I just now, minutes ago, was drifting between awake and asleep. slowly becoming aware of the lightly falling rain outside, while also in a half-dream state, contemplating random mathematical ideas, when I caught a glimpse of the 1/2 which completes the sphere of infinity. I have glimpsed it intellectually many times, but this is the first time I've seen it intuitively, actually seen the shape and how it relates to a sphere and a circle and a 1/2.
The idea moved like lightning, not for its brilliance, but for its speed. Gone immediately. I was contemplating a sphere, but I was looking through the top downward and studying its bottom side for a brief moment. The idea disappeared and became irretrievable right away, but I saw it! I saw the 1/2, realized a moment later what had just happened, and tried to reconstruct the thread of random ideas which led to it appearing in my mind. Too late. All I had were fragments I could no longer name, but I saw it, yes I did!
This means I will see it again. It may be hours, weeks, months, or years before my subconscious brings it around again, as surely it is elusive, like a unicorn for how hard it is to find. If I am right, it has taken centuries of effort to see, and I'm only the latest one to glimpse it, and Gauss and Riemann and many others preceded me in the line of thinkers contemplating it.
In my attempts to reconstruct it (while still half awake), I soon came upon 1/3, thirds, and the p-adics. Then I remembered a passage I read a few days ago which opened up a new world of insight, having to do with the state of the art in mathematics a little over a century ago when Wittgenstein was forming his key insight into language, the "Grundesgedank" I think it was called, which led to him writing the Tractatus Logico-Philosophicus. (Someone wrote a book about the path of his discovery, which included the work Gauss did to break open a centuries-old riddle on the nature of quartering the circle... hold on, I've got the book right here, I'll find it and show you the passage...)
No, wait.
Before I do that, let me finish circumambulating the tergiversation, so that I can include the clues which are tumbling around in my subconscious right now, before the moment passes. I was looking through a transparent sphere. I think I was drawing a circle, or closing a sphere, which results in a circular shape getting smaller and smaller. Maybe. Then I saw that, to complete the idea, I needed to make a 1/2, a perfect bisection of a circle, it seemed. It was a leap across a chasm, of sorts. But it was graceful, not awkward, as a leap would be. And even as I write these words, it's not feeling just right, so I think the idea has gotten away, but some of it is right.
Trying to recover the idea, I contemplated other ways of dividing a circle, having just discovered that the 1/4 that Gauss used when inventing imaginary numbers was, in fact, 1/4 (that is the insight from the passage in the Wittgenstein book), and that it was arbitrary, that it could just as easily be 1/3.... or even the p-adics.
If it were the p-adics, would it be an infinite series, not a static bisection -- a leap -- in the way that 1/2 works, but a continuous motion over time?
I was trying to recapture the idea. Putting similarish ideas into my imagination in hopes of triggering the unicorn sighting again, I imagined looking downward, through the center of an invisible, spherical torus, which represents the shape of the universe, and seeing its motion. I made the flow go upward through the center, as I have seen in a few vortex-math-related illustrations, and wondered about it, but could not by these means re-connect to the beautiful, intuitive, 1/2 that I had glimpsed a few moments earlier.
My full consciousness finally realized what I had just seen. I was very excited, and turned immediately and began writing these words (I had fallen asleep before a computer). I'll fish around for that passage in the Wittgenstein book and add it as an illustration to these words...
Hold on, the plot thickens. While writing, I heard my daughter dreamily awaken, and her first words were "I've just had the most interesting dream!" So when I stopped, I turned and asked her what the dream was. At one point, it involved a circular ring of metal, which she was holding on to...
"We dug up the whole back yard 12 feet deep looking for a treasure, but all we found was sand. So we filled it back in again. Then there were three of us, and we were on a bridge over some water. I fell off the bridge and was holding onto a circular ring of metal for awhile, like a bar in a circle. Then I fell in, and [my sister] told you so you dived in and rescued me."
Very curious reference to a circular ring which my daughter dreamed at roughly the same time I was dreaming of a circle at the bottom end of a sphere...
Oh my, the plot is even thicker. While my daughter was telling her dream (as is her nature when telling dreams, she ambled around for a long time), my other daughter came in and said she had cool dream! We let her sister finish, and then she began telling her dream:
"We were in a Minecraft world, and we built a very high tower, that was so high, we built a ring of iron bars around it every few feet or so..."
"Wait, a ring of iron? Was it square, or round?"
"Square. But it had... two faucets. Each iron ring had two faucets, one on each side, pointed inward, and water was flowing from the faucets. The tower was very very tall."
So all three of us had a dream involving a roughly circular shape, two of them made of iron, two others involving a duality with two equal halves, all at approximately the same time?
Very curious indeed.
I asked for more details on the Minecraft version, and she went into Minecraft and created it to show me. Imagine a tower, a square shape 3 blocks across and deep and many blocks high. The central block is empty, leaving a hole through the center of the tower. About every five blocks upward, there is a ring of blocks surrounding the 3 of the tower, making it five blocks across for a single row. On top of this ring of blocks is an iron fence. Then continuing upward another five blocks, there is another such ring of blocks with an iron fence on top of it, and so on. She said there were about twenty of these, or even more. In the dream state it was possible for there to be two faucets on each ring of blocks, pointed inward, through which water was flowing toward the center.
So not only did she see a ring, but she also captured the 1/2 idea (two spigots, each being a half), as well as the flowing, and movement upward. This arrangement was surprisingly similar to the fragments of my dream, although hers was missing the outer sphere which would have contained the tower.
Now I'm eager to see the next stage of this insight.
[Update] While recounting the curiousity of this synchronicity to my two daughters a few days later, the youngest one corrected me, that it wasn't a "ring of iron," but a cylindrical bar of iron that was straight -- a rod of iron -- and several feet long. So the circle doesn't come through as clearly, but still. Interesting.
[Update] I see I never did find and include that passage about Gauss in the Wittgenstein book, which is pictured above.