|What seems to be happening in these ill formed areas is that there is a prominent minibot nearby which is robbing these neighboring sites, where you would usually expect to find a minibot, of sufficient coherence or symmetry to produce one. Instead the area is split into three dissimilar areas, and there is a much smaller minibot in the central region. This three-fold non-pattern is repeated as you go to smaller scales.
The canonical pattern may be seen near the minibots out in front of the Mandelbrot along the negative real axis. Each one is found at an apparent cross-roads with a fourfold symmetry. It is not a real cross-road because the filaments cannot intersect, they can only branch out. This has to do with the fact that the Mandelbrot set is simply connected. This is related to the fact that we are dealing with a flow system in 'complex dynamics' (q.v.). Your arteries never intersect, they only branch out. If you look along the 'crossing' filaments, you will see 'S' curves, with two main branches on each side. This is what gives rise to the threefold symmetry observed above. There is a confusion between the three and four-fold symmetry. The same confusion is seen in the threefold Mandelbrot, the cardioid portion having two halves, at the intersection of the real and imaginary axes. What I am coming to here is a rationalization for the Mb. Its generator, z' = z^2 + C has a quasi-fourfold symmetry. The generator would like to twist the Cartesian axes into an 'S' curve, but that is not allowed, and the result is the Mandelbrot. The cardioid is performing the function of the heart in attaching the arterial to the veinal system. Also, recall what I said above about the ritualization of the intersection of the sexes. All of this is being recapitulated chez minibot. A critic would say that I am using the Mandelbrot as a Rorschach. Perhaps it is a cosmic or zodiacal Rorschach.
It is said that the Mb is a connected set. That is true, but only in a technical sense. A primary connecting 'link' is a non-link. It is the ubiquitous infinite inward double spiral. The two arms of the spiral never meet, yet there is no topological way to separate them. It then becomes a mathematical convention to suppose that the two parallel arms meet at infinity, like the two parallel lines. It would be more intuitive to say that the Mb is inseparable. The suggestion is that the Mb has a microcosmic aspect.
Here is a ubiquitous feature of the minibots. As you step down in scale each succeeding generation tends to dissolve into own lace work of veils. There is an apparent trend toward self-censorship of the successive generations. I was able only to go out to the fourth or fifth generation before reaching the 10^14th magnification limit of Neal Ziring's explorer. To see this effect, simply pick up the filamentary thread in any spiral or lattice structure. Because of the simple connectedness, there will be bottlenecks/bridges through which the single filament must pass. It is near the center of such structures that you will find a minibot. But then within each bottleneck, by self-similarity, there will be several smaller bottlenecks/bridges of perhaps one-tenth of the width; however, the central minibot in these substructures will be orders of magnitude smaller relative to that bottleneck. Is there any larger significance to this fact? Can it be rationalized?
There are significant trends to observe on venturing into seahorse and elephant valleys. The seahorse valley/canyon walls are studded with minibots. With each minibot is a associated an increasingly dominant triplet structure consisting of two distinct objects. One of these is a spiral and the other is a circular lattice also decreasing inward, let's call it a 'millipede'. On one side of the canyon there is a millipede sandwiched between two spirals, with the reverse arrangement on the other side. In elephant valley there is, on both the floor and ceiling, a looser triplet structure consisting of a relatively large minibot with a spiral on one side and a millipede on the other.
With the disappearing minibots noted yesterday, they seem to be attempting to turn themselves into millipede, without a central structure. This trend is seen continuing down to a magnitude of 10^25 in the series of snapshots taken by Bengt Månsson. Please forgive me if I now give in to the temptation to try my hand at deep zooming. The main issue is the degree to which novelty appears at the deeper levels. Or will it just be variations on the established themes. I suspect the latter.
Using Fractint, the next smaller generation of minibots that I could locate was at a magnification of 10^23 employing a 28 decimal floating point precision. This was the first minibot I could find in the vicinity of its much larger parent. This tend to confirm my suspicion about the disappearing minibot. Fractint runs mainly in DOS mode, with marginal graphics, but it can calculate up a storm. The final image took about half an hour to generate at 1.5 gig.
Yesterday I gave up on the DOS running of Fractint, and have since been using an evaluation copy of Ultra Fractal, which is easily worth its sixty bucks. What a relief! So now I'm eyeball to eyeball with the Mandelbrot.
(Udo of Aachen may be a fiction, but I suspect the hoax will outlive its author. Do I then have to give credit to Ray Girvan?)
The Rorschach test continues. I am not the only one to be fascinated by the Mb. What is its psychological charm? Clearly it is something more than a mathematical tool. There is a game of hide and seek. The minibots are quite clever at hiding. In the process of looking for them in all the strangest places, it is hard not to get the impression that there is some kind of intelligence behind the game.
Yesterday my nephew, Toby, discovered a whole new class of bots. His first one was at the center of what looked like a spiral from a distance, but at high magnification the spiral split apart, with a bot between its two branches. For some spirals, then, the two arms are connected well short of infinity. Now we have more places to look. I'm calling these bots the 'missing links'. This class of bots evolves from a linear triplet of minispiral-bot-minispiral that gets sucked into a master spiral, and in that process, the two companions effective rotate around the bot, swallowing it, while they in their turn are swallowed by the main spiral. Now I am using 'mini-spiral' to designate both the canonical spirals and its cousins that I have variously called amoebas or gopher holes.
What is there to be learned? Toby's discovery bolsters the cosmic interpretation of the Mandelbrot. The minibots' effectively being swallowed by the Mandelbrot is very much in keeping with our old acquaintance, the Ouroboros. The Mb is being both head and tail, Alpha and Omega. In the process, it is turning itself inside-out.
In parallel with the above action there is the more visible bot 'parade' out of elephant gulch and back into seahorse canyon, with considerable mutation and evolution along the way.
Here are some good Mandelbrot pedagogy sites: Mb vs. Julia Set applet using iterations, an iteration of the Mb boundary (the previous applet can also do iterations), Julia star trip, Math World, Wikipedia. Be sure to see the Julia star trip on the Ibiblio site.
There is astonishing detail in the Mb. It is coordinated as far as the eye can see. From whence does the order come? Is the PSR at work? Can we speak of causation? Is it upward or downwardly acting? Is it just one darned iteration after another? What might the appearance of this order tell us about the world, keeping in mind the difficulties we have in explaining the spontaneous emergence of order.
Outside the Mb the Julia set is disconnected. It is at the points of disconnection of the JS where one finds the minibots in the Mb. As one progresses toward greater detail, the minibots grow more hair, while become smaller relative to their corona of filaments. The central ordering feature of the Mb is to be found in those minibot halos. The halos have either circular or polygonal shapes in powers of two, alternating without any obvious sequence. How these larger structures are formed out of the microscopic filaments is the issue here.
Relative to the filaments, there are sources and sinks. If the minibots are the sources, then the spirals are the sinks. The filaments retain the option of spontaneously ending in thin air, this being their most common fate. Thus are the coiffures maintained. The spirals, however, will consume whatever comes their way, including even the most smartly tonsured minibots. Any measurable segment of filament includes an infinity of minibots.
A very significant source of order is the singly connected attribute of the Mb. There are no holes. The filaments can only branch outward, or die-off, but they can never reconnect with anything. This restriction ensures considerable regimentation. In other words, there are no puddles in Mandelbrot land. All the convoluted 'streams' must flow into the ocean.
The proof of single-connectedness relies on the over-arching principles of complex variables. Almost all mathematical proofs are deductive in nature. The four-color map theorem was a notable exception in that it relied on a computer to perform an exhaustive tabulation of all possible configurations. Such a negative requirement is a long way from providing a positive account of the complexity. In exploring the Mb, one finds clusters of minibots of similar size. There is no reason to limit the complexity of these clusters as one progresses to greater detail. The computing power limits Ultra Fractal to a magnification of about10^33on present-day pc's. The dimension of the visible universe is about 10^28 cm. If the Mb were the universe, our viewer could discern the period at the end of this sentence.
My interest in the Mb lies in its microcosmic aspect. Mb is the Creator. The minibots are the creatures, and the external portion represents the rest of the world. This demonstrates how the creatures act as the co-creators. What is outside represents what is inside. The world is elicited from the cooperative effluence of the creatures. It demonstrates how an overarching coordination could be obtained in a relatively democratic fashion. Is this the BPW? Well, it is the mother of all fractals. There is also the ouroboric quality, with the spirals serving as the tail of the cosmic serpent.
(Permit me to speculate that the M-bulbs represent the Zodiac, in which case the Cardioid is the Matrix, and the primary bulb is the X-factor. Each bulb is a perfect circle, and this shows the place of Pi in AZO/X/QRP. The Mandelbrot is then the archetype of archetypes. The unit circle and triune syzygy, e^i*pi = -1 (and here &ff), is implicit in the complex construction. It appears that the fictional 'Udo of Aachen' may have the last laugh.)
In an immaterial, holographic world, every part necessarily represents the whole. The generator, z' = z^2 + C, creates a blank space or vacuum that can only be filled by the cosmos, in accord with the PSR. This should be no more controversial than pointing out that every number is a microcosm of mathematics, with Pi holding the place of pride. This, in turn, is a recapitulation of Quine's holism. We stir in the anthropic Monster Group (MG), and we have the primordial Matrix soup. We are left to wonder how the multiplication of two numbers can have cosmic significance. Is this not the same problem of how the mind can emerge from atoms swerving in the dark? Where is the hidden hand? We might even settle for a 'God of the Gaps' if there were any gaps. What could have fewer gaps than the multiplication table?
How then do we explain e^i*pi = -1? If we didn't know any better, we would never guess that the multiplication table could generate the Mb or the MG. Then again, the Pieta was carved out of a block of marble. The microcosm of numbers, syzygys and all, is laid bare by an indefatigable process of chipping away at the |z| < 2 block. What is left is the coherence of the syzygous numbers. These are the numbers that have stable orbits under the fundamental operation. So does the solar system carve itself out of the primordial dust cloud.
Mathematics, generally, is a testimonial to the complexity hidden in the multiplication table. As to the provenance of this complexity, we have no explanation. Mathematicians merely catalog it. Srinivasa had an unsurpassed numerical intuition, but asking him about the source of that intuition would be like asking a fish about the source of water. The Mandelbrot is one more demonstration of that hidden complexity.
Should we suppose that the source of mathematical beauty differs from that of truth and beauty generally? Is it an accident of nature? Did it evolve? Do we ascribe to a correspondence theory of truth or a coherence theory? The coherence of the Mb is evidence against the former. The anthropic principle and the role of mathematics in it, should already have settled this argument.
The skeptic would argue that it is we who have carved coherence out of the numbers, but this explanation falls far short of explaining the Mb or the MG and mathematical physics generally. In retrospect, we need not have been surprised by the Mb. We can see now that it fits the cosmic pattern. And is this not the heart of the matter? Platonism is inescapable. Theism is then just a fact of the living, shining Presence of the Forms. When it comes to incarnation there are only three historically viable choices. The first two are the Buddha and Mohammed, and neither of them claimed to be that. This is not rocket science or brain surgery. This is waking up and smelling the coffee.
How can even the simplest of integers not be integral to the cosmic coherence? Integers can only reflect the integrity of the world. This is not a play on words, it is integral to the cosmic play. Think about it, my friends. Think about one and zero and e^i*pi = -1. Is this the AZO/X/QRP, or what? Does the Mb not incarnate the AZO/X/QRP? Does anything else come so close? How could the cosmic coherence fail to fill that void?
We don't need to invoke a deus ex machina, or a God of the gaps to explain the Mb. It is rather the hidden hand of mathematical coherence. It is the same as the order of the seemingly random prime numbers that is exposed by the Riemann hypothesis. It is the hidden intricacy of Fermat's last theorem, exposed in Wiles' proof based on elliptic analysis. All of this intricacy had to be incarnated for the world to witness. That is an essential aspect of the observer principle. And what better place? What better raison for our computer technology? It is all wrapped up in the strangely familiar beauty of the Mandelbrot: Exhibit B!
That's my story, and I'm sticking to it. Is there a better explanation? The only alternative is for us to poke out our eyes like Oedipus. But we, unlike Oedipus, are able to know that we are forgiven. It goes with the territory of truth and beauty.
How does the observer principle work with the Mandelbrot? This seems now to be the biggest issue. Is there any way the observancy can be incarnated within the Mb? I would guess that it has something to do with the single connectedness of the mini-Bs. I suppose, also, that the notion of the microcosm is tied up with observancy. This, in turn, has to do with direct perception. We can perceive only that which is of us. We are God perceiving herself. And there is only one God, as there is only one Mandelbrot, and it is the best of all possible such.
The optimality is reflected in the simplicity of the generator. What would be the optimal such? Would it be simple or complicated? It would be the simplest non-linear generator, which is what we have. And there is nothing that is not contained herein. Anything more complicated would be our arbitrary imposition on God's, and on our own, freedom.
My experience with the MG and Mb leave me suspecting that there is something basic about numbers that we are missing. This something has to do with the problem of perception and anthropics. Counting is deeply imbedded in our observational and linguistic capacity. This capacity may most simply be described as labeling. Labeling brings to mind indirect perception and representationalism. This may be the point at which our epistemology has gone astray. The map becomes confused with the territory.
With mathematical physics and computers, we take numbers to be the the substratum of reality, with language and mind being epiphenomenal. The Platonic forms become divorced from our lived world, and the Cartesian dualism of matter and mind, quantity and quality raises its specter.
Yes, I am attempting to regress from quantity to quality, and, of course, this will take me uncomfortably close to the hazards of numerology. We have been there before. We are acquainted with numerical coincidences and sacred geometry. This is why the archetype of Pi figures prominently in the AZO/X/QRP(i). I am suggesting that the relational qualities of Pi are ontologically prior to its quantity. The irony is that we depend on the mechanism of the computer to demonstrate the organicity of the Mandelbrot.
Pi has an ouroboric quality, especially in the context of e^i*pi. The ouroboros has a lot to do with observancy. This quality of the ouroboros and the Mb is captured in the observance of the reflections of multiple mirrors and in paintings of paintings. It has to do with Godelian self-reference. We should not have been surprised by the Godelian construction. Is there evidence for any of this that can be found in the multiplication table? How is it imported into the Mb? How is the spark of life imported into the chaos of the primordial soup? An immaterialist does not have to play that game, but we do have to play the Mandelbrot game. How can the multiplication table be viewed as a microcosm, and not as a pure artifact? Perhaps we should be looking at the dialectic embedded in the imaginary i = sqrt(-1). Perhaps there is more to the iota than mere quantity. Can the iota sustain such an ontological burden? Perhaps only in its natural (sic) triune context. The Mandelbrot is the imago dei, especially in the triune form of the latter. Does this quasi explanation of the Mb smack of circularity? Well, welcome to the real ouroboric world! Mathematicians, for all their jadedness, have not quite outgrown the magic of the iota. And where would any of us be without Iona? They just neglect to stop and smell the flowers. The iota merely adumbrates the (observancy?) magic inherent in numbers. It is the salt in our primordial soup.
The iota is the magic mirror that we can hold up to the numbers to bring out their self-referential quality. The Godelian self-referential construct has a similar effect, but it is much more cumbersome. It should be no surprise that the iota is an essential ingredient for quantum physics, where it appears necessarily in the all-important commutation relations between complementary variables like position and momentum.
Our basic ontological dynamic is MDX(/Z), which is reminiscent of e^i*pi, also of the mother, spirit and son, if you are a reformed traditionalist. The iota here is something of the shape-shifting Trickster. It is the cosmic wild card. It mediates between mother and son, and this role captures its Dialectic quality. It is the basic mechanism of the bootstrap. The Zodiac, as a psychic circuit, bootstraps itself out of the Matrix. The X, with the help of the Trickster, sacrifices itself to break the circuit to form Creation between the Alpha and Omega. Thus the cardiacal cleavage in the Matrix, with the Zodiacal procession emanating from and returning to the nearly cleaved Matrix. Pardon me while I free-associate on the Imago Dei.
The dialectic is our biggest mystery. It is the primal generator of form. If the iota can shed any light on it, then more power to the iota. The sqrt (-1) is not just imaginary, it is impossible. Yet, it is easily the most powerful agent in mathematics. The iota provides an extra dimension to the real numbers, making visible their hidden structures, as in the Riemann Hypothesis relative to the distribution of the primes. The iota elicits hidden structure. The iota is to the reals as the dialectic is to the matrix. Is it fair to say that the functionality of the iota is dependent on the participation of an observer? It is an observational tool or operator or artifact, not unlike the projection operator in quantum physics.
Recall that the dialectic juxtaposes a thesis with its antithesis to produce a higher order synthesis. It is all about transcending the box. That juxtaposition implies an external agent. Pi bootstraps itself out of 'e' using the iota. The Mb recapitulates this process, somehow. e^i*pi is the generator of the numerals, somewhat as the primary bulb in the Mb seems to generate the lesser ones. The same bulb also seems to generate the prime minibot on the negative real axial filament. That creature is the Adam Kadmon of Creation. Like the Mb, it is apparently hermaphroditic, being in the Imago Dei. The cleaving of the cardioid/female component of the Mb was aboriginal, as in the zim-zum. (BTW, the Mb should have reminded me of the polyandrous cosmic model that appeared on my first website. The Mb is more monotheistic.)
Another point needing clarification is the relation between the logos and the dia-logos. The latter seems to be a conflation of D & X. In e^i*pi, it is the iota that bends the inflating 'e' into something cyclical. The Pi is the measure of those cycles. It projects the cyclical onto the lineal. It is the iota that may be responsible for the cleaving of the cardioid and the emergence of the first bulb; however, that point is censored, even mathematically. It is lost in the cusp singularity at (1/4, 0i). In general, the formula for a cardioid is r = 1 - cos(theta), deriving directly from e^i*pi. The Mb cardioid is exact, as is the circularity of the primary bulb. The other bulbs depart slightly from circularity, with only a fractal pattern for those departures. And, yes, we are talking about the heart of the matter.
In the MG and the Mb, numbers seem play two very different roles: ontological and epistemological, respectively. Proper names speak to individuality. Nouns and numbers speak to commonality and identity. Individuation precedes categorization. Before that there is a shining present, and perhaps a stream of consciousness. Enumeration is founded upon the experience of space and time. Then come the Cartesian coordinates and the physics that flows therefrom. That physics turns out to be derivable mainly from symmetry, which is based on identity. The Mb is the most complex object in mathematics. The Monster Group is the most complex of all mathematical groups. We have complexity with a vengeance. Do we not see that complexity reflected in the world? The BPW puts a premium on that complexity. The implication is that what we have is the optimal complexity. Anything more or less would be suboptimal. Is it possible to have too much of a good thing? Yes, but not in the BPW.
On the other hand, I supposed the Mb to be infinitely complex. I was about to say that there could be no non-cognizable complexity. Do we now have to posit an infinitely intelligent deity? Is there no limit to the discoverable coherence in the Mb, using now a more specific definition of complexity? In this case, either the Mb cannot be a microcosm, or the world is infinitely complex. The Mb is part of the world, so its level of complexity becomes that of the world. We then need to consider the ontology of that complexity. Is it actual or merely potential? How might it figure in the eschatology. Might not this conundrum force us to separate ontology and epistemology in the usual Platonic fashion? That would push the theism back toward deism. Can we posit an objectivity for the Mb that is less than that of the MG? Can the role of one be less fundamental than the other?