Gecko Tech, Fractals, and Human Interface

by Swinton Roof

an ongoing addition to the Fracman discussions

Dec. 18, 2000

Deric posed the following problem to me - come up with a good name for the concept of human/tech/culture interface and all it implies in terms of meta-level recursion, fractals, etc. My first irreverent thought was 'Borg Zone'. 'Fracman' is another. Upon further thought, I realized that technology itself IS the interface between humans, between humans and nature, between humans and society etc. It is our technology that to a very large degree determines our culture and vice versa. It is of note perhaps that Deric himself placed it in the middle between 'human/' and '/culture'. The connotations of the word, however, do not fully express it's complete role in human affairs as I would like to approach it in this essay. I will therefore leave the challenge in effect. I would instead like to explore some of the thoughts that this question inspired. Part 1 will deal with the basics. Part 2 will discuss the human drama.

Part I

First I will assume the reader has some degree of familiarity with the concept of 'fractal' but maybe not its implications. The first concept I will explore is interface as fractal. Then I will talk about Geckos. The Gecko lizard makes an excellent example of the importance of fractal interface in the real world. To get there I feel the need to build a series of key concepts required to fully understand how important these ideas are.

Consider the concept of interface as a boundary. Take a triangle. It separates 2d space into an inner space and an outer space. The triangle itself is an interface between these two spaces. Notice also that the triangle is composed of lines. Lines are linear i.e. one dimensional. Interface, as thus defined, has a lower dimensionality than the space it divides! This is a very important aspect of interface. In three dimensions an interface becomes essentially a two dimensional surface. Now that we have established this essential feature of interface let's complicate things a bit. Reality is not composed of ideal mathematical points, lines, etc. How can we approximate mathematically how things really are? Also, how can we understand how interfaces naturally emerge in the real world.

This brings up the subject of fractals. The prime precursor to fractal as concept is size invariance or scaling. An equilateral triangle as concept is a perfect example of absolute scaling invariance. No matter what its size, the relations that define 'equilateral' remain true. It demonstrates perfect or exact linear scaling invariance. So long as we do this in Euclidean space, the properties of side and angle relate to each other in an invariant way. We can introduce translations and rotations without affecting the invariance. Now if we take a triangle and divide its sides up into smaller segments of equilateral triangles, and then take those smaller segments and divide them equally, we end up with the famous Koch Curve. If one carries the process out to infinity (in your imagination please!) the overall length around the interface so defined becomes infinite! In real space there of course is always a real limit to how fine you can draw lines. Limits, both upper and lower, are very important when dealing with fractals in physical nature. We will talk more about limits and their implications in a while.

Now the really interesting thing about the above example of a fractal is that the interface between inner and outer space wiggles all over the place and seems to cover more than the simple one-dimension we characterized at first with the triangle. In fact there is a mathematical dimensinality called the Housdorf dimension (Hd) for such a fractal. The Hd value for the Koch curve is 1.26... Now we said that interface must have a lower dimensionality than the space it divides - in this case 2dim. A Koch curve therefore has an interface dimensionality somewhere between one and two, or 1.26... to be more precise. Different sorts of fractals have a different dimensionality, but the essential nature is the same. When one talks about three dimensions and real space these properties can have astounding impact. A simple filter for example functions much better the greater the surface area of the interface. Fractals provide a parsimonious way to achieve such higher dimensionality in an interface. It's like something for nothing. More surface area for the same size orifice! The interesting thing about fractals is that a very simple scheme or process repeated over and over again can achieve this sieving action. Mathematically they are very simply defined, yet they are capable of extraordinary summary results.

Scheme or process is very important when talking about the physical world, because so much of what we call nature is the end result of cyclical process repeated over and over again at different scales and different parameters involving atomic,molecular,chemical, cellular, etc levels of scale. It's a question of physical logistics. Grab all the concepts you have of mass, energy, levels, layers, scales, process, recursion, interface, interaction, feedback etc and let them react in your mind like the prebiotic stew of ages old and you will begin to get a glimpse of where I'm headed with all this. Interface it seems is an emergent property of non-linear process (ala positive and negative feed-back).

The above example involved exact scaling, but in general, real world scaling is not exact and the forces, relations etc. vary to some degree with size. The scaling is not exact. Limits both lower and upper apply. At the extreme lower we have quantum uncertainty. At the extreme upper we have relatavistic shifts and singularities. In between these extremes, real world interfaces manifest non-linear effects leading to non-linear scaling invariances. The fractals in this class are to me the REAL fractals of which the Koch curve and others are just degenerate or limiting forms. Such fractals produce an imperfect similarity of forms. They produce what is called 'self similarity' or partial scaling invariance. The general concept of scaling, however, provides a model upon which non-linearity can hang its hat. The Koch curve in result is in fact non-linear and leads to a fractional dimension (i.e. the term fractal). When the the scaling itself is non-linear though the resulting fractals can become monstrously more complex and difficult to decipher.

The Mandelbrot set is the classic example of non-linear fractal scaling. The feature I wish to point out is that the limit boundary of the set while densely populated, and it is believed connected also, achieves a staggering complexity of form. This limit boundary (actually it is the set itself) is an interface between what is called a sub-critical inner region and the the outer super-critical region. Since the formulation or scheme or process for mapping the Mandelbrot is the simple non-linear iteration of a complex number ( Z = Z*Z + C ), we have a second order (power of two exponent) non-linear scaling rule. Starting points inside the interface scale downward toward a locus or 'attractor'. The iterative arcs are bound. Points outside the Mandelbrot will diverge out into the super-critical region and the iterative arcs are unbounded. The Mandelbrot is so complex that it doesn't display self-similarity of the whole, yet self-similar shapes repeatedly emerge unpredictably at varying scales. In fact, when one talks about real world physical process, it is discovered that non-linearity can lead to self-organizing features (read INTERFACE) which themselves form self-similar features, layers, levels, and boundaries which repeat in a quasi periodic fashion through space and time. Interface is an emergent feature of non-linear self-organizing systems.

Speaking of quasi periodicity. There is the well-know period-doubling route to chaos demonstated by the chaos bifucation diagram. The simple rule x = x(1-x) is able to produce, by iteration, a non-linear scaling that repeatedly moves from periodic behavior zones into wilder fluctuations and ultimately into chaos where an infinite amount of time or process occurs between repititions of state value. The important feature to note is that when one looks at these plots (checkout it out), he is struck by the fact that it resolves into pronounced demarcations or boundaries at the edge of chaos. The non-linear effect erupts into chaos in a most sharply defined way. This is interface that we are talking about. Interface ala non-linear fractal nature forms a boundary between more or less linear (or periodic) behaviour and non-linear aperiodic chaos. Interface represents quasi-periodic behaviour. Self-similar scaling.

Thinking along these lines one can see that a particular metabolite within a cell's wall (interface) is bound and useful in a controlled way for periodic linear process scaling (cell metabolism) in a sub-critical region. Once it exits the protection of the cell wall interface however it enters a super-critical region where its behavior is subject to free radicals and a whole host of chaotic conditions. It becomes unbound and of no immediate consequence to the cell. By regulating the flow of molecules across its interface, a cell is able to move, adapt, grow, and regulate its behaviour. Evolution and principles of self-assembly from the nano-scale and up, have arrived at fractal structure, fractal process, and foremost of all - interface as functionality.

Fractal interface maximizes useful throughput. Trees and lungs provide inverted examples of how a fractal structure maximizes* surface area of interface. When one is trying to filter out and absorb specific nutrients, the greater the fractal dimension the greater the surface boundary area and the greater the selectivity. This gives greater throughtput. Two birds with one stone. An elegant solution to a problem in logistics.

Interface defines function. The above example of function (osmosis, respiration, filtration etc.) is just one of the many possible. The Gecko example which I promised to get to will demonstrate interface as a 'grasping' function. Before we get to geckos, however, I want to further delineate the concept of function.

Lets start with the math first. A math function has an input or argument, also called an independant variable. It has an output, result, or dependant variable also. The mathematician feeds the inputs into the function, crunches the numbers, and gets a result. The way the function is defined determines the result one gets from any given input. Now I said above that interface defines function. How so? In math, the symbols used will prescribe a set of operations or proceedures which the mathematician must use to get the result. y = 3 * x + 4 would be such an example of a 'symbolic' interface to the functionality of multiplying 'x' times 3 and adding '4' to that intermediate result. The issue of interface is a bit clouded by the fact that the mathematician himself is part of the interface. A representation in computer technology might help clarify what a functional interface is.

Early attempts at programming computers involved a series of statements in linear fashion which prescribe a series of steps, or proceedure for achieving some sort of functional result - i.e a deterministic law abiding behavior. Things get complicated very quickly under this scheme so subroutines and naming conventions evolved which attempted to cordon off areas for better conceptuality. What was needed was a sort of blackbox system for creating functionality. So long as a black box function was defined as having a certain interface for getting its results, one no longer needed to be concerned with the internal details. The function once defined would always have the same functionality. This concept was finally achieved with the advent of C language programming. The invention of the 'state stack' enabled functions to be defined* with a true interface. One takes all the argument values to the function and a jump vector to its proceedural code and copies (push) them onto a stack like dinner plates. The function is thus initialized and ready to call. This 'calling by value' achieves an abstract separation of the internal details from the outside code. The function then lives in its own little universe and can reliably perform its functionality flawlessly without any outside influences to monkey with the values. Outside influences introduce error, feedback, coupling and all sorts of unwanted effects. The functionality is safe because the values are COPIED onto the stack and not affected by outside events. In complex multitasking computational enviroments it is essential to create interfaces to provide such separation and shielding. Conceptuality is enabled in the process. The evolutionary development of such interface modularization has led to object oriented programming, abstraction modeling, common object model protocols across platforms etc. It is hoped that the role of interface in defining functionality is now appreciated in an abstract sense. From here we can go to natural examples in the real world.

At last we get to Gecko technology. A bit later we want to explore the human technology interface. Geckos are small lizards with amazing powers. The adhesive action of their foot pads enables them to hang effortlessly by one toe from just about any surface. It has only recently been discovered how they do this. It turns out that the pads on their feet are fabulous fractals. Each self-similar toe has a pad which is itself divided into self-similar pads. There are two columns of pads arranged in a dozen or so rows each smaller but similar to the original. On each of these pads their are thousands of self-similar hairs. Each of these tiny hairs bifurcates at its tip into very many tinier hairs - split ends if you like. At the very end the hairs are so tiny that the contact they make with a surface can explore every nook and cranny. They are so extreme in fact that the tips produce attractive VanderWaals forces between molecules. The net effect is an astounding adhesive action which is hundreds of times better than any modern adhesive tape. Since the attractions are purely induced and do not involve molecular alteration, the pads separate exquisitely clean from the surface on retraction. No trash adheres. Separation is easily achieved because the pads and hairs are arranged in rows. Starting at one end it is easy to pull one tiny row at a time one after another from the surface. Exquisite logistics. Acting in unison they produce an outrageous ability to grasp almost any surface* without ever becoming clogged and useless.

The key feature about this amazing gecko technology is that it is fractal* interface which enables and defines* the functionality. The elegant and simple logistics of the structure means that it was something that could evolve over time from simple growth processes. As form follows function so did the structure* of a gecko's feet. The monstrously complex and variagated internal processes of the gecko's feet are hidden within its structural interface. The exterior interface in turn is almost pure functionality - the 'grasping' function* if you want to give it a name. Recursive metalevel interplay between the need to hold onto vegetation tightly and also achieve mobility and precision while hunting have led to an evolutionary miracle - simple and sweet - but profoundly articulated at smaller and smaller scales. Need I say more?

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Part II

Perhaps we are interested in how humans relate to all this. At the very start I said that technology itself IS the interface between humans, between humans and nature, between humans and society etc. It is our technology that to a very large degree determines our culture and vice versa. Foremost I regard language as the first human technology.

Let me explain what I mean by technology. I wish to differentiate tools from technology. In its broadest sense I guess technology is the time bound proceedures a society uses to ensure its survival, growth, and enrichment. They must be time bound via memory, passed on tales, or writing etc. Tools are very important extensions of our functionality and hence interface but they convey very little information about how to make them and pass them on to our descendants. It is technology that serves this purpose of time binding. If the appropriate technology is lost, all the tools in the world wont help because you wont know how to use them.

Language was the first enabling technology I believe. Other animals use tools such as sticks and stones to achieve limited survival goals, but they really dont have a language technology for passing this information along. They do not have active time binding technology. The techniques are simply observed and imitated. Time binding does occur in a limited fashion albeit passively but it does not transcend immediate clan groups easily. Language enabled the ability to discriminate, specify, communicate, and preserve.

Language itself, however, is specialized and confined to groups. What was needed was a greater generality. Mathematics, I believe, provided the next technological leap. The simple concept of counting transcends the particulars of regional language. Competing groups with different language constructs would have limited ability to communicate, but the ability to count is an abstraction which can easily transcend language differences. The math has to work out the same regardless of the language in which it is represented. One stone - one sheep , two stones - two sheep is a concept* that easily translates across language boundaries. Therefore I regard mathematics as the second great leap of human technology. Tool making and using were a proto-typical technology, but language and mathematics enabled a staggering change in the way humans relate to their enviroment.

All sorts of interfacial layers and functionality emerged as a result of human technology. Hand tools, weapons, clothing, housing, the list goes on. The feature of all this is that interfacial functionality both separated and clarified the ways humans interact with each other and with their enviroment. A boundary layer emerged between humans and raw chaos. One can even perceive the emergence of mind and self awareness out of this chaos as an interfacial process. As the bhuddas said, out of the first distinction spews forth the many from the one.

Further evolution led to speciation into different cultures based on their essential technology: hunter-gatherer, agri-culture, stone age, bronze age, industrial, etc. The base cultures in turn reflexively shaped the tools and technologies to be bound in the future of a given group.

Now it is appropriate to discuss how such time-binding technology evokes echoes of our intial discussion about fractals, recursive process, and scalability. I believe that for technology to become time binding it is necessary to have scalability of process. Simple one to one transmission is not efficient enough for time binding. Language is scalable up to a limit because one can speak to a group as a whole and hence formalize proceedure into ritual. One is limited by the number one can speak to however, and when other language groups are encountered scalability reaches a wall. If mathematics did indeed provide the next step, we can envision translation and trade of technology across language* and group barriers. There's the interface again only now it was scalable in size. Recursive iterations of functional exchange across interfacial boundaries enabled the evolution of large group structures - hierarchies, tribes, towns, nations etc. Scalability of process is the key. Scaling differences also lead to emergent interfacial boundaries based on functionalities and technologies beside the core cultural differences. Supercriticality emerges also as some regions have technology that has reached a scaling limit. Limited resources or in some cases overcomplexification may be the nonlinear force producing such criticalities, but the results can range from tighter interface control to chaotic breaches of protocol - suppression, war, pestilence, famine - you name it. I'm not even going to bring religion into this whole discussion, but suffice it to say that it might represent the ultimate in time binding.

A few more thoughts about scaling are in order. A very general sort of graph common to depicting all sorts of physical process is the so called S-curve. An s-curve in general starts out rising quickly as an exponential, flattens a bit in the middle, and then tapers off toward the end. It is typical of growing things to start quickly, enter into a linear phase, and then gradually peter out. The beginning phase is non-linear exponential rise or acceleration. The middle phase marks a linear region where doubling the inputs more or less doubles the outputs. The end phase is law of diminishing returns where growth decellerates inversely exponential. The important point to notice is that the non-linear phases are at the lower and upper bounds of scaling. Process in general is bounded by thresholds. There are variations on this theme for example* critically balanced process wherein the s-curve is rotated 90 degrees such that there is a rapid rise followed by a stable linear region followed by another rapid rise. One can also have well shapes and saddles, upright or inverted. The point again is that well behaved, unstable, superstable, or linear regions are bounded by non-linear behaviours. These non-linear areas can give rise to unpredictability and chaos or divergent behavior. Stable areas are usually designated sub-critical while divergent areas are super-critical. Interface functionality arises out of these boundaries. The functionality itself might depend more on the linearity as in a transistor amplifier or it might depend more on the criticality as in a transistor flip-flop gate, but it is this boundary* that makes useful distinctions possible.

The above scaling laws have a profound influence when the process is recursive i.e. feedsback on itself. Oroborus, the snake eating its tail. Feedback can produce non-linearities, fractals*, and indeterminate behaviour. This gives interface the potential for infinite degrees of functionality as well as the potential for chaos.

Well, what about modern man and his technology. Technology is such a new word. What does it mean to us today? The two most important technologies of the past century - electronic media and computers - have for the past decade been converging into a greater* technology, the internet. For the first time in history individuals are able to interface with other individuals globally at a multimedia level. They are able (or will be) to access virtually all of mankind's timebound knowledge. Interestingly the internet or world wide web is an almost invisible technology. It is almost pure interface in a sense - a ubiquitous and invisible technology that puts us into direct contact with as much human culture as our own personal bandwidth can handle.

Ok, how does our previous discussion of scalable process* and interface functionality* apply here? Let's talk about scalability. Alan Turing and Von Neuman fathered the computer age by formalizing the conceptual model of how a machine could compute. The model itself was in Turing's words a model of how the mind computes. Computers thus became a real interface between machines and the mind of man. The initial problem was scalability. At first it was all wires and hand plugging, then punched tapes, then stored programs, then transistors, then chips etc. Eventually we ended up with millions of transistors and multitasking. But the interface was still limited to one person - one machine. A scaling* limit had been reached where adding more speed, transistors, software etc. had reached the point of diminishing returns in terms of its functional effect on individuals and society at large. The next leap occured when we started linking computers together in networks where a central server allowed many to share a common interface. In principle this was scalable up to a point. Next came switches and routers to connect networks. Finally we had a truly scalable internet capable of spanning the world. This was not the original intent of the internet. Bomb resistance was the original idea, but packet sending technology* enabled scalability as a byproduct because it detached the interface from any particular hardware or structural scheme. The infrastructure became free to scale up from the local level in a more or less linear fashion. Non-linearities are now building again as bandwidth bottlenecks occur, but new technologies will arise as needed because the system now represents a global interface* between man and himself. Oroborus. Homunculus scaled to its logical conclusion. Culture is both free to speciate and free to merge. It remains to be seen whether the system achieves the stagnation of closure. That is another aspect of the above discussion which hasn't been talked about. The differences between* closed and open systems, autopoesis etc. would make a good next topic.