W A R N I N G !
VERY DULL MATERIAL!
DO NOT ATTEMPT TO READ THIS CARD!
It's been nearly two years since my last update of my ongoing research into a mysterious mathematical structure I call the Starmaze (see VC 18 John 8).
That research has continued unabated even through all the turmoil of the last two years, and although I have still not settled on a I Ching assignment for the maze, I have made good progress. I now believe that the solution I was trying to find back in volume 18 is not possible. But I still can't PROVE it's impossible.
I've been working on this particular problem for five years now, and I've attacked it from at least 50 different angles! Last spring I wrote a very innovative program that moved through the topology of all possible assignments like a marble rolling at random through a countryside of hills and valleys.
The "marble" was programmed to seek "sea-level" by rolling into ever deeper valleys. Every time the marble found a new valley, my trusty Mac made a trumpeting sound. The program ran for two straight weeks, and gradually the trumpets came less often. Sometimes the marble would get stuck and I'd have to nudge it along by rewriting a subroutine. Finally, when it had reached 7/8ths of the way to a perfect solution, it bogged down completely and nothing I could do seemed to help.
I had experienced this 7/8ths barrier before, in many different forms. Although I still can't prove it, I now suspect that 7/8ths is a theoretical limit. That is, no matter which patterns I assign to which rooms, no more than 7/8ths of the 512 rooms in the maze will exhibit all of the properties I was looking for. On the plane ride to my wedding I had an insight which might help explain why this is so.
Until then I had been focusing on the 512 rooms of the maze rather than its 2048 outer passages, trying to assign the same hexagram to all eight passageways of each room. I decided to change my focus and find assignments for the passageways that utilized all 4096 hexagram transformations and were, at the very least, reflexive. By this I mean that if a passage from room A to room B was assigned a hexagram which represented X becoming Y, then the same passage from B back to A should represent Y becoming X. I dropped the condition that all passages from a given room be assigned the same base hexagram.
Even meeting this far less strenuous set of conditions is not easy. On the plane ride I stumbled onto a method of representing both directions of each passage as a cell in a 64 by 64 cell triangle. By assigning a number from 0 to 2047 to each cell (in any order) I could guarantee reflexivity.
It is interesting to note that under this system, trying to meet the third condition would be like trying to fill the triangle with clumps of cells that are eight units wide. It can be seen by inspection that the best possible fit will only fill 7/8ths of the triangle. This is not a proof that the third condition is impossible - some other system might allow a perfect fit - but it does suggest that the 7/8ths barrier is real.
Since then, I have shifted my focus to finding the best possible passage assignments. Recently I've made progress on two fronts: finding ways to enumerate all the passages of the maze in a sequential order by contructing "Eulerian Tours", and arranging the 4096 ways that 64 base hexagrams can transform themselves into other hexagrams into orderly sets of number sequences.
I'm beginning to see that some passage assignments can reveal a hidden "maze within the maze." It is possible to assign the hexagrams in such a way that the passages form a continuing trail. I should be able to create occassional forks in that path. The resulting structure would still be vast (about 2/3rds as many decision points as the starmaze) but would have only two or three branches from most points instead of nine. The research continues.
I have also done work in two entirely different areas of starmaze research. The first involves the problem of assigning an evocative name to each of the 512 rooms. For help in this endeavor I turned to a most unlikely source: "Zolar's Book of Dreams." I now have a spreadsheet with approximately 500 names; this is a good start.
The other area involves finding new ways of visualizing the maze, especially in three dimensions. You may recall that I have shown that the maze can be represented as one-way passages along the edges of a nine dimensional hypercube, and I have also found a two dimensional representation that has been enormously useful in my research.
One of my dreams, though, is to construct a three dimensional version that one could actually walk through. In order to enforce the one-way directions, I could use slides or fireman's poles and construct each passage so as to include at least one elevation drop. There are many different ways of arranging the rooms in 3-space, but the complexities involved in a structure this big are somewhat overwhelming. I have started to tackle this problem and already have a few interesting sketches.
It would take millions, perhaps even billions of dollars to actually build a real 3-D starmaze. It would be a project on the same scale as the ancient pyramids. So until recently I could only dream of someday walking through the maze. But with the astonishing progress being made in virtual reality I now believe that a crude virtual 3-D starmaze is already possible, and a more sophisticated version could be within my grasp by the end of the decade.
Here, then, are my goals:
Here, then, is my ultimate ponarv. If I am successful then even this dry history of my efforts will be interesting to someone. Someday.
- To find an acceptable I Ching assignment for the Starmaze and to provide a name and set of fixed properties for each room. When this process is complete I would like to self-publish a reference book with basic information about all 512 patterns along with a brief introduction to the maze and a summary of its properties.
- To design a detailed, three-dimensional representation of the maze and create a virtual reality interface that would let me experience a walk through the maze.
- To create (and possibly sell) a hand-held version of the puzzle using either a circuit board with lights and push buttons or one of the pen-based palmtop computers like the Newton which will soon hit the market.
- To create (and possibly sell) a quality poster-sized map of the maze with instructions on how to navigate through it.
- To publish a fantasy novel set in the starmaze and/or a book of wisdom and meditation patterned after the I-Ching which would allow other people to experience the beauty and power of this structure.