mandelbrot set as addition squared

15 05 2013

I think this makes sense.

disappearing dimensionality

I’ve been staring at this for a while. Looking at how 1 has the same value as 1 squared and has the same value as 1 cubed, etc.

1^n = 1

At least in terms of integers. IF we ignore the dimensionality of it. But that’s what’s annoying me. Since, obviously, a line of length one is different from a square which is different from a cube. 1^billion is a pretty dense multidimensional space. But when we just look at the integer value, it is just one.

When we multiply it, even by a tiny amount, the power function takes our number to zero if lower than one, or infinity if higher than one — both at an incremental rate.

eg x2: 1, 2, 4, 8, 16…

eg x1/2: 0.5, 0.25, 0.125…

But look at what is happening. I can understand that if I double a length, I get four times the area, eight times the volume. When we half the length, we get a quarter of the area, and eight of the volume. But don’t you think that an eight of the volume of anything has got to be bigger than half a line? Surely half a line is smaller in some way than a quarter of a square?

When we pull the same trick of forgetting about the dimensionality, by reducing the multiplier, we simply get smaller integers.

I sit and stare at things like this for a while. I stay awake in bed, thinking about this.

mandelbrot set

The Mandelbrot set is defined by the following iterative function:

z -> z^2 + c

It should read z_subscript_n+1 = z_subscript_n_^2 + c. That is, the next value of z will be the last one squared with c added to it.

I re-examined this, because it is somehow related to the mystery of disappearing dimensionality.

Let’s take c to be 1 (on the complex plain, c = 1 +0i).

We assume that we start with z = 0, so our first z = 0 + 1. Actually I am not sure if this is the first, or not. But after this, it get’s easy.

first z = 0^2 + 1 = 1

second z = 1^1 + 1 = 2

third z = 2^2 + 1 = 5

fourth z = 5^2 + 1 = 26

And so on… rapidly escaping to infinity. Which is why we don’t see much around the point (1,0) on mandelbrot set.

what’s the function again?

What are we actually doing in the iterative function that generates the Mandelbrot set?

It looks like we square then add to get each new version of z. This is how the transformation is written. But is there another way of looking at it?

Well, we add c (the original number) to a square of the previous addition.

The original script emphasises the function of squaring then adding. But in iteration, it doesn’t really matter what order it is, since it is squaring and adding, squaring and adding, squaring and adding and so on.

We could just as easily say that we are adding and squaring, adding and squaring, and so on.

This gives a slightly different sense, and can be written as follows:

z_subscript_n+1 -> (z_subscript_n + c)^2

The numbers that come out of this transformation are:

first z = (0 + 1)^2 = 1

second z = (1 + 1)^2 = 4

third z = (4 + 1)^2 = 25

fourth z = (25 + 1)^2 = etc

Notice our answers from the other way of parsing the operation are hidden in our calculations within the brackets, ie 1, 2, 5, 26…

So what, you might say.

Well,  the Mandelbrot set is the set of complex numbers that do not escape to infinity, and gives us the nice shape on the complex plain. We focus on the static numbers. Does 1 escape? Yes, so it is not in the set. Does 1+i escape? Yes, and so on.

Mandel_zoom_00_mandelbrot_set

However, in our alternative way of parsing the iterative function, we see that it is not the static number which is iterated, but the addition. It is the operation of ‘addition’ that is being squared repeatedly. Without that little addition being iterated, we wouldn’t get the fractal quality.

This is how the fractal is created. Without the addition, we simply get a unit circle. I think.

conclusion

There isn’t much of a conclusion in these explorations. Perhaps only the pleasure of coming up with an alternative interpretation of something I have been looking at for the last twenty-five years. That’s a buzz in itself. And the fact I don’t think anyone else on the planet is poking around this material in this way. Applying XQ scrutiny to math.

The implications are… well I don’t know. We will see if there are any.

It’s like teasing apart a structure, and by doing so, a little at a time, over the years, we have enough flexibility, enough alternatives, that new associations are made that depart from a path that math has taken us historically.

What would I like to see?

Well, there is something about dimensionality appearing in the Mandelbrot set. The increased detail, with a Mandelbrot zoom, indicates something about dimension. And this zoom is performed by increasing the accuracy of the initial number, the number of decimal places. What appears to be in the Mandelbrot set at 0.5 + 0.3i or whatever, turns out to be actually not in it at 0.55 + 0.30i. The precision, gives us the interesting boundary.

This fractal boundary is in the depth of the number of decimal places. There is uncertainty there. We don’t know after 1,000 decimal points for some points, whether it will be in the Mandelbrot Set or not. This is remarkable. It is like saying there is depth between 0.5 and 0.6, and it is not regular. Peaks and troughs of depth. Between numbers.

I am more interested in the movement through a Mandelbrot Set, rather than a straight dive, which I shall cover in another post. Movement beyond arithmetic iteration, or scale. Movement ‘across’ the set. I am thinking about the bulbs on the set, and the ravines.

The Mandelbrot set is too… static. I think there is something that we may find that is dynamic in it. And I would like to tease this out.

Why?

Because I think it is related to the shape of consciousness.

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therapeutic math

30 04 2013

Recently, I have had the opportunity of experiencing hell. My previous offerings (2020worldpeace, eco^2) seemed abstract or detached, but they weren’t, they just weren’t down and dirty with emotional problems. Now, I might be able to provide a means which may actually prove directly useful in our daily lives. Transforming negative emotional states, as well as dealing with psycho-active agents which cause us so much internal and social turmoil.

When I first explored XQ way back in 2008 in Thailand, I wrote a section which suggested that if the premise is correct — that there is a subjective side to math — then the act of performing certain math actually performs certain processes in the mind which may be useful to us within our internal mental space. I think I am approaching a time where this exploration is now possible. Surviving hell has its benefits.

Unknown

I have finished (messily) three movements of a book I am writing called GIFT, and approaching the final movement. As with all my books, there is always something dramatic at the end. I set up up the book as I write, and create a mental space at the end which is entirely empty, hoping that inspiration will provide a fitting conclusion. Not so much a logical conclusion, but an emergent one.

With GIFT, the narrative is about an older man who is living in a period of time where society is approaching a massive global transformation, of which he is part. I do not know how this is to be written, the content, the drama. I do not know what format, first person, third, whether to write more dialogue, or to simply describe. I simply do not know.

However, because of my personal relationship, the hell I have been going through, I have undergone a form of transformation, and though math has been partly responsible, it is not entirely clear. I have had some projections into the verbal field, noticing how our mental environment matches our ecological one, or how self-denial matches our social-denial. But when I start to describe them, because of their nature, they tend to multiply in word and story. Hence, the desire to capture it in concise mathematical form.

Funnily enough, I started to write an article which goes into multiplying by negative one as well as multiplying by i, the square root of negative one, but I thought this material was too much for my 2020worldwalk blogpost and transferred it here. I continued writing it, but got bogged down in detail and have not returned to it yet.

I simply wanted to write this post to indicate where I am at the moment. Midway between a social, verbal description, and an XQ mathematical description.

XQ — a rigorous path to personal and social happiness!

And if I manage it, then not only will it align to my current mental trajectory, but it will fulfil my intuition of a therapeutic maths, as well as provide people with direct testable material which will not only improve the quality of their own lives, but will naturally lead to us all improving the quality of our lives collectively, globally.

images

But these are just words, not fitting for this blog. What is needed is math.





difference over time

3 02 2012

The late great Gregory Bateson drew attention to the simple equivalence:

What he meant was, if you close your eyes, and run your finger over a crack, you can sense the crack only because your finger is moving. If it is still, whether on the crack or not, there is no differential, and as such, there is no sense — you can’t tell whether your finger is on the crack or not.

It has made my mind go in a few explorative directions, some of which may test the reader’s ability to conceive as it may transgress certain conceptual boundaries.

Consider:


That is, value is an estimation of the difference between state of mind at time t and at t+1. This subjectively derived enumeration is a measure of experience. In a crude way, it could be used to determine the learning experience, how much I have learned over a period of time.


That is, the experiences of person A and B are different, given the same time. There is something multiplicative about this. Two different experiences, or indeed as many experiences are there are people, of the same time.

Combine this with the squaring observation above, and consider a group of seven people experiencing the same “meeting”, and the reflective evaluations of each other’s experience. Things get out of hand rather quickly, and not simply in terms of the mathematical number of handshakes (7! is it?), but in terms of the psycho-dynamics. There are orders of complexity here.

Which leads to a rather intuitive leap to map it against buddhist “piles” or “heaps”, which I have considered to be emergent levels of complex organisation:

form inorganic matter .
sense ∆ form awareness of form
perception ∆ sense awareness of change of form, and yet maintenance of “form identity”
volition ∆ perception awareness of change of sense, “making sense”
consciousness ∆ volition awareness of change of perception, continuity of “perceptual identities”
“social”? ∆ consciousness awareness of change of consciousness, true acknowledgement of equivalent “other”

I’ve no idea how useful this is. And it is too linear for my liking. Bees and ants, for example, seem to suggest to me that their “perception” may be embodied at a collective level rather than at the individual. It is too easy to think of hierarchy here, of progress. The buddhists give clear warning to the illusion of our projective mind, the thought center, and seem to seat consciousness at a lower level, heart-mind. Having the simplicity of an animal, a self-aware animal, rather than an animal that runs after mental objects and thinks of themselves as something different.

This kind of thinking might enter into wise^0 territory. Just wanted to include this in order to exercise the possibility that any mathematician will tend to import math that has been applied to the world of objects, whereas there may be a different form of maths, or application of it, that is better suited for the exploration, and perhaps modelling, of consciousness and social dynamics. With this in mind, the table above suggests rates of change, eg

Which is simply reminiscent of acceleration, as d/t^2. And if you find your mind responding sceptically, simply consider your experience of music. Distinctions in time. And the different layers of complexity. It is in this are, subjective experience, that math needs to evolve a simplicity and subtlety that is lacking in its application to physical systems.

Can we allow ourselves to even think this…?





the maths of uncertainty

3 02 2012

This is definitely peculiar. Absolutely no idea how useful it might be. I used to be a math teacher, and only recently did I realise that an equation is not read like a normal sentence. To keep things simple, consider the following expression, or sum:

Any school kid will tell you, you don’t start out from the left, like it was a sentence, with eg 2×3. You follow the procedure, BODMAS, in order to unpack the expression.

Consider what this means in time. The procedure of BODMAS sets the order by which the equation is unpacked in time. The mental sequence of processes. So, in the above expression, you complete (4+7) and (6-3) simultaneously, then multiply them respectively by 3 and 2. and then you subtract the second from the first, and then multiply the whole lot by 2, which gives 54. Or I guess you can multiply all the brackets and then simply sum them up. Whichever way you do it, the answer is the same, 54.

Now consider the following statement:


If we consider it from left to right, we get (10+4) then multiply by 3, which gives 42. Or, if we follow our usual BODMAS protocol, we do the multiplication first (4×3) and then add 10, the correct answer.

But what would a maths look like that holds both correct, at the same time?


That is, where both answers, 42 and 22, were correct?

And if this doesn’t strike you as being particularly useful, you are in popular company. However the following expression does pique a curious line of inquiry:


Which derives a possible simultaneous answer of 0 and 1. And because the digit in the middle can be a 1 or a 0, it might be more clearly written as:


And if we multiply this expression by itself:

And the answer to this is the truth table for the AND function:

AND (x)

0

1

0

0

0

1

0

1

And if we add it to itself:

We derive the boolean OR function:

OR (+)

0

1

0

0

1

1

1

1

This is not entirely unsurprising, since our original expression simply changes the order of the operations, thus producing the effect of sequenced logic gates.

What is interesting, at least notionally, is that this is contained in a single arithmetical expression. I know that variables can perform this function in maths, and set theory, but to have it in a simple sum, is… peculiar. And I can’t help but relate this to chaos equations, which after a suitable number of iterations, tend to a constant, or two constants, or a set of constants. That, and qubits.





discrete numbers

9 10 2011

Had more of an idea about this a few days ago, but never got down to writing it. So, here is the vestiges of it.

The very act of numbering cuts up the world. Two as distinct from three. It is less a distinction, between this and that, and more of a boundary. Drawing a boundary around a thing, much in the same way a word might be used to package a thing perceived.

The structuralist might necessitate a distinction, of opposites. And there is, of course, a difference between two and three apples.

I’ve already remarked on sameness as being important for counting, or multiplying. There was something in here about multiplication…

3×7       7×3

Three times seven. Three groups of seven. Seven times three. Seven groups of three. Quite different descriptions of reality, or indeed different situations in reality. When calculated to the “answer”, 21, there is a further loss of information. Three boxes of matches with 7 matches in each, versus seven boxes of matches with 3 matches in each… same number of matches, different numbers of boxes.

It wasn’t this… it was a different tack. I just can’t remember… it was something very very simple about this notion of discrete numbers. What this means in terms of our mind’s processing. What the last observation seems to be about, is our mapping of number to “things”, discrete mental objects that might match some material situation, eg matches and boxes. This is happening at a slightly… later, or higher… aspect of consciousness; what comes first, is the discrete mapping of number like a word to a thing. One, two, three…

Wait a mo — another thought. Counting is one, two, three… the pointing at a new thing and including the old… two includes one, three includes two and one. Whereas, there is the whole-image form of counting (can’t remember the correct name for this… it’s not “counting”), where eg 7 things are immediately recognised as seven things even if the perceiver doesn’t have a word for “seven”. Hmmm three as in third…

For some reason, this makes me come round to music. Thinking about counting in time. Counting objects, I think, comes later. It’s a combination of this pattern matching, immediately taking in a pattern, and noticing a pattern in time. Combine those, and you get counting things.

But this is miles away from my initial thought about discrete numbers.





positive and negative infinity

5 10 2011

Just noticed, perhaps for the first time, I can’t remember, that Reimann’s trick was to conflate positive and negative infinity to the same point. This is quite remarkable. This actually connects in my head to the higher-dimensional twist, the mobius loop in time, that I intuit is going on with consciousness. Ho hum, but there you have it.

It is strange. Start with zero, then the number line going off to infinity to the right along the positive axis, and going off to infinity to the left along the negative axis. And strangely, these two opposite directions meet at the same point.

OMG, as I write this, I am amazed I haven’t seen this. No I am amazed that Reimann saw this. I think this is what he saw.

It’s actually pretty simple. In terms of fairly standard human, enlightenment thinking, we are standing on the planet, let’s say at the equator. You point off east with your right hand, and west with your left hand, and sure enough, if you follow your pointing around the globe, you are actually pointing to a thing at the opposite side of the planet. And indeed, you could point through this and continue until you are pointing back along the opposite direction: your right finger pointing eventually meets the end of your left finger pointing in the opposite direction. In a way, you are pointing at yourself.

But this is predicated on the curvature of the earth. Now imagine this pointing is not bending. So you are standing in space. You point off in one direction, and you point in the other. You point off to infinity. The only way this could possibly make sense, is if you think there is a similar thing going on with the universe. That your right pointing eventually ends up coming back to meet your left pointing. This may or may not be the case with the universe. But what Reimann does, is suggest that it is, in effect. That is, be bends infinity. That is, infinity is not this endless thing, but converges. That is, he captures our ability to conceive of infinity. Or, more prosaically, our ability to label it, “infinity”.

We have taken a step from received understanding, I suspect. We are definitely performing within XQ space here now.

It is not so much that the infinity used in the reimann sphere has anything to do with existence, with the physical universe. It is to do with our human immersion in it. But not so much our physical embodiment, but our subjective orientation within it. Eg, left and right.

Returning to the math of Reimann, the strange thing is, positive and negative infinity meet at the same point. If you visualise this, it is a simple circle, as we have circumscribing the equator as in our initial thought experiment. This is ok if we are thinking of planets which are curved. This is a bit stranger if we are talking about subjective qualities, eg good things and bad things. A super extreme bad thing ends up meeting a super extreme good thing. The more they are apart, the more they converge on the same point.

It may be a mistake to think of this as a point, but it is definitely done with the Reimann sphere. Anyhoo, however you think about it, the reimann sphere is useful because it allowed mathematicians to map arithmetical functions to geometric functions. Multiplying by 1 and -1 and i and -i result in rotations of the Reimann sphere.

What I was trying to get around my head is how this is related to my notion of o being the centre of the individual and 1 representing the individual. I sometimes play with 1 as being the human, and in the Reimann sphere, that would be the equator delineated by 1, i, -1, and -i. I’d like to square this (might have to be careful with my language here;) with notions of internal states, which would be represented by fractions as aspects of being approach zero, or negative numbers as the mind’s reflection of what is, which takes zero as the centre point of the individual.

There’s something like a torus floating around here somewhere, but I can’t find it. Three dimensions of circularity, versus the two of a circle. Not sure if this is correct. Torus can be created from rotations of two circles of different scale. A sphere can be created by a single circle rotated around one axis. That is, a two dimensional shape rotate through the third dimension. A torus can be created by created by rotating a circle through a disconnected axis through the third dimension. So they are the same, except for the connection/intersection with the axis. That is, whether the zero is inside (circle, sphere) or outside (circle, torus). Interestingly, as far as I can conduct the transformations in my head, the first transformation can be conducted with a 180˚ turn, while the second requires a complete 360˚ rotation to complete the torus. But these are artificial means of producing torus, whereas I am more interested in how they form in smoke, for example, or dolphins producing air rings in water.

This was all sparked when I started to thinking of the mathematics of emergence. First spread consisted of systems, circularity, fractal in time, simultaneously iterative, Second spread has one equation/function/expression that simultaneously operates on several different processes which have different periods in time; which also may map to presence, mental, emotional, physical as basic levels of being. Third spread jumped to 0 = 1 – 1, which could be translated as

0 as the centre of the equation/function/expression, if it is to represent consciousness

0 or 1 and -1, if we wish to explode 0 into duality of mind

0 or 1 and not 1, which is kind of a description of the duality of mind

which leads to

e^iπ -1 = 0, euler’s equation

which means that

e^iπ = 1

and this has something to do with period, if i remember correctly, and the Reimann sphere.

There’s a lot in here. A lot. The revelation for today remains, that the positive and negative infinities meet at the same point, at least can in terms of maths. And if we are to take an XQ interpretation, this means the Reimann sphere is more a description of how we bend subjective thought space, eg the notion of infinity, to a well behaved point, eg the word “infinity”.

Somewhere along here is the math of emergent systems. It is do with nested systems in time.

And interestingly, I noted as I started out that my mind approached this from consciousness outwards. I started with zero, and then attempted to derive the other numbers, and found myself with euler’s rule, and thereby to Reimann’s sphere. That is, it is not about trying to work it out from the details, and integrating some kind of theoretical sense, but it is deriving details from some simple starting point. Kinda like the buddhist methodology, and like einsteinian physicists working it out from insight, first principles, and so on. This is quite promising. It suggests, that it is about appreciating simplicity, and from this derive all kinds of complexity. Can we intuit the field equations for consciousness?

Another thing to note, is that I am making progress, albeit slowly, on the notion that we may need a different form of maths. I am not sure about this at all, but it is simply a hunch when appreciating the invention of calculus by newton. A new maths had to be invented to capture the mapping of physical objects, that of functions. A new maths may be necessary to capture the mapping of mental objects. My mind might also have met with some encouragement when I read somewhere recently that the whole path of functions, the entire realm of maths, is limited in some way. Sadly, I can’t remember where I read this, heh. I shall have to wait until it pops up on the radar again, and again almost by random. Still, it will give me plenty of time to explore/prepare other areas.

ADDENDUM: after going back to include the image of the  Reimann sphere

What if the infinity point was the asymptote I keep thinking that mind is, not zero? And zero is the touching point with existence, the flat plane? I know this doesn’t make any sense, but all the numbers are contained in the sphere, and the plane below the sphere is just an illusion. Well, it is all an illusion. No, that’s not quite it. I keep getting fooled into thinking the  Reimann sphere is an object in space, heh, and I have thought about what is inside the Reimann sphere, for example, or outside it. Nope, the extra thought wasn’t this.

It was something about rotating the sphere through one of the axes, eg the real axis. Not the axis that is wrapped around the Reimann sphere, but the axis of the plane surface below. This should create a torus, which touches at zero, or rather does not touch at zero. But all mind is capable of is not really rotating the sphere but the great circle, marked by o, -i, infinity, i, 0, if we are rotating around the real axis. To rotate a whole sphere through space leaves what exactly? And an empty sphere at that?

Ho hum.





mapping social dynamics

26 09 2011

OK, when I wrote XQ Conditional, I made some faultering steps on mathematising consciousness. Very, very basic steps. Nowhere near an equation in sight. For some reason, as I approach the end of my fling with a Facebook community, The Next Edge, thoughts have occured regarding social dynamics. Perhaps spurred by this two hour video on Artificial General Intelligence, which to me holds some rather scary potential. Our system understanding of social emergence is pants in comparison.

Remember, this is after Wisdom, where the psycho-social concept actually revealed itself to be two different systems: the system going on in our heads, and the system going on out there. The out-there bit is predicated on our action, and consists of social objects. The in-here bit is predicated on our thought, beliefs, and so on. What we have in-here partially determines what happen out-there, but it doesn’t really matter what we think is going on.

Here’s a couple of thoughts regarding rates of change.

We are all on a continuum of how much change we think is possible. The more we are engaged with the current system, the more fixed we are about what can and can not happen. The top dog is essentially fixed in position, responsible to maintain it. The further out we are, the more flexibility in our thinking, the more we can see the bigger picture perhaps, unbounded by any specific institutional directive. There is a relativity of sorts going on here, socially constructed.

Things get ugly when two people from different systems engage, each embedded in their own. This is just about understandable for two companies competing in the same sector or domain, what about different companies from different silos? What about job system versus a governmental system versus the natural world system? Embedded systems. Messy.

Can’t be bothered writing this in html… gotta wait until I am off the ipad…. jeeez, the hassles…








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