“Idiotic” because I wake up at 6am thinking these things, get all excited I am exploring a new space, and when I run a search under “reimann sphere mandelbrot set”, it throws up this blog, which demonstrates a real mathematician — and Chris is only a teenager! He’s actually playing with maths, actually got the mental plasticity and some computing skills to be able to explore it. His posts on education are sharp, and he’s doing things I have barely thought, regarding cyclic derivatives. Incredible! I love it! And yet, when I reflect on my night’s mindflow, I can not help but feel like a denk. So much of what I do is intutive, with very little backing in terms of real maths. Same was true I guess with my peace projections, based on so little evidence in my classes. Still… given the apology, here are the four or five snap shots in the area of investigation.

First, mapping actuality. I’ve been approaching this a few times over the last two years. Representing an actual object, and mind object. My first exploration while writing XQ Conditional in 2008 was that mind was in the negative space. It seems like a sensible… convention? But it did derive from thinking about subtraction and the mind’s capacity to filter, as well as imagine things not there. Earlier, as a teacher, holding up an empty hand and saying to the students, “I am holding up two oranges, exactly two oranges,” as a way to teach negative numbers, and asking where these “not oranges” were. So, here is the current mapping, or at least the mapping of last night:

positive numbers — discrete aspects of Actuality (actually undifferentiated from everything else), eg this cat here (3), that table there (eg 7)

negative numbers — the sense-object of mind (sense, married to an internally comprehensible meaning unit, possibly perceptual node, found in animal mind, me thinks), eg this cat-shaped perception (-3), that table-shaped perceptual object (eg-7)

imaginary number i — the linguistic label attached to the mental object or thing out there, eg the word “cat” (3i), the word “table” (3i)

imaginary number -i — the linguistic label attached to a mental object that does not exist, eg the word “darkness” (-1i), or “peace” (-9i)

To construct this, I see I have bodged meanings of negative as being not, as well as opposite, especially in the interpretation of negative i. That is, darkness as an absence of light, as opposed to a thing in itself. And yet we find ourselves thinking, the darkness spreading. That is, purely imaginary objects of the mind. So, in this way, it is the same relationship between a thing out there and the mind (positive and negative 1), and a word and the concept it applies to (positive and negative i).

Interesting consequences of thinking this. Multiplication by -1 has the effect of mapping actuality to the sense-object, of word to concept, and the converse. Multiplication. The number 4, and it’s two roots +2 and -2,

(An aside: roots, as nudged by reading Arthur Young, can be found in the complex plane too. But just sticking to positive and negative roots of eg 9, we get +3 and -3. Notice, not +3 OR -3. They are both roots. And then it came to me. +3 x +3 is +9. -3 x -3 is +9. Same time? Two branches. But in terms of a specific case, most physics, they look only at the positive solutions, because these numbers refer to things… And… +3 x -3 is -9… which suggest to me that the root of -1 is +1 and -1, not i. Or rather, it gives weight to the notion that i is both positive and negative, the elevated station, the both-and of buddhist logic, that which is above happiness and sorrow, or true happiness. I know… this sounds odd: I am mixing too much up, but we are mapping not only objects, but subjective entities, so we are entitled to metaphor to whatever field in our heads… the buddhist space is at least minimal.)

Second, map this onto the Riemann sphere. What does this do to your head? It does funny things to mine. Honestly, if are getting this far, you might as well enjoy the sights from your own mind. (Who am I fooling, no-one is on this path. Not no-one. I am enjoying the vista of a new land unseen by human eyes. Or if it has been seen, such individuals have never returned to humanity… they’ve gone feral, mentally. I wonder how many loons populate our university math departments…? I suspect most do not step off the well-worn tracks at the regular stations, after all, who questions addition, or even counting?). For some reason I think of the riemann sphere as being a thing, not just a surface, a single point manifold, but a substance. If the numbers are the surface, what on earth — or in mind — is mapped inside the surface? Some odd answers pop out, like words. That is, the thin film that is the number sphere is the minimal language set, and so what exists outside and inside that layer are words; perhaps only outside… words lose integrity at a certain level of internal, subjective, depth.

Also, what constitutes the central axis of the riemann sphere, in maths, and in our mapped version? It is like taking zero and pulling it apart into positive and negative numbers, thus creating the space between 1 and -1. Same goes for i and -i, thus creating a 3d space within the equator. Interesting. And then considering the spindle around which this spins — and that is why the riemann sphere is useful, because it relates the arithmetical operations to be related to geometric transformations. Multiplying by i rotates the sphere through π/2 I think. Astounding. Rotation around what? And for some reason, another answer pops into my head, and alternative — time. Because nowhere in this, do we have time, yet.

(Another aside: riemann sphere is a trick to “positivise” all the numbers. Look at the sphere. The bottom is zero. That means that the entire thing above it suggests positivity. In the same way we often represent all our graphs of economics and the world falling apart in the positive quadrant. The point is, it jars with how I usually conceive of a sphere, with zero in the middle. Is this why it is valuable? This contortion gets maths back in line with what I consider to be a mistake of considering negative as opposite, that we should derive a maths which considers -5 as being “not 5”? Who knows… another thread trails its way into the dark. I feel like an old man wandering a vast hall with only a candle to light the way.).

Third, somehow relate the riemann sphere to a torus. This involved quite a few mental mappings. From the second mapping, if we consider the central dimension to be time, then another part of our brain comes up with time cones, which is something I related to being the centre of a torus (with the hole of the torus being a point).

How do you go about relating the time cone shape to the riemann sphere shape? And what comes to mind is that the riemann sphere is rotating. The rotation is the funny thing that makes consciousness stable. We are out on a limb here, but remember I was thinking this while in bed in the early hours and the mind has more freedom to think the absurd. So, consider the riemann sphere — hmm, can’t manage that, just the equator bit, -1, i, 1, -i, as the cross-section of a torus, the centre of which is a point. Which makes one wonder what the dimensionality represents. The upper part of the torus is a circle of i, the outer part a circle of +1, the inner bit the circle of -1, and the lower bit a circle of -i. But this is rotating, like a standing wave, in time.

Ho hum. I am not sure. None of it. But when it gets this far… It’s to do with simulating a moving thing. Most of maths is about capturing a dynamic thing in a fixed thing. Hence, mapping an object’s trajectory on graphing paper, deriving change of change of position, ie acceleration. I’d love to get my hands on modelling software — or better, a person who can actually manipulate, has facility, manipulating computer images.

(Further aside: computers are entirely positive. They have bits which represent negative, etc. They have to simulate “negative”. What does negative mean in terms of computer engineering? Again, pointing at purely positive computation. With negative just being a type of number. Then again, number as computing thing??)

Fourth, mapping the mandelbrot set onto the riemann sphere. This should be possible, but I haven’t been able to find a map of it yet. Looks interesting, in my mind’s eye. And I have always thought of the mandelbrot set to be a bit thin. I mean, 2-d…? And as far as I can guess, the magnification we usually apply to “zooming in” on the mandelbrot set, is actually a magnification of the universe outside of the riemann sphere, from an internal point of projection. That is, being inside the riemann sphere, looking at the mandelbrot set on the surface of the sphere, and then looking more closely at the point at -1, for example, and magnifying the space where the first bulb meets the core base of the mandelbrot set which happens to run along the equator, at least on the negative side. And why doesn’t the mandelbrot set colour as a complex number tends to zero? It usually only shows maps where it colours how fast a point escapes as it tends to infinity.

All of this, these snapshots, somehow come together in a nice way. They do. Just now, they are snapshots. It is all my feeble brain can come up with. I will never be able to pull it together. I don’t have the mathematical precision, nor fine awareness of mind. But a team of people might.