Fractals, Graphs and Transportation Networks

In the summer of 1998, I was contemplating entering a PhD program in the school of Arts and Sciences as I attracted the attention of Thomas A. Breslin, Professor in Department of Politics & International  Studies arisen from my interest in fractals and a couple of research papers I wrote about them as they  relate to many aspects of evolution.  Following this line of thought--by introducing a small change in a natural or man-made dynamic process the result over time, would be in both cases, an entirely new organism.

My view then was that in a space of time where dual processes are taking place, one natural and random and the other planned and man-made would lead to the conclusion of two independent processes that could either lead to entropy or continue to change ad infinitum with just an infinitesimal amount of variation on each cycle.

Probably, the first person to observe this dynamic process was Edward Lorenz when working at MIT using a three variable model for forecasting the weather in the 1960’s. At this time, computers were pretty slow and cumbersome, so, when he decided to extend his weather forecast for a few more days he rounded off some numbers into his model’s equation expecting only minor change in the model. He ran the model again and to his surprise the minute differences had made a startling difference in the forecast. Lorenz’s discovery revealed that in the immense dynamic s of a weather system all parts are connected by feedback to all the other parts affecting the way a system ends up after interacting with its component parts.

Subsequently to Lorenz’s discovery many researches embarked in trying it out in other dynamic systems from printed electric boards to brain functions and in the process they found new laws.

In the 1970’s an IBM researcher, Benoit Mandelbrot conceived a new geometry which he named “fractal” to suggest perhaps fractured or fractional with its main focus on scaly, flaky and uneven surfaces. Fractal geometry was meant to describe the roughness of nature, its dynamic process and its use of energy. As fractals patterns are studied other secrets are revealed such as scaling and self-similarity when visuals of an image at a given scale are reproduced as the scale is change up or down.

Nevertheless as previously mentioned, if a fractional change is introduced in a pattern its interaction with the whole will eventually render a new pattern or a new scale of the same pattern. This suggests that a nonlinear co-existence is taking place in a circular continuum with no beginning or end in a multidimensional space.

Modern computers have made possible the visualization of many fractals the most famous being the Mandelbrot set and also the most famous object in modern mathematics and a leap departure from the tradition of Euclidian geometry.

“The set itself is a mathematical artifact…clustered in a complex number plane.”  As said by John Briggs, in his 1992 book “Fractals the Patterns of Chaos”. As we know from the study of the mathematics of complex numbers there are two parts to it: One real and the other imaginary.

During the 1980’s Mandelbrot and others were using simple iterative equations to observe and study the behavior of numbers in the complex plane. Many were using a three slot equation described by :

[Changing number]+ [Fixed number} = [New number} where this new number is plugged into the first slot to become again the changing number, i.e. 0+1=1, it follows that 1+1=2 and so on. This simple equation of number iteration can be accomplished with whole or imaginary numbers each rendering different visual representation when running in a computer.

In a previous paragraph it was mentioned that my personal take on fractals was twofold one natural and random and the other planned and man-made. Since randomness is all around us and self-explanatory only man-made planned change will be consider for now.

Having in mind a sample of a digital photograph or an idea that can be digitized for study and observation, and considering the components of this picture/idea made out of pixels that can be viewed over a interval of time: From (zero time) to (completion time) then the pixels can be spread out along a dimension of time-distance. 

As a planned iteration among the pixels takes place a complete photograph or idea emerges at the end of the time slot selected. This action plan once executed will render a completed objected in the tradition of Euclidian space of shapes that model nature.

Fractal geometry provides a closer view at nature’s subtlety change in the un-noticeable slow motion of time. Notwithstanding, we all know that if an event can be fast forward the slow change can become possible and visualize its entire process to its terminal entropy.

During this period in my pursuit of fractals, my wife died and I became too depressed to continue with academic work. As years went by my curiosity was aroused again only this time I thought applying my background on transportation networks to my previous fascination with fractals I started viewing the two as one neural system that could be graphed and be placed in a system of inputs and inputs with varying attributes throughout the network that could provide a utilitarian mission for international transportation.

So, from iterative non-linear equations I moved to the study of graph theory and its mathematics.  This time I became convinced that such neural system could be built over a geographic information system such as Google Earth and run in real time in layers of ground, water and air transportation networks in Euclidian spaces using vehicular speed displacement in the graph attributes to adjust for the topology of space transforms.

Since I am not a computer programmer I needed to enlist the expertise of one qualified to build the idea into a tangible object. After several months of frustrating research on the subject I came across Kevin Chugh a Ph.D. computer scientist and his team. At first, he was reluctant that such a system could be built let alone function  but after several sessions of discussion with me and his team he decided they could develop such a system using the parallels of graphic interaction embedded in social networks commonly used today.

In fact and in theory transportation networks should be easier to build as only vehicles move but not individuals. Also, all vehicles move in a predetermined time-space within the confines of our planet. Other components of the networks such as seaports, airports, cities and depots exist in a fixed location and can easily be graphed.

The question that remains is whether a utilitarian application is feasible to solve many of the problems burdening the transportation industry today? We’ll try to answer this question and others in a future paper once a model is available.