Jon Adams rifles through the instrument cabinet of the man upstairs by way of William J. Jackson’s Heaven’s Fractal Net. Adams finds more problems than solutions in Jackson’s position that fractals are a fundamental and universal structure of life - a position Jackson stakes out by vacillating between scholarly proof and speculative guruism.
Inside God's Toolbox
Inside God's Toolbox
An illustration from an article in The College Mathematics JournalSee John Ewing’s “Can we see the Mandelbrot Set?” (92). shows an image of the Mandelbrot Set. Beside it is a magnified image of part of the boundary. It is labelled “Figure 3. Part of the boundary, size of hydrogen atom.” This second image is equally detailed, and with a sufficiently powerful computer, a successive series of such magnifications could be made. The detail would continue, even as the diameter of the original image exceeded the width of the known universe.
The Mandelbrot Set has been called “God’s thumbprint.” The thumbprint is a good point of comparison, for while every thumbprint looks different, every thumbprint also looks like a thumbprint. While such fractals never repeat themselves, nor do they ever (in any meaningful sense of the word) change. So if the Mandelbrot Set is the thumbprint of God because it possesses a complexity that never precisely repeats, then it may also be the map of hell, for it possesses a complexity that never advances, no matter how hard you push. Everywhere is different; everywhere is the same. We always know where we’re going; it’s just that we’ll never quite get there. They are the visual equivalent of a Zeno paradox - never quite finishing, but not really changing either.
Before Benoit Mandelbrot called them “fractals” in the late 1970s, they were known as “monster curves.” These pre-fractal fractals include the Koch Curve (or Snowflake), and the Hilbert and Peano Curves (one-dimensional lines so complex they can fill two and even three dimensional space). Appearing less rigidly geometric (although in construction no more elaborate) is the Dragon Curve, which begins as a simple right-angled hook, hung on a replica of itself, but with successive iterations widens quite suddenly into a spiral of organic complexity. (Readers of Jurassic Park might recognise this as the pattern Michael Crichton employed to illustrate how order and simplicity rapidly escapes into chaos and complexity. As the monsters in the book spiralled out of control, so did the monster curve.) Alongside them are such oddities as Cantor Dust, the Menger Sponge (like an enormous office building, the footprint of which is called Sierpinski’s Carpet), and Pascal’s Triangle, an isomorphic grid of numbers which, when shaded, yields the Sierpinski Gasket.Along with black and white images throughout the text, the book contains the obligatory section of colour pictures, and further fractals are included on a DVD (although these are at a surprisingly low resolution). Considering the variety of fractal visualisations now available, the overall content is repetitious and unimpressive. There are many websites with far more impressive material, and the reader would be advised to look here for more interesting examples.
The mechanics of actually drawing fractals requires such laborious calculation (and such vast canvasses) that it was only with the advent of the computer that mathematicians were able to graph all the products. Out of the 1960s came visualisations of the more complex images; the Lorentz Attractor, the Julia Set, Henon’s attractor, and, of course, the Mandelbrot Set. As computing power increases, so do the possibilities, allowing visualisations such as the magnificent “quaternion Julia fractals” which look like loops of torn dough, like shorn metal, like the layered skin of a wasps’ nest. That this mess has a precise mathematical construction and could be perfectly recreated, time and time again, seems impossible. But much about fractals seems impossible. In the fractal, infinite detail is contained within a finite boundary.Measured with sufficient exactitude, and taking in every turn, every nook, the length of a coastline approaches infinity. So does the outline of a tree. Or a leaf. Of course, that being said, like in a Zeno paradox, the “infinite” is an illusion produced by mapping the mathematical realm over the physical realm - for the detail in a fractal is only infinite in the number-theory sense, and there is a big difference between infinite values and infinite things. Infinity occupies a curious ontological position as a mathematical entity that, like e, i, and π, has no place in the real world of integers but seems to do some work there nonetheless. Of course, these aren’t really problems for mathematicians to think about, they are problems for philosophers of mathematics. The questions that actual mathematicians ask are about the relations that these numbers have to each other. They especially like dense relations, like ei π + 1 = 0. But the philosophers of mathematics - along with the applied mathematicians, engineers, economists, statisticians, and so on - are interested in the relations that may or may not hold between these numbers and the world. Those who think that the way the world works is mathematical are mathematical realists. Those who think that mathematics is a closed system which describes only the relations between its components are mathematical formalists. If you ask mathematicians about this problem, you’ll probably find a split between the formalists and the realists, and a blur of positions in between. For the everyday business of being a mathematician, it doesn’t really matter very much. Mathematical truths are true analytically; which is to say that ei π + 1 = 0 holds in virtue of the meaning of 0, 1, =, +, e, i, and π. Whether it has any relation to the physical world is as irrelevant to its truth as whether the king on a chessboard has any real territory, or whether the bishop is really limited to moving diagonally. But if you’re making a claim for the application of mathematics to the world, it matters quite a lot which side you are on.
Most people think fractals are significant largely because they think fractals are beautiful. Most people would claim that fractals were about the most beautiful branch of mathematics. But the idea that fractals are the most beautiful branch of mathematics is one put about by people with little understanding of what beauty means to mathematicians - who, on the whole, are less interested in the appearance of the fractal than the mathematics that generate this appearance. The fractal (as some put it) isn’t the image at all, it’s the mathematics. Most mathematicians find ei π + 1 = 0 equally beautiful, probably more so.Mathematicians and physicists frequently describe formulae as “beautiful,” and have their own distinctive aesthetic criteria. In October 2004, Physics World published a list of “the 20 Most Beautiful Equations.”The difference, of course, is that there’s no obvious way to show this beauty to a non-mathematician. The fractal is something everyone can see because its beauty has been “translated” out of mathematics into a visual representation of astonishing complexity. Fractals are interesting because everyone can “get” them, in a way that only a few people can “get” ei π + 1 = 0. In contrast to the Mandelbrot Set’s status as modern iconography, not nearly so many people could recognise a Fourier transformation, a normal distribution, a Fibonacci sequence - although each have significant and profound connections to the machinations of the natural world.
One of the curious things about the Mandelbrot Set is that while its appearance awaited the advent of the computer age, seeing it for the first time everyone felt a strange sense of familiarity, and this familiarity is where William Jackson begins Heaven’s Fractal Net. Jackson thinks that the fractal has been intuitively known to humanity for millennia, and that evidence for this implicit understanding can be found in most civilisations in all places and at all times dating back to the origins of art and culture.
It’s an interesting hypothesis: fractal forms do indeed seem resonant with the interwoven loops within interwoven spirals seen in the illuminations of the Lindisfarne Gospels, with the towers upon towers of certain East Asian architecture, with the logical structures of certain Zen koans. And these in their turn with the easy complexity of nature: a head of broccoli, the outline of a fern frond. How much of this mathematics was known before, how much has since been lost? And if our ancestors did possess a pre-theoretical understanding of the complexity of such forms, what might we gain by their rediscovery?
Heaven’s Fractal Net presents the reader with hundreds of examples of fractals, and fractal-like forms, and forms which seem fractal-like. This apparently endless sequence of cases ranges across religious beliefs, religious art, literature, and architecture. Like the fractals, the detail is seemingly without end. Unfortunately, also like the fractals, it doesn’t seem to take us anywhere. In the “introductory reflections” Jackson wonders, “what is the best pattern to use in presenting an explanation of patterns?” and decides that “the most appropriate is fractal-like. My ideas and findings are not confined to a conclusion at the end but are spread throughout the book, reiterated and illustrated in a variety of ways” (6). The foregoing perhaps explains the slightly chaotic structure that results.
It’s clear that Jackson has spent a good deal of time preparing the research for this book. What’s less clear is what he wants the book to do. For a start, he seems unsure of quite who he is addressing (perhaps various sections were written independently for different audiences). In one chapter he will present a balanced and well-researched scholarly argument (such as the interesting case for Cartesian dualism being a consequence of Descartes’ own sedentary habits [194-200], which will surely appeal to followers of George Lakoff’s increasingly strong position on the necessity of embodiment); only to spend much of the following section engaged in what he calls his “playful riffs.” Here is a typical fragment from towards the end of the book:
The One God is Life Breath, Brahman, That. The One consciousness, atman Self, exists in all beings, making its form manifold; the wise find it in themselves and find endless joy. One who knows the One that’s in fire, heart, and sun attains the Oneness of the One. (221)
What could such passages mean? “One who knows the One…attains the Oneness of the One.” When you try to parse this, sense collapses in a semantic vacuum. The self-referentiality that creeps in to the language here - the thing is like itself - is reminiscent of Gertrude Stein’s a-rose-is-a-rose-is-a-rose patter. The aim is to refer past the word to the world, to show that a thing is a thing before it is a word, and Stein makes a brief (if typically frustrating) appearance in the chapter on fractal literature. At a stretch, language which identifies itself as language does seem to share some of the properties of self-similarity that we find so intriguing in the fractal images. But there’s an important difference between what Stein is doing with her attempts to damage our habits of analogy, and what Jackson is supposed to be doing writing an ostensibly scholarly text in which ideas are to be explained, not performed.
Jackson trades on his professorial status, but at the same time, distances himself from the academic community. Fairly early on, he declares “epistemological crises” a “fancy term” (19) only to use “holotropic consciousness” on the following page. It seems quite certain that “epistemological crises” has a clearer definition and makes more sense to more readers than “holotropic consciousness.” (The latter arising from a misunderstanding of the significance of holograms.) Consequently, in tone the book sits somewhat uncomfortably between scholarly academic work and the type of vague theorising often propounded by new-age gurus.
One gets the sense that Jackson would quite like to be seen as a guru, especially during his “riffs.” But although these sections may perhaps be treated as creative writing, even in sections that maintain a relatively steady academic stance, there are lazy factual errors. What, for example, are we to make of this:
Comprehension (understanding of meaning) and comprehensiveness (wholeness of the One) are intertwined. From one confused point of time and space you see nebulous randomly scattered stars; from another point you can see spirals of stellar orders - nebulae. (236)
It seems to be the case that Jackson has confused nebulae (which are amorphous clouds of stellar dust and gas, usually lacking firm structure) with galaxies (which do sometimes form spirals). Later on the same page, he reinforces such inaccuracies by writing, “Far-off nebula lights are many, their spiral is one.” Such terminological errors are important when the subject is the alleged fit between structural patterns on different orders of scale. Elsewhere there are muddled blendings between bona fide science and what is presumably Jackson’s own spiritual belief system: “the ‘soup’ of neutrons (which were undifferentiated in the beginning of the universe)” seems like a feature of orthodox cosmology, until we are told that “protoplasm’s creative potential was enfolded in those neutrons” (135), at which point, it all becomes very confusing. But it’s typical of his attitude to the sciences: helping himself to the authority of its claims when it suits him, but when it contradicts him either ignoring it altogether or drawing on some other source. Although it is not explicitly stated, Jackson is apparently an adherent of intelligent design, declaring his belief that “something like ‘mind’ or consciousness is at work in both selective processes: survival of good ideas and survival of life forms” (176). (And this in spite of the fact that fractals are one of the many natural mechanisms which overdetermine ID.) Meanwhile, unperturbed by the general drift away from theories that rely on the inheritance of acquired characteristics, he endorses a version of the Jungian collective unconscious:
the shadows of the unconscious…bringing figures from earlier days of the human race, the days of caves and mysteries, the days of developing the wits to use fire to preserve the spark of human life. All the past still smoulders in our unconscious, flickering when we sleep at night. (174)
There is further sloppy thinking here. His criticism of Wilsonian sociobiology throws up the claim that “instead of interdependence it is cause and effect that seems most important” (177). Cause and effect being interdependent, it is not clear what is the substance of the alleged contrast. Subsequent criticisms of Wilson are increasingly baffling: “Wilson is a great authority on ants…. He is not so adept at imagining the depths involved in the experience of a shaman, a yogi, a poet-visionary, or a Buddha” (179). Surely the writings of E. O. Wilson are not so error-free that we have to resort to criticisms of his shamanic powers?
Yet elsewhere, science is an authority we should trust - “science has been telling us that…” (274) - being a typical construction that relies for its persuasive power on our consenting to the reliability of scientific opinion. The problem is one common to much radical/unorthodox thinking, and that is a vacillating relationship with academic authorities. “Radical thinkers” of all persuasions do something like this (see, for example, Rupert Sheldrake, Graham Hancock, and Erich Von Däniken): science is cited as a reliable authority on one point in order to support an argument whose truth relies on most other scientists being wrong. Jackson wants science to support him at some times, and at others, he wants to dismiss science as slow-witted, close-minded, insufficiently “deep.”
In opposition to those thinkers he considers scientific reductionists, Jackson sets himself up as someone alert to natural harmonies and open to “oneness.” (“Adults habituated to a deadening rigidity, having killed wonder, go on to build walls through which mountains, rivers, stars, and wind cannot penetrate. Spiritual paths…can sometimes offer ways out of this sad impasse” (54).) Some of the writing consequent from these claims makes for very uncomfortable reading:
Woman’s body mediates between the disconnected male and the realms of nature and the beyond. Entering the warm split-up-the-middleness of elusive beauty, awakening man traces depths and learns how his longings are involved in cosmic correspondences.” (146 - note also the habit common to many mystical thinkers of eliding articles, as in “awakening man.”)
Does Jackson say this type of thing apropos (or worse, during) sex? It makes toes curl for all the wrong reasons. And what could it mean? That intercourse awakens men to the unity of the universe? Women, presumably, are already sufficiently attuned to the cosmos. Such indulgent material has no place in a book that purports to have an academic or educative agenda. Although classified as “popular science / philosophy / religion” and published by a University Press, Heaven’s Fractal Net is not aimed at the academic or general intelligent reader.
More likely that the target audience here are the credulous followers of alternative medicine. Acupuncture. Crystal healing. Ear candling. The laying on of hands. Trial by fire. Jackson even includes a diagram, unaccompanied by any explanation in the main body of the text, that demonstrates the affinities between the shape of the human ear and the shape of the human embryo. For anyone who hasn’t seen it before, it is a striking image. A special branch of acupuncture exists dealing directly with this. It’s called “auriculotherapy.” The inclusion of the diagram is supposed to indicate another level of self-similarity in the human. One thinks of the illustration of the “body-politic” that served as the frontispiece for Hobbes’s Leviathan, and later as the cover illustration for Shapin and Schaffer’s Leviathan and the Air Pump. It all seems very intriguing; and yet, with even the briefest reflection, it should be immediately apparent that the putative affinities between ear and embryo are entirely coincidental.
This coincidental affinity becomes clear when we try to extend the connection to non-human species, only to discover that (unfortunately for would-be practitioners of veterinary auriculotherapy) it doesn’t work for many other animals. Mammalian embryos all look strikingly similar, but mammalian ears do not. The embryonic elephant, for example, doesn’t look much like it’s adult ear. But perhaps humans have a special relationship to embryology? It seems unlikely. We don’t look like our cells, and the “body politic” doesn’t really look like the Hobbes frontispiece. In organisational terms, it’s perhaps true that we clump together in “organs,” comprised of individual “cells” with common purposes. But, again, this social structure doesn’t exist for solitary animals, despite the biological (and moreover functional) similarity of their internal anatomy. The links between the parts of the body, the individual, the social unit, the society, the population, and the ecosystem may well be complex, interesting, and interdependent, but there’s little here to suggest that they could be accurately or even usefully characterised as fractal.
Some cases are quite interesting, but if Jackson is to succeed, he must persuade us that the relationship between fractal geometry and cultural/religious history is a somehow special case of the wider relationship that obtains between mathematics and culture (i.e., distinct from pictorial numerology). A Chinese illustration shows a figure in meditation, from whose head sprout five more figures in meditation, each of which spawns another five meditating figures (35). These images do indeed resemble the manner in which the edges of the Mandelbrot Set are studded with miniature replicas of the whole, each miniature in turn adorned with even more miniature versions, and so on. But is the Chinese illustration really a case of fractal geometry? It seems to have more in common with the riddle of the man met on the way to St Ives. The principle in both cases (figures upon figures; kits, cats, sacks, wives) seems to be the same as in the story of the rice on the chessboard (whereby one grain on the first square is doubled on the second square, then the same rule applied for each of the sixty two remaining squares successively, yielding a huge quantity). Are all three cases fractal? Surely, the common thread isn’t that each demonstrate recursion, but that each illustrate the power of a simple geometric sequence.
Making the broader case that number theory has impacted upon culture is a worthwhile endeavour. There are many examples of how myths and religious stories can possess mathematical structures (for example, the Christian fable of the loaves and the fishes is another version of the Zeno paradox), and many unexpected ways in which number theory has impacted upon culture (John Barrow’s popular history of mathematics, Pi In The Sky does a good job of presenting some of these). There are also many ways in which the peculiar properties of recursion seem to have deep affinities in both art and nature. But these are better explored in Douglas Hofstader’s Gödel, Escher, Bach. Hofstader’s book was also capricious, but his playful sections were also frequently profound. Anyone speculatively interested in Heaven’s Fractal Net would be well advised to divert their attentions to Gödel, Escher, Bach instead.
We don’t learn enough about fractals to be persuaded that the relationship Jackson feels he has identified has been properly established. Jackson does not give the impression of knowing enough about fractals, and his definition of fractal is so labile that it sometimes includes simple nestings or one-fold repetitions. A typical sentence will insulate the connection behind two or three layers of analogy: “The model of the ultimate One being found in all the beings generated, at various scales of recognition, seems to have fractal-like aspects” (222). Elsewhere he employs switcharoo arguments whereby set membership is implied then suddenly withdrawn, leaving a thaumotropic ghost of sense. Too often, one reads a construction such as, “Although not a fractal…” - such cases should not be necessary if, as the book claims, the human mind in general and the religious mind in particular have such an affinity with the fractal. Even after the glut of examples we are offered over the 250+ pages of the book, the connection between fractals and the logical structure of religious belief and the decorative arts remains too loose and speculative.
The most compelling cases are those that don’t appear to have been consciously modelled on natural forms. For example, the fact that certain African villages are laid out on a plan that, when viewed from above, reveals a pattern of fractal-like organisation is startling (especially given that such an intricate pattern would doubtless be invisible on ground level - which is the only level available) (193). But such a case is hardly representative of the central thesis, this being the correlations between religious thought and imagery and the fractal. Closer to the theme are the Indian temples that in cross section reveal a nested sequence of rooms-within-rooms. However, in an age before I-beams, these constructions are surely the result of structural necessity as much as any spiritual affinity with the construction methods of the universe.
The additional and overriding problem with Jackson’s search for fractals in art and culture is the question of causal priority. If fractal geometry really is the mechanism through which natural forms are created, then we might well expect that the decorative arts will have employed a similar symmetry or patterning - not because the artists have a pre-theoretical understanding of a sophisticated branch of pure math, but simply because artists copy first what they know. The decorative arts imitate nature, and if nature is fractally constructed, then the decorative arts will presumably retain some of that structuring. In other words, what’s being created isn’t directly fractal; it’s transitively fractal. The resonance isn’t with fractals; it is with natural forms. The difference is between an explicit and a tacit understanding. That this distinction matters becomes clear when we ask what, exactly, might a “pre-theoretical understanding” of fractals actually amount to? Would it be anything more than the claim that we can spot a natural form; that we have a sense of what looks organic and what looks artificial?The notion that there is something in the human mind that facilitates recognition of (and perhaps even preference for) fractal-organic patterning is an interesting one - and the evidence for the existence of pattern recognition as a program in our innate cognitive software has been fruitfully explored for some time now (e.g., the environmental aesthetics section of the Adapted Mind, 552+ or the discussion in Wilson’s Consilience regarding a preference for a twenty percent level of detail redundancy in visual art - something found in Chinese ideograms and genuine Mondrians - see Consilience 245-46). But Jackson is not interested in this, and when he grazes this argument, it is in favour of a Jungian account. Would it amount to a claim any more startling than the truism that a folk theory of inheritance (like-father-like-son) preceded the genetic theory of inheritance? In other words, what’s remarkable about fractals is not they have pre-theoretical antecedents, but precisely that our understanding of them has graduated from a pre-theoretical to an explicit formulation.
In the end, the actual mathematics of fractals are more impressive than the (sometimes quite distant) approximations Jackson unearths from comparative religious studies. Here’s a quotation from philosopher of science Bas Van Fraassen that says it more clearly:
There is a reason why metaphysics sounds so passé, so vieux jeu today; for intellectually challenging perplexities and paradoxes it has been far surpassed by theoretical science. Do the concepts of the Trinity, the soul, haecceity, universals, prime matter, and potentiality baffle you? They pale beside the unimaginable otherness of closed space-times, event-horizons, EPR correlations, and bootstrap models. (258)
This, in short, is the problem that Jackson has. Religion is presented alongside mathematics, and there is an asymmetry in the available wonder. Once you have begun to think about fractals and mathematics seriously, then the history of religion seems parochial and artificial. A Zen koan remains logically impenetrable, but only because that’s what a Zen koan is meant to do. The fractal, on the other hand, and more generally, the world of mathematics, seems to offer something far more magical: access to the actual mechanisms of nature, to god’s toolbox. Unfortunately, there’s only a secondary trace of that magic here.
Barrow, John. Pi in the Sky: Counting, Thinking and Being. Oxford: Oxford UP, 1992.
Ewing, John. “Can we see the Mandelbrot Set?” The College Mathematics Journal 26.2 (March 1995): 90-99.
Hofstadter, Douglas R. Gödel, Escher, Bach: an Eternal Golden Braid. NY: Basic Books, 1979.
Lakoff, George and Mark Johnson. Philosophy in the Flesh: The Embodied Mind and its Challenge to Western Thought. New York: Basic, 1999.
Van Fraassen, Bas C. “Empiricism in the Philosophy of Science.” Images of Science: Essays on Realism and Empiricism, with a Reply from Bas C. Van Fraassen. eds. Paul M. Churchland and Clifford A. Hooker. Chicago: U of Chicago P, 1985. 245-308.
Wilson, E. O. Consilience: The Unity of Knowledge. London: Little Brown, 1998.
Reviewing Ira Livingston’s book on autopoiesis, Stephen Doughterty marks a convergence between fractal complexity and imaginative writing by pointing out how words are themselves fractal in their organization and evolution.