Mapping the StarMaze

June 4, 2006 9:10 PM

Mapping the StarMaze A tale of mathematical obsession: "Before I can explain my decades-long quest to map the starmaze I must acquaint you with a small puzzle...I have a habit of seeing everything (cities, organizations, computers, networks, brains) as a maze, so I named this puzzle the starmaze....The first problem I ran into was that there were a lot of rooms...I invented wacky names for each room...But something funny happened...In that instant I finally grasped that the starmaze was arranged on the edges of a nine-dimensional hypercube..."
posted by vacapinta (38 comments total)

I don't quite follow . . . perhaps because I'm not a genius (or perhaps borderline autistic?) in the way this fellow is. Fascinating and amazing stuff.
posted by aladfar at 9:31 PM on June 4, 2006

borked
posted by netbros at 9:33 PM on June 4, 2006

aw crap. I just read that through, and now I can't see the images. was this slashdotted or something similair?
posted by WetherMan at 9:36 PM on June 4, 2006

aw, works now. this is very cool. I don't understand on a deep level what he's talking about exactly, but it's a cool insight into how this man's mind works.
posted by WetherMan at 9:39 PM on June 4, 2006

Well, alright then.
posted by delmoi at 9:42 PM on June 4, 2006

forgot to add. via mathpuzzle.
posted by vacapinta at 9:43 PM on June 4, 2006

Mapping personal memories into a state machine is interesting as a real-world implementation of existing ideas in cognitive science but what does his work have to do with (pure) mathematics, exactly? I honestly don't see it.
posted by Mr. Six at 9:48 PM on June 4, 2006

I wayyyy too stupid fer dis.
posted by overanxious ducksqueezer at 9:57 PM on June 4, 2006

Ok, so the only thing I don't understand is where he found this puzzle that he then mapped graphically and mentally. Is it something he also made up, without fully understanding the rules he used, or is it an old puzzle? If the latter, where did it come from?
posted by odinsdream at 10:06 PM on June 4, 2006

There is no spoon...?
posted by peeedro at 10:19 PM on June 4, 2006

I'd pay good money to see this guy locked in a room with the Time Cube guy for a half an hour.
posted by jimfl at 10:22 PM on June 4, 2006

What a pleasant confusion.
posted by TwelveTwo at 10:29 PM on June 4, 2006

You people scare me. Now help me with my calc homework.
posted by 517 at 10:32 PM on June 4, 2006

Metafilter: I don't understnad, hope me
posted by blue_beetle at 10:43 PM on June 4, 2006

I would need to break out my book to be sure, but I think this guy may have spent years re-creating basic group theory. His text is a bit disjointed but all his talk about symmetry and four-cycles is giving me severe flashbacks to my 4th year of math in college.

It would be almost sad if that were the case. All along he could have just cracked open a textbook and looked up most, if not all, of the answer from theorems from the late 1800s.
posted by Riemann at 10:47 PM on June 4, 2006

I couldn't completely follow everything he said, by I had the same feeling, Riemann.

At the same time, even though I would guess that he's familiar with a fair bit of mathematics already, I don't really think he did this to prove anything new. It seems to me that this starmaze stuff is purely the result of his personal curiosity and creativity. Whether he could have identified his problem with the axioms and theorems of group theory seems somewhat irrelevant.
posted by Alex404 at 10:56 PM on June 4, 2006

this starmaze stuff is purely the result of his personal curiosity and creativity obsessive-compulsive disorder.

He named almost 500 configurations.

83: The pessimist's perch
130: The meadow of starflowers
156: The cockleshell boat dock
204: The Green Dragon tavern
212: The Bureau of Needless Paperwork

This man is living in Borges' Ficciones. Probably in the table of contents.
posted by anotherpanacea at 11:17 PM on June 4, 2006 [1 favorite]

Is this something I'd need a brain tumor to understand?
posted by quite unimportant at 11:58 PM on June 4, 2006

I find it difficult thinking in three dimensions, let alone nine, but nevertheless I found it fascinating how Mr Cartan constructed such a vivid memory palace to visualise the maze.
posted by misteraitch at 12:06 AM on June 5, 2006

I'm with Riemann and Alex404 here. I don't know if he's missed or dismissed it, but there's a way of mapping N-dimensional grids onto a 2D page fairly simply:

(0)
[_] point (0D) - no range of possible values

(1)
[_] [_] [_] line (1D) - possible values -1, 0 or 1

(2)
[_] [_] [_]
[_] [_] [_]
[_] [_] [_] square (2D) - possible values (-1,-1) through to (1,1)

(3)
[_] [_] [_] [_] [_] [_] [_] [_] [_]
[_] [_] [_] [_] [_] [_] [_] [_] [_]
[_] [_] [_] [_] [_] [_] [_] [_] [_] cube (3D)
- possible values (-1,-1,-1) through to (1,1,1)
- note that this can also be represented by (0 to 9, 0 to 3)

(4)
[_] [_] [_] [_] [_] [_] [_] [_] [_]
[_] [_] [_] [_] [_] [_] [_] [_] [_]
[_] [_] [_] [_] [_] [_] [_] [_] [_]

[_] [_] [_] [_] [_] [_] [_] [_] [_]
[_] [_] [_] [_] [_] [_] [_] [_] [_]
[_] [_] [_] [_] [_] [_] [_] [_] [_]

[_] [_] [_] [_] [_] [_] [_] [_] [_]
[_] [_] [_] [_] [_] [_] [_] [_] [_]
[_] [_] [_] [_] [_] [_] [_] [_] [_] hypercube (4D)
- (-1,-1,-1,-1) to (1,1,1,1)
- note that this can also be represented by (0 to 9, 0 to 9)

The next step of course is three copies of diagram 4 arranged in a line, and you see where the coordinate system is going: in 4 dimensions, to (0 to 9, 0 to 27).

Or alternatively:
1: 0-3, 0
2: 0-3, 0-3
3: 0-3^2, 0-3
4: 0-3^2, 0-3^2
5: 0-3^3, 0-3^2
6: 0-3^3, 0-3^3
7: 0-3^4, 0-3^3
8: 0-3^4, 0-3^4
9: 0-3^5, 0-3^4 = (0-243,0-81)

So, if we were to bother to print out such a grid (it can be constructed in Excel quickly enough), and include in each step of the grid a description of the contents, the "room" if you like, we would have a map of his magic castle.

As to how useful this is, I don't know. It's a mathematical curiosity, like Sudoku, which can be similarly mapped onto a 4-cube in which the "rows and columns" of a Sudoku puzzle become the 3x3 square edges of the hypercube. Which may lead to an easier algorithm for solving Sudoku - I don't know.
posted by aeschenkarnos at 12:09 AM on June 5, 2006

Hmm. For some reason MetaFilter's text input system has taken it upon itself to remove the spacing between horizontal squares in diagrams 3 and 4. Oh well, imagine them in. :)
posted by aeschenkarnos at 12:11 AM on June 5, 2006

I got through the maze twice with no idea how I did it. I'm hobbling back to my samurai now.
posted by overanxious ducksqueezer at 12:16 AM on June 5, 2006

I'm with odin's dream - he said he came across the puzzle in the 70's, what form did the puzzle take back then? Who made it?

I'm wondering how you could come across a puzzle like this in the 70s and not have access to info on how it works.
posted by -harlequin- at 1:07 AM on June 5, 2006

It would be almost sad if that were the case. All along he could have just cracked open a textbook and looked up most, if not all, of the answer from theorems from the late 1800s.

Oh, I don't know. This has become a meditation for him, a way to dream really strange and beautiful dreams. I read some more of his pages, and he's obviously a sane, happy, productive guy both socially and professionally. People like him make me wish there were such a thing as brain tourism, because I would love my mind to move on such clean and starry paths.

Thanks, vacapinta -- this was really very wonderful.
posted by melissa may at 1:14 AM on June 5, 2006

I feel like I just read how he has built a scale model of Devil's Tower in his living room.
posted by missbossy at 2:26 AM on June 5, 2006

Wow, that made my brain very happy. Great post. Thanks vacapinta.
posted by intermod at 4:26 AM on June 5, 2006

Damn, long day at work today and not enough sleep the night before. The above should be:

1: 1-3^1, 1-3^0
2: 1-3^1, 1-3^1
3: 1-3^2, 1-3^1
4: 1-3^2, 1-3^2
5: 1-3^3, 1-3^2
6: 1-3^3, 1-3^3
7: 1-3^4, 1-3^3
8: 1-3^4, 1-3^4
9: 1-3^5, 1-3^4 = (1-243,1-81)
posted by aeschenkarnos at 5:44 AM on June 5, 2006

First, I felt the same as Reimann with regard to the group theory, but I don't think it's sad that he might have recreated a good deal of the beginnings of abstract algebra. The fact that he was interested enough to try this for so long and work at it is really cool.

I think this is a poset (as some elements are unobtanable, and thus not a group), but honestly, it's 6am, and my abstract skills are failing me at the moment.


Reading his site leads me to think that the best way to simulate this would simply be a DFA (Deterministic Finite-state Automaton), especially when some of his graphs are oftly close to a DFA. I think what would be really neat would be to model his map with a Regular Expression (correct me if I'm wrong, as I think Regular Expressions and DFAs are of the same power, but I took an introduction to comp sci course six years ago that included about half theory, but I've forgotten a good chunk).

I think it's a shame that there is so much more interesting math that this man could be doing with this problem, but instead he has been working on a special case to the exculsion of seeing if there is anything interesting about puzzles like these in general. (eg, given a puzzle of this type of any size, does it still take only four jumps to come back to any state with the exception of the unobtainable, the inescapable and the start and end? does the n-cube work for a 5 x5 square? 2n+1 x 2n+1? Getting into higher numbers for one of these could potentially lead to some interesting graph theory.) Although given how much basic theory he probably had to develop (inventing terms and such) I can understand why he may not have taken on some of the more ambitious ideas I mentioned above. (I just graduated as a math major, can you tell?)

I did not intend to write that much when I started. But thanks for the post, for all the above rant, it was really inteteresting. And for all the editing I did, I hope it's coherent.
posted by Hactar at 6:22 AM on June 5, 2006

As someone who's studied a lot of discrete math I have to say he didn't explain it very well.
posted by delmoi at 7:40 AM on June 5, 2006

It's all words. It's all English. I can understand every single word. But I've read the whole page and I understand none of it.
posted by talitha_kumi at 8:45 AM on June 5, 2006

I love how he went exploring in this Universe, constructing for example a Grand Tour which visits all the cells once and imbuing it with literary qualities:

When the clock strikes twelve on New Year's Eve in the upper chamber of the Room of Awakening, the first room in the House of Darkness in the east deep of the starmaze, a ghostly cavalcade of hooded pilgrims begins a year-long journey..

It is like a short story with mathematical underpinnings except that he first started with the mathematics and then overlayed his discoveries with narrative. A fascinating creative project.
posted by vacapinta at 11:35 AM on June 5, 2006

I hope I have aeschenkarnos with me when the government puts me in the Cube.
posted by Smedleyman at 1:11 PM on June 5, 2006

It's interesting - he has recreated some aspects of group theory, but has also managed to somehow mingle his own "philosophy" among his mathematical meanderings. I'm always curious when I see folks independently discover such things, especially in this day and age where information can be easily disseminated. I wandered into multidimensional constructs as part of my job - I used to work in the business intelligence arena - and watched as folks who never had a reason to contemplate such things before not only embrace them, but change their problem-solving skills and approaches as a result.

People like him make me wish there were such a thing as brain tourism, because I would love my mind to move on such clean and starry paths.

Makes me wish for the POV Gun.
posted by FormlessOne at 2:48 PM on June 5, 2006

That was delightful! Thanks for the great post.

Here is a man who has found a beautiful realm that delights him and wants to share it with others, and is doing it while reflecting the radiant charm it has been his pleasure to experience.

No sadness. No wasted time here folks. Just humanity.
posted by buzzv at 3:42 PM on June 5, 2006

anotherpanacea: "this starmaze stuff is purely the result of his personal curiosity and creativity obsessive-compulsive disorder."

This guy makes ye ordinary OCDer look positively ADHD. He spent years—decades—mapping and describing a silly puzzle he probably found in a used book store or in the back of a magazine. Impressive as hell, and pretty in its graphical manifestations, but "scary" doesn't begin to capture it.

Hactar: "I think it's a shame that there is so much more interesting math that this man could be doing with this problem, but instead he has been working on a special case to the exculsion of seeing if there is anything interesting about puzzles like these in general."

Well, he does say the following, which he links to this page:
I established a larger context for the starmaze by studying other dimensions of the maze from a 5-D starmaze to a 25-D starmaze and beyond. And when I realized that the starmaze was just one example of a larger class of hypercube puzzles, I broadened that context even further by studying a whole range of variations, from "Neutral Key" puzzles to "Minimum Coverage" puzzles and many others in between.

The linked page has a few general propositions about the whole class of such puzzles.

buzzv: "No sadness. No wasted time here folks. Just humanity."

I wish that I were capable of even a small fraction of the dedication and persistence he's displayed, that's for certain. But I can't begin to understand what such a life would be like.
posted by dilettanti at 3:57 PM on June 5, 2006

buzzv said it well.
vacapinta- thanks for the trip.
posted by pointilist at 11:17 PM on June 5, 2006

I wish that I were capable of even a small fraction of the dedication and persistence he's displayed, that's for certain.

Dilettanti indeed. :) Me too.

But I can't begin to understand what such a life would be like.

As one who's gone for a paddle in that ocean once or twice, I can try to give a written description of the sting of the salt and the cold of the water. What is it like to have OCD? It's like this - everyone has things they have to do. OCD is nothing more than the overemphasis out of control of an element present in everyone's character. If you were to truly look over your own shoulder while you go about your day to day life, you could find your little rituals.

How you brush your teeth, for instance. Your manner of doing it is fairly well fixed by now, in your semi-unique way; other peoples' tooth-brushing habits are fixed differently, but still fixed. I expect if you had your teeth brushed by another person--a dentist's assistant, for example--then even though you know intellectually that that person's brushing habit is perfectly fine, and your teeth should be clean, they won't feel clean to you. Now after the dentist's assistant is done, you'll probably lick your teeth and champ your lips and, in a few minutes, forget about it. My point is that, for that moment of not feeling clean, you've touched the surface of OCD. You felt, for a minute or two, the "itch".

Another example is a "hot-button issue". One's perfectly rational, normal acquaintances have things that must not be said to them, because they will not be able to help themselves, they must respond. ("Don't mention Microsoft to Charlie, he'll rant on about it for hours.") They may even be fully cognizant of the fact that this is an aberrant response - still, if something takes an act of will to suppress, that something counts as a compulsion. You "need" to respond. It "itches" you, not to.

That "itch" is a physical sensation for me, an unpleasant skin-crawling which can even raise goosebumps. But I'm a very low-level OCD "sufferer". I wouldn't actually want to get rid of it totally, as it's useful to tap into it to do unpleasant or repetitive tasks, like dishes or proofreading. I can imagine that for someone with serious OCD, the "itch" could get far, far worse, and become physical pain.

Another way of making the itch happen, if you've any OCD in you at all, is to look at your fingers for loose bits of dead skin. Worry at one for a little, so you're aware of the sensation of it being there. Now leave it for a moment. I've only tested this on a couple of people, but it worked on them - they felt a distinct "need" to pluck it away, or clip it off, or otherwise get rid of the offending thing. Other small acts of physical self-maintenance (a hair missed when shaving, a blackhead, etc) can produce the same "need". Some people have a compulsion for order - if the person has colored in one of the 'o's on a page of text, they may feel a need to color in all of them. When engaging in acts, physical or intellectual, driven by OCD, there can be both a negative reward (the removal of the itch) and a positive reward (a feeling of calm, centredness). These can overcome pain or tiredness, as in the classic example of scrubbing hands until they bleed, or induce enough

Other examples: "One more turn of Civilization III." "One more coin in the slot machine." "One more cast with the fishing line." OCD is strongly tied in with the repetition and reward motivation.

The itch is the same as the denial of the need. Dysfunctional OCD such as Tourette's syndrome, which I believe is a verbally-manifested form of OCD, occurs when the need/itch to do trivial things is overpowering to the point of pain, and/or when the need/itch to do inappropriate things overcomes the social consequences.

Now a person lucky enough to have OCD which drives them to attempt to solve intellectual puzzles, and who possesses the necessary intellectual power to succeed at that task, can hardly be considered a "sufferer" in the same sense as a sufferer of, say, psoriasis (the Singing Detective's skin condition). It is true that they do experience pain and pleasure for bizarre reasons, though.
posted by aeschenkarnos at 12:12 AM on June 6, 2006

Thanks for this. This guy sounds like he belongs in a Borges story.

"The simplicity I had sensed from the beginning was shining before me in a radiant light.

In that instant I finally grasped that the starmaze was arranged on the edges of a nine-dimensional hypercube. This may sound complicated to those of you used to living in three-dimensional space, but in fact it couldn't be much simpler."

Must be nice to suddenly make the leap to R^9 mental conceptions like that. Really wish I could do that.
posted by archae at 12:26 AM on June 6, 2006