Ergonomics and the Anatomy of a Fractal Landscape

May 1, 2000
-sar

Natural systems interface ergonomically over time because of the principle of least action or geodesic logistics. Deviations or protrusions of interface wear first until ultimately interface fits like a glove between interacting systems. Subsystems which are already geodesically arranged wear the least and form the static structures of interface.

Interface between natural systems has a minimal dimensionality of one less than the dimensionality of the two interacting systems or subsystems i.e. the interface between two 3dim bodies is minimally a 2dim surface & the interface between two 2dim areas is minimally a 1dim line. I say minimally because natural interface being inherently fractal will in general evolve into a fractal dimension somewhat closer to the dimensionality of the subsystems. This maximizes energy throughput for the least amount of change ( "action" ). Elegant logistics is the keyword. Each rearrangement of interface at one scale opens the door to a self-similar rearrangement at smaller scales until the natural limit of system granularity becomes dominant and random forces take over i.e the chaos limit.

Interface resolves into a relation between static and dynamic poles of interaction. A subsystem interacting across interface finds itself playing either a dynamic role or a static role on a moment by moment basis depending on the direction of mass-energy exchanges across interface. The dynamic shapes the static over time, while the static gives form to the dynamic at any given moment in time. Interface is a marginally thin zone which maximizes throughput via impedance matching and sub-dimensional patterning. Impedance matching bridges the transactional or logistical bandwidth differences between the static and the dynamic. Subdimensional patterning (fractal) maximizes cross-section area or the number of one-to-one points of contact between subsystems.

Now I would like to add a quote from "Fractals, Chaos, Power Laws" by Manfred Schroeder. Schroeder says, "Yet, among all these symmetries flowering in the Garden of Invariance, there sprouts one that, until recently, has not been sufficiently cherished: the ubiquitous invariance against changes in size, called self-similarity..." Scaling invariance is quite simply the very essence of what a fractal is. One can have translational symmetry, rotational symmetry, and scaling symmetry, but it is scaling symmetry that creates fractals. Fractals form an interface between regions that is fractional in dimensionality. This fractional dimensionality, somewhat less than the dimensionality of the interacting subsystems, maximizes the throughput of interaction and minimizes the logistics.

Given the above, we can now analyze how a coastline comes to be fractal.

First let's define the interacting subsystems. A continental land mass in general will form the static pole of the interfacial relationship. The water system becomes the dynamic pole or susbsystem. We will ignore land shifts and variations in density as being causitive for the purpose of simplicity, but they can likewise be incorporated into a complete picture. Within the total three dimensional system, interface is minimally a two-dimensional surface defined by riverbeds, gullies, rivulets etc. and ultimately a ocean, sea, or basin. The global surface feature of this interface is that of static land mass rising above the lower basin area. This static subsystem forms the structure upon which the dynamic water system will act. The driving force in this system is the influence of gravity. The logistics of this global force are that it is essentially the same strength at all points and has the same direction - down. Hence "water flows downhill".

Having defined the interacting subsystems, we are now in a position to seek for the scaling invariance. Notice what happens when a bit of water is input into the top of the system. It immediately makes a beeline toward the lowest point it can reach directly with the least amount of travel. The direction of travel can be mathematically described to perfection as the vector gradient of the interfacial surface height. Now interestingly the kinetic energy gained in the process is basically equal to the gravitational potential energy lost, and the change in gravitational potential is directly proportional to change in height and height only. This means that the velocity at which a river or rivulet flows is dependant only on the rate of descent or gradient of the static susbsystem. Area and depth of flow relate in this case only to throughput. It is the rate of descent that is the prime mover here. The ultimate points of contact or change within the subsystem, are causality impacted via sudden velocity changes i.e. impulse forces, therefore velocity is the variable that determines change and velocity is determined solely by gradient of descent. As a rivulet twists and wanders around geodesically fixed land masses it wears a groove into the static subsystem. The efficacy of this structural change is determined mostly by the gradient. Steep slopes wear more quickly.

Now look at what happens after the first groove of rivulet is worn. The sides of the groove become watersheds themselves with their gradient pointing inward toward the center of the main bed. Since the sides of the groove were worn locally at the same gradient, they will spawn rivulets of their own which have a similar gradient of action. These smaller rivulets will be scaled down versions of the main rivulet and they will locally share the same scaling factor of local gradient. The hardness of the riverbed etc. will factor into the actual resultant gradients but on the whole, it is the gradient of descent itself that determines the scaling factor locally. Steep mountain crags spawn steep canyons. The prime mover, gravity, is invariant in this process, and the scaling factor is invariant locally. This means that rivulets that form on the banks of rivulets that form on the banks of rivulets all more or less exhibit scaling invariance. Self-similarity is the result. As the landscape changes, of course, the scaling invariances will change also, but the local rules will still invoke a fractal interface that is at least self-affine if not self-similar. This process continues on down to smaller and smaller scales till the granularity of the underlying geodesic changes abruptly and a new fractal dimension or interface opens up. The final end to the process must likely be a colloidal realm at the microscopic level wherein the two subsystems merge and the forces of gradient energy are not sufficient to penetrate further. At this level, one would characterize the interface as chaotic or randomly controlled by local factors outside of gradient descent.

The coastline becomes an elegant illustration of the process because it provides a simple crossection. The full three dimensional process goes on above water and below. The water in the basin was the very dynamic that spawned the fractal structure above water level. The water level represents the static end to the dynamic subsystem. It forms an equipotential plane wherein all water particles in that plane reside at the same gravitational potential. Water has achieved its final level of lowest energy. Unless more energy or water is put into the system, no more water will flow downhill. This plane bisects the three dimensional system interface and presents a two dimensional crossectional view of the fractal interface. The original fractal ranging from 2d fractionally up to 3d becomes a fractal interface whose dimension ranges fractionaly from 1d up to 2d. The dimensionality of a coastline is typically on the order of 1.33.

Behind all the above process lies an exclusion principle and a cohesive principle. Water and earth basically are different enough to be excluded from mixing. The cohesive nature of earth due to its geodesic arrangements beomes the static pole of interaction. It also provides the gravitational cohesiveness for the water system too. The fractal interface between land and water is a natural 'best fit' between such disparate subsystems. Water meets land and the fit is ergonomic.

It is interesting to point out that the complete natural system is cyclic and involves an atmospheric return process that re-energizes the system. From a global viewpoint, the ultimate chaos of atmospheric moisture becomes entrained into rain which becomes entrained into rivulets which entrain into tributaries which entrain into some final river basin. As the gravity gradient subsides near the final ocean basin the river opens up into a new fractal called a river delta which finally opens up into ocean. The whole picture is of a fractal source emerging out of chaos into convergent branches which blend into a unified and energetic singular trunk which then divergently forks in a fractal sink back into chaos. The natural boundary between chaos and order is fractal. Being a complete cycle, order emerges out of chaos, and chaos emerges out of order. In the natural water system, sunlight is the energy source, and gravity is the sink.

It is also interesting to point out that the upward branching fractal interface tends to remove random elements from the process as order emerges. The process becomes law abiding and ruled by the law of gradient descent. The downward branching fractal in contrast, introduces randomness into the process till the chaos limit is finally met. What we have are two scaling laws. Order is the result of upward size scaling. Chaos is the result of downward size scaling.

"Scaling invariance is the key that unlocks fractal relativity" ...rasputin11