There are times when I get tired of all of the major sites and apps that make up a lot of the seen Internet. Sadly, I was born too late to experience the wonder of Geocities and intricate handmade websites run by just a single person. But, on some evenings, I still search for those smaller websites that no one else normally sees. I find the wonder of coming across hidden blogs and pages really fun.
It was a night like this when I stumbled across John Cartan's writing about the Starmaze last year, and I quickly became fascinated. Unlike every other corner of the web, Cartan's website wasn't trying to sell me a useless product or tell me about news I've already heard a dozen times. It was instead a series of journal-like recordings about some strange discovery that he called the Starmaze.
It started with a game he had come across on a computer years ago that involved a square divided into nine sections where each section could be turned on or off. The game began with the middle section of the grid turned on, and the outer eight sections turned off. Clicking one of the sections would turn on/off other parts of the grid in a predictable way, but you could only click the sections that were already turned on. The goal was to click the sections in a way to end up with the outer eight squares turned on, and the innermost square turned off.
Being interested in games myself, this sounded familiar to me immediately. I've seen, in several video games before, puzzles just like this — the player starts with a blank grid and has to fill in all the sections by pressing them in a specific pattern. The difference this time, though, was that in Cartan's puzzle, only the "on" sections could be pressed. This small change made the puzzle far more difficult than the ones I'd encountered in the past. You can try it yourself, if you want.
I couldn't figure out the solution to the puzzle, so I continued reading, expecting an explanation of the rules and methods of solving it. Maybe, I thought, the webpage would end with some interesting insight, and I would be done. In the end, I was correct that Cartan would offer up an explanation of the rules and methods, but I never expected how far it would go. It wasn't a simple explanation, and it wasn't even one that he knew immediately. It wasn't a puzzle that could be understood in one night.
Cartan saw the puzzle as a maze, with the "on" sections being doors that lead to new rooms with new doors. This approach made the puzzle — once seen as just 9 sections — seem instead like a new, vast world. So, he started trying to map out how the rooms connected to each other, but it didn't seem to function like a normal 2D — or 3D — world. I recommend that you read his own documentation about him discovering this, but he eventually came to the incredible insight that the maze wasn't easily translating to a 2D map because it wasn't a 2D maze. It was, as he calls it, the Starmaze — a maze that exists on a ninth-dimensional hypercube.
Once math became involved, my interest, which was already quite high, managed to double. I won't spoil everything here, as that defeats the purpose of Cartan's wonderfully crafted pages, but I can tell you that he writes about every step of his discovery in a greatly intriguing way. There's lots of material to get through, and, like the Starmaze, I didn't go through it in a linear way. The site's pages are connected through hyperlinks spread across Cartan's writing, so the viewer can directly read more about whichever parts of the story interest them. Because of this, I spent hours reading the parts that fascinated me most, and still hadn't covered everything. I covered enough at this point, though, to grow a fascination for higher dimensions. I was especially inspired by his 3D rendition of the 9D puzzle, as it made it feel like a tangible world. Of course, I had to learn as much as I could.
A portion of John Cartan's 3D map of the Starmaze
As a person who loves games and math, the Starmaze was a perfect challenge for me. I wanted to be able to understand and visualize it, so I got started learning about the next dimension up — the fourth dimension. Now, everyone who I talk to about this tells me to read Flatland, and I know, and I should, and I will. Instead though, at this time, my father gave me the book The Fourth Dimension by the great writer and professor Rudy Rucker. I also got a copy of Fantasia Mathematica for Christmas since I asked for it after reading about a story in it on Cartan's website. Much later, Cartan would eventually tell me that "There is a long history of mystics who have gotten hooked on higher dimensions and maybe spent a little more time on it than they should have. To an actual mathematician, a 9-dimensional hypercube is ordinary to the point of being trivial or mundane. But if they are honest, and remember why they became mathematicians in the first place, they will admit to experiencing the same wonder that you and I have felt contemplating things just beyond our reach."
He would be correct — at this point in time, I was most definitely experiencing great amounts of wonder towards the Starmaze and towards higher dimensions in general.
School — before the pandemic — always took up several hours of my weekdays. Of course, I appreciate my education, and I work as hard as I can, but there will always be times where I've finished my work or my exam and can't use my phone or leave until everyone else has finished. This happened weekly, if not daily. After reading about the Starmaze, that was all I could ever think about during these times of sitting in a soundless classroom in the middle of the day. So, I decided to use these times productively. On his website, Cartan describes a mathematical triangle that allows you to figure out how many lower-dimensional cubes are in a higher-dimensional cube. For example, a 3D cube has 6 square (2D cube) faces and 12 line segments (1D cubes). The triangle could tell you that. It could also tell you things you don't know, like that a 14th-dimensional hypercube contains 1,025,024 5th-dimensional hypercubes. Cartan calls this "Cartan's triangle," and has one available on his website to see, but I know from my own experience that I only absorb so much information through visuals. So, I decided to handwrite my own in my math notebook at school, which I think was a good way to spend my time. I enjoyed making it much more than I'd enjoy anything else I had access to in that environment. And by making it, I was actually able to find patterns and surprise myself. It also gave me an excuse to ramble on to any poor kid walking by who happened to ask, "What's that?" Some of those kids have come back later to talk more, though, after they too got interested and started researching higher dimensions, so I regret nothing.
My Cartan's Triangle
Even after I finished my Cartan's triangle, I wasn't satisfied. I was really happy, of course, but I wanted to create more and I wanted to learn more. Then, I thought of something that made me surprise myself. If you'll recall, I mentioned before that Cartan found that it wasn't exactly easy to map the Starmaze onto a 2D space. Again, I recommend that you read his own documentation of this, but he eventually did figure out a way to do it. He created a couple of rules and a great system to translate the 9D maze into the second-dimension, and it was really beautiful looking. Using simple red and blue lines, you can traverse this hypercube several dimensions higher than we could ever really imagine. So in 1987, Cartan printed out the map on a scale that took up 16 pieces of paper put together, and he spent days meticulously coloring in each and every line. Now that my triangle was done, I could think of one other thing I could do — I could make my own starmap.
John Cartan's starmap
Although an image of the starmap like the one that Cartan had himself wasn't available on his site, he did still have a different image of the starmap available that was used to show the solution. I was happy with it, so I printed it out. I didn't have the courage to construct it on the scale of 16 pieces of paper like Cartan did, so I settled with the scale of four pieces of paper. Next, the red and blue lines were essential, so it was my job now to color in those lines. Unlike the original starmap, though, the map I printed out was black with white edges, meaning normal markers or pencils wouldn't show up on it. So, I decided to use red and blue gel pens to color in the lines over the darkness. It was a Friday night and my parents had just left town for the weekend, and I didn't have plans myself, so I figured that I could finish it by Sunday evening.
This was very, very incorrect.
Using a guide on Cartan's site, I spent hours coloring in the map that weekend, and by the end, I was not halfway done. I had to go back to school again on Monday, but I would devote a couple of hours the next weekend to coloring in the map, and then again the next weekend. In the end, as I had worked, I'd managed to listen to 9 hour-long episodes of my favorite podcast, the entire audiobook of The Great Gatsby, lots of music, and more before I was finished. The months it took me to finish my 4-page map gave me great new respect for Cartan coloring in his 16-page map. At one point, I had to order new pens to finish, as mine had run out of ink from the coloring. But in the end, this tracing activity taught me more about the maze than I ever could have gotten from just staring at the map. There were many patterns and rules and symmetries, and I understood how the map worked in the second-dimension now. I would be slightly surprised when Cartan would later tell me that besides me and him, there was no one else who had printed and colored the starmap, to his knowledge.
My starmap
I don't think I'll ever be able to fully wrap my head around the Starmaze, but I am always grateful that I stumbled across it one night as I clicked through links and pages. It's inspired my thoughts and given me a whole new interest (and new reading material). It's brought me closer to ones who know more about higher dimensions than I do, and it's given me a really cool poster to hang next to my desk. And maybe, one day, if the universe collapses and we get swallowed into a ninth-dimensional hypercube, I can put my geographical knowledge into the new society we create. Thanks for everything, John Cartan.
