The idea that there could be a mathematical theory of why cows, tigers, and tropical fish have stripes or spots was something that biologists had never really thought about, and neither had the mathematicians. People had simply not put those two things together.
The theories of British mathematician Alan Turing set the scene for much of what occurred in science over half a century later. The “universal machine” he described in a paper in 1936 was purely hypothetical, but it laid out the fundamental principle underpinning all computers. He applied his theory in the development of code breaking machines during World War II, but after the war, Turing returned to contemplating systems in biology. He decided that mathematics could, in principle, explain a process called morphogenesis, the way shapes and patterns emerge in living organisms as they develop, from whorled leaves on a plant stem to the stripes on a zebra.
He approached the problem from a purely mathematical point of view, and in 1951 published a paper entitled “The Chemical Basis of Morphogenesis.” His equation described how patterns formed, demonstrating in principle that chemicals following incredibly simple mathematical rules could spontaneously create “binary” markings on living creatures. The stripes or spots are not inherent in the formula, but once the equation is set in motion as a system, out they come. Turing was now decoding nature.
Ten years later, another mathematical maverick, Polish-born Benoit Mandelbrot, was recruited by IBM in the hope that he could provide a new perspective on a problem they needed to solve. “White noise” was interrupting the flow of data transmitted over telephone lines, and as a visual thinker, Mandelbrot instinctively bypassed the established analytical methods. He looked instead at the shapes it generated, an early form of the practice now known as data visualisation. Intriguingly, graphs of the disturbance, whether representing a second, an hour, or an entire day, all displayed a similar pattern. Clearly, there was an internal logic to the turbulence, one which resulted in the same shape at multiple levels. Mandelbrot made the connection between this phenomenon and the obscure but fascinating theories of “mathematical iteration” by a pair of French mathematicians, which revolved around the simplest of equations: z = z² + c. Basically, the equation’s first output becomes its next input, much like pointing a video camera at its own signal on a monitor, resulting in patterns known as “video feedback.” The equation never changes, but the two-way relationship between z and z² enables the multiplication, escalation and variation of c, the circulating “content.”
It was not until 1980 that Mandelbrot was able to use the resources of IBM to “iterate” the equation a million times over. When the results were graphed, Mandelbrot noticed that the strange shape which the equation produced had rough edges, and that these rough edges consisted of shapes that were very similar, but not identical, to the “parent” shape. This was the first glimpse of the possibility of an entirely new school of geometry, one that existed beyond the limits of the Euclidean principles suited to the smooth, orderly world constructed by humans. Nature, in contrast, is formed of shapes which are immeasurably rough. Now it was possible to mathematically explain the apparent irregularities of cloud and wave formations, or measure a coastline. As Mandelbrot said, “In the whole of science, the whole of mathematics, smoothness was everything. What I did was to open up roughness for investigation.”
Mandelbrot’s discovery did not receive legitimacy beyond the sphere of mathematical science until 1982, when he published The Fractal Geometry of Nature. The science not only revealed the internal logic of the structure of trees, but also described almost all of the physiological processes of the human body. Since then, fractal geometry has been applied not only to biology and healthcare, but also to more efficient antennas in wireless devices, to computer graphics that mimic natural forms and processes, and to statistical analyses of the formation of galaxies, the behaviour of the weather, and variations in the stock market. It has even been used to quantify the carbon dioxide output of a forest.
If fractals are foundational to efficiency and beauty in both the created world and the modern technological marvels of Man, would it be beyond reason to discover them in the Word of God? After all, God making Man in His image was the supreme act of “self similarity.” Fractal geometry is similar to the practice of “literary homology” inherent in systematic typology. The Bible’s structural “matrix” is no different from the process of infinitely self similar iteration which defines the patterns now familiar to us in physics, technology, and the natural world. However, this also means that the practice of systematic typology shares some features with machine learning in that it requires some talent for pattern recognition, the identification of regularities in data and the labeling of each step in repeated sequences. Not everyone is going to get a handle on it straight away. We must learn to read the Bible in the way that Turing read the “visual language” of the patterns in flora and fauna, or Mandelbrot noticed both the similarities and the differences between the “parent” shape and its subsequent “generations.”
These similarities and differences, and indeed the same apparent “roughness,” are exactly what we observe in the Bible as God guides sacred history and its literature. To illustrate, all humans bear the same image but no two are completely identical. The Bible’s fractal algorithm is deterministic but only to a limited degree. It allows for unlimited variations within a limited literary “kind,” all of which remain capable of textual compatibility. The history of the Law of Moses and the houses constructed for God are also wonderful examples of iteration.
The Mosaic Law was given to Israel as a pair of tablets. These were similar but not identical, much like Man and Woman. They were divided that they might be reunited in obedience. Like the single legal “code” given to Adam in Eden, this tenfold “multiplied” Law was a code which would bring maturity, authority and prosperity to Israel, a greater kind of expansion than the “natural” multiplication of the number of Hebrews in Egypt (Exodus 1:7). Like Adam, they were given greater authority, and with it greater accountability. Like Adam, the people broke the Law. A second set of tablets, a new iteration, was given as a kind of new covenant (Exodus 34:1). Two sets of two tablets had been cut, but both were given on Sinai, now constituting a single iteration.
When the exodus generation of Israel was disinherited by God for unfaithfulness and given up to death in the wilderness, Moses gave the Law a second time. The name “Deuteronomy” means “second law,” and this time Moses himself spoke the words, with slight modifications tailored for new circumstances. This signifies a growth in wisdom and authority, a process which God intended. Just as the two tablets constituted a single code, and the two sets now constituted a single giving of the Law, the givings of the Law by Moses in Exodus and Deuteronomy are both part of the Torah, the books of Moses.
Each time the people turned against God, the old covenant was broken and a new one was required. This is why the return of Israel from exile was referred to by the prophet Jeremiah as a “new covenant,” and he refers explicitly to all that went before as a single “old” covenant (Jeremiah 31:31-34). Although every iteration of the Law was for the purpose of inscribing the words on stone upon hearts of flesh, Pentecost brought the ultimate shift from “external law” to “internal law,” the final circumcision of heart. The writer of Hebrews does exactly the same thing, quoting Jeremiah not because the prophet’s prediction was fulfilled in the first century, but because the same process was occurring again in a fresh iteration. The New Covenant in Jesus’ blood rendered everything that went before “old.”
Similarly, we can see this process in the various houses of God throughout the Scriptures. The Mosaic Tabernacle was torn apart through the disobedience of Israel’s priests and resurrected as the Tabernacle of David. The Temple of Solomon was torn down through the disobedience of Israel’s kings and resurrected as the Temple of Ezekiel. In a greater picture, the combined Tabernacle era “died” and the combined Temple era was a resurrection. But then, both Tabernacle and Temple, combined as earthly houses, were slain for the sake of the greater glory of the New Covenant temple, a spiritual building made of obedient people inhabited by God. The growth to maturity of Law and House throughout the Bible is as organic and as formulaic as the fibonacci pattern in a nautilus seashell. Each new era of growth requires the vacation of the old.
Thus, since this process of iteration occurs in multiple layers, and in three dimensions, the Bible functions less as a text than it does as a place. When the Bible is understood visually, or rather, spatially, it becomes not only texts to meditate upon, but also a place in which to meditate upon them. Line by line, the inspired texts construct and beautify a sacred architecture, so that every story brings us into the court of God. This is not achieved as bricks in a two dimensional wall but as triangulated steps in a ziggurat, or the branches, twigs, leaves and fruit on a three dimensional “family tree.” To the inspired authors and the saints, the Scriptures themselves were a living enclosure in which to dwell and grow, a tree of righteousness, a source of food and a place of shelter. History itself is a magnificent temple, a house of many rooms (John 14:2).
As in nature and technology, this fractalline method of construction provides not only strength and beauty, but also efficiency. As one traces the trunk, branches and twigs (or arteries, veins and capillaries) one can develop a mental map of the entire body of Scripture, similar to the concept of a “mind map,” a technique for memory enhancement which uses visualization to organise and recall information. Although the terms “mind map” and “mind palace” originated in the 1970s, the method can be traced back many centuries as the “method of loci,” a mnemonic device adopted in ancient Greek and Roman rhetorical treatises where information was arranged “spatially,” as shops in a street or places on an imagined geography. As an extension of the fundamental “creative algorithm,” a Bible mind map is also a map of the mind of God. The Bible is not arranged in this manner merely for the purpose of easy recall, but for the purpose of reshaping us in its image. Its literary structure and historical progress replicate the pattern of growth and maturity found in every sphere of the Creation.
Moreover, familiarity with its basic formula makes the Bible not only easier to comprehend, but also easier to teach. It not only allows us to understand the complexities of the Bible’s history, architecture and theology in a simpler, more integrated way, but also to express them more efficiently. In some sense, systematic typology may become the theological equivalent of the Feynman diagrams, pictorial representations which revolutionised nearly every aspect of theoretical physics. But old habits die hard. For Feynman, who possessed a frightening ease with the substance behind the equations, persuading the physics establishment to give up pages and pages of complicated calculations for simple diagrams was not so simple.
If this particular thesis holds water, the benefits of the fractalline nature of systematic typology for hermeneutics cannot be understated. An intuitive understanding of the Bible’s textual architecture as a multilayered “waveform” emanating from the very nature of God would necessitate a new school of theological thought. It would mean that the days of hurling proof texts at one another in longstanding theological debates are over. It would mean that many or even most of the questions which have dogged Christian theology could be solved with a diagram scribbled onto a folded napkin. It would also likely be rejected by a theological establishment set in its ways, content with academic wool gathering, revelling in insoluble complication, and lacking the skills of imagination and visualisation required by such a shift in the paradigm.
Like Israel, the old school will be left in the dust, while the “new word” is grasped immediately by their children. The fact that we did not expect to discover a geometrical basis for all literary form in Scripture will be no excuse for neo-Luddism in hermeneutics. We must prove all things, certainly, but once proven we must get with the program. History shows us that, in every sphere, God gives us gifts as we are ready for them. Perhaps it is no coincidence that mathematical biology and the science of fractals are only a few decades old. Turing had an almost physical feeling for how equations moved and played together. To him, morphogenesis was a giant, beautiful, mathematical dream. Like a composer, he could “hear” the entire glorious structure in motion. To others it was just a series of scary, mathematical functions. To these “sacred equations” we now turn, and your eyes must become the eyes of a child.
Mike Bull is a graphic designer in the Blue Mountains of Australia and author, most recently, of Inquiétude.
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