 rooms in the maze in a "consistent" manner. The solution could be written as a list of 512 numbers arranged in a certain order. All I have to do is find the right order.
I have already written a program which can inspect any such list and determine whether or not it is consistent. Why not just try all possible orderings and have my program test each one until it finds the right order?
The problem is that there are an ENORMOUS number of possible orders, almost all of them wrong. In fact the number of possible arrangements is almost too large to comprehend: a 1 with about a thousand zeros after it. This means that if every single quark in the entire universe could be turned into a Cray supercomputer testing a billion arrangements per second, and if the calculations were to continue from the big bang to the end of the universe, all the quarks put together in all that time could not possibly test more than a teaspoon full out the ocean of all possible solutions. For all practical purposes, the number of inconsistent arrangements is infinite.
To make matters worse, it's not at all clear to me that a consistent arrangement is even possible. In fact, under a strict interpretation, the problem is clearly impossible. Here's why.
If every possible I Ching transformation occurs somewhere in the maze, the null transformation (a pattern turning into itself) must also appear, which means that at least one room with a given hexagram on its ceiling will lead to another room devoted to the same hexagram. Since each hexagram appears in 8 rooms with 8 doors, each hexagram has 64 available doors to lead to other hexagrams. If a hexagram is adjacent to itself, this uses up 2 doors (1 door in each room). This leaves only 62 doors which must reach a total of 63 other hexagrams. Clearly impossible.
There is a way around this, however. Suppose I arrange the rooms so that a hexagram is never adjacent to itself. Since we now have 64 doors leading to 63 other hexagrams, this means that 2 of the 64 available doors will lead to the same hexagram. On one of these 2 doors I can hang a sign reading "Do Not Disturb." This door will represent the null transformation. With this alteration in place, I have been unable to show that my problem is impossible. I have also been unable to show that it is possible (by finding a solution). I am in Limbo.
You might say that it is hopeless to even look for so small a needle in so large a haystack. How can I hope to find something that all the Cray-Quarks in the universe cannot? Actually, problems of this magnitude are solved every day (or at least every decade). The human mind is capable of succeeding where brute force techniques are doomed to fail. And I have met a problem like this once before, and, after SEVEN years, I was finally able to find a solution (a map of this very Starmaze). So I have hope.
Success against an all-but-infinite foe is possible because of the existence of beauty. Because the solution is beautiful, it stands out from all the many inconsistent arrangements, as a perfect diamond shines forth from a great heap of coal. In this very abstract mathematical setting, beauty can be defined as symmetry. And symmetry can be used to drastically simplify overwhelming complexities.
By now you have probably given up all hope of ever discovering what any of this has to do with the rather odd picture to the left of this text. The picture is a map of Cubeville, population 8. Cubeville consists of 8 houses occupied by 4 newly divorced couples:
The divorces came about all at once as the result of a delicious and improbable scandal far too involved to explain at this time. After the initial excitement died down, the denizens of cubeville held a town meeting and agreed to swap houses. They wanted to arrange things so that each citizen would not have to run into his estranged spouse on a regular basis. And yet each couple wanted to retain every other couple as neighbors. Specifically, here are the conditions that needed to be met:
- Abe and Abby Anderson
- Bill and Betty Brown
- Cal and Cindy Carter
- Dave and Debby Dawson
First, each citizen was to be assigned his or her own permanent house. No more double occupancies or summer homes.
Second, no citizen wanted to live north, south, east, or west of their former spouse. Because they tended to stay on one plane, being above or below a former spouse was of no real concern.
Third, cubevillians want to arrange things so that at least one member of each couple would live north, south, east, or west of at least one member of every other couple. For example, there should be at least one Anderson house next to a Brown House, at least one Anderson House next to a Carter House, and at least one Anderson house next to a Dawson house.
In order to make the third requirement workable, it was necessary to stipulate that one pair of neighboring families could occur twice, and every other pair could occur only once.
After much thought and debate, the citizens of Cubeville settled upon the arrangment shown in the map at left. Each person has his or her own house, so the first requirement is fulfilled. And since all the men are on the lower plane, while all the women are on the upper plane, no one will ever run into his spouse by travelling north, south, east, or west, and so the second requirement is also satisfied.
The Andersons and the Browns are next to each other twice. That is, Abe is south of Bill and Abby is west of Betty. The Andersons are next to the Carters and Dawsons only once. Abby is south of Cindy and Abe is west of Dave. Thus condition 3 is satisifed as far as the Andersons are concerned. If you examine the map you will see that the condition is also satisfied for the Browns, the Carters, and the Dawsons.
There. We have now found a solution for the 3 dimensional version of my starmaze/hexagram problem. The starmaze is essentially a 9 dimensional structure. Thus to find the solution I am so desperately searching for, all we have to do is leave Cubeville, which is little more than a wide spot in the road, and travel to the nearby town of Hypertesseractville, the county seat, with a population of 512.
Here we find an even larger scandal with much more bitter divorces. The divorces are more bitter because there are children involved. There are a total of 64 families in Hypertesseractville, with each family consisting of mom, dad, 3 sons, and 3 daughters. The divorces sweeping through this town are SO bitter that every person in every family wants a house of his own. Fortunately, there just happen to be 512 houses in Hypertesseractville.
In cubeville, the notion of adjacency was easy to understand. Two houses were said to be next to each other if one was north, south, east, or west of the other. Up and down didn't matter. In Hypertesseractville, up and down still don't matter, but all the other directions do, and there's a lot more than four. In Cubeville, each house was adjacent to two other houses. In Hypertesseractville, each house is adjacent to EIGHT other houses.
I would have liked to show you a map of Hypertesseractville, but there's no way I could make it fit on a voice card. However, the two towns were laid out by the same anal-retentive street designer, so by studying cubeville it is possible to get a feel for how Hypertesseractville is laid out.
The houses in cubeville are arranged on the corners of a three dimensional square (or "cube"). The houses in Hypertesseractville are arranged on the corners of a NINE dimensional square.
What's that? You say there's no such thing as a 9 dimensional square. O ye of little faith! Granted, it is impossible for human beings to visualize anything beyond 3 spatial dimensions. But through the power of analogy, we can PRETEND that such a thing is possible, and we can even figure out what it's properties must be. For example, if a square has 1 "side" and a cube has 6 sides, then a Hypertesseract (as I call my 9 dimensional square), must have 4608 sides.
I am including a gift with this issue (described in the next card) which may help you to grasp the concept of higher dimensional spaces. One approach is to consider how squares are formed.
Consider a single point. A point has 0 dimensions; it doesn't take up any space at all. Now stretch this point one inch in any direction. You now have a line segment, a 1 dimensional entity with 1 edge and 2 "corners." NOW, grab this line and stretch the whole thing one inch in a direction at a right angle to the first direction. The result is one square inch, a 2 dimensional entity with 1 side, 4 edges, and 4 corners. NOW grab this square and stretch it one inch in a third direction which is at right angles to both of the previous two directions. The result is a cube, a 3 dimensional entity with 6 sides, 12 edges, and 8 corners.
Now comes the hard part. We need to grab this cube and stretch it one inch in a fourth direction that is at right angles with the first three. The problem is that we live in a three dimensional universe and our poor little three dimensional brains cannot conceive of such a direction. Don't let this bother you. Just pretend that such a direction exists and stretch away. The result is an imaginary shape called a tesseract, which has 16 corners, 32 edges, 24 sides, and 8, uh, cubes. It looks sort of like one cube inside a second cube connected by 6 more cubes except that all the cubes are the same size, which seems impossible to us.
Now, to visualize the map of Hypertesseractville, you just have to perform this stretching maneuver 5 more times. If you try to draw a 9-square on a two dimensional piece of paper, you will probably end up with tangled mess, but on a NINE dimensional piece of paper, our map would be simplicity itself.
So all I have to do is get out my 9 dimensional map and arrange the 512 members of the 64 different households so as to meet the 3 requirements listed above. Incidentally, I have shown that these requirements cannot be met in 4, 5, 6, 7, or 8 dimensions. It appears that they COULD be met in 9 dimensions and then not again until we reach 33 dimensions. And there are over 8 BILLION houses in that city! So it's nine or never!
You and I are hard pressed to come up with a 9 dimensional pencil, let alone a 9 dimensional piece of paper, but out trusty Macintoshes are not so limited. A computer exists in a world which transcends space. In my many years of starmaze exploration, my Mac has been a kind of magic carpet, carrying me to places my feet cannot reach. Although it can't really "visualize" much of anything, nine dimensions are no more bizarre to a Macintosh than two or three.
So I really don't have a good excuse. Like Ahab, I have a good ship and crew, and I grow daily more convinced that the whale I seek really is out there somewhere. Again and again during the last two years I have seen him spouting on the horizon, but every time I approach he is gone into the deeps where I cannot follow. My crew grows ever more restless, but the harder my whale is to catch, the more meaningful he becomes, and the more I want him.
After dozens of programs and dozens of yellow pads filled with strange equations and dozens of sleepless nights, I have become dimly aware of the following paradox. The only way I have any hope of capturing something this large, is by applying some simple ordering principle, a pattern applied over and over again, to cut it down to my size. And yet the beast I seek is, in a sense, PERFECTLY disordered. Every regular pattern sooner or later causes a hexagram to be adjacent to itself, or to violate some other condition. Only the most wildly, weirdly scattered arrangement can succeed. So, since I am too small to grasp the solution with a single blinding insight, how then can I find an orderly path to perfect disorder?