\documentclass[]{article} \title{Congestion - Mimesis of traffic flow} \author{Peter Todd} \date{April 8 2010} \usepackage{graphicx} \usepackage{url} \usepackage{setspace} \usepackage{amssymb} \newcommand{\BibTeX}{{\sc Bib}\TeX} \addtolength{\oddsidemargin}{-.75in} \addtolength{\evensidemargin}{-.75in} \addtolength{\textwidth}{1.5in} %\oddsidemargin 0in %\evensidemargin 0in %\textwidth 7.5in %\topmargin 0in %\parindent 0in %\textheight 11in %\parskip 0.25in \doublespacing \begin{document} \maketitle \begin{abstract} Taking heed of the approach of physics to examine reality, construct laws, and deduce consequences, here we explore the process of creating an algorithmic work of art, through an iterative approach based on the steps of examining the underlying logic of traffic and traffic networks, constructing rules inspired by this logic, and examining the consequences of those rules. However where in physics the goal is one of prediction and explanation, here we strive to create rules that are an impression of reality. \end{abstract} \section{Background} \subsection{Discovering equations} Physics is the application of math, to predict and explain reality. In practice, the advancement of physics is process in which two main actions happen: experiment, and theorizing. Experimentation, done by experimentalists, usually consists of a controlled situation that the experimentalist sets up a scenario in reality, and then collects data as they observe what happens. A theorist on the other hand considers what is known about physics already, and tries to imagine further rules that could potentially describe reality better. Now suppose an experimentalist decides to collect data on what happens when objects are dropped. After dropping a large number of objects and recording the details of each fall, he will notice first that the objects always seem to move towards the ground. He will then notice that the time taken to fall a given distance appears to be related to the distance, $t=kd$, where k is a as yet unknown constant. Playing with the numbers further he notices that the relationship between time and distance is not linear; the amount of time taken does not increase proportionally and indeed for decreases as d grows larger, $t=k\sqrt{d}$ fits, although the constant is still required. He publishes a paper at this point, "On the behavior of various kinds of falling fruit", (title translated from the original latin) and comes up with tables showing k for apples, oranges etc. Our theoretician hears of this work being done and he starts thinking. The table of constants in the experimentalists paper are quite simply ugly in his mind; they represent hacks to make an equation fit, when no-one really understands what is going on. He is also aware of others published work attempting to describe how much an object pushes downwards when at rest, and other such experiments being done by his contemporaries. He starts to think that perhaps the heavyness, the downward push, objects have at rest is somehow related to them falling. Similarly, if one pushes an object along a floor, it will move. This push feels to ones hand no different from the push the object has on ones hand when one is holding the object. He decides to call this push sensation force. He notes that objects seem to have an intrisic force downards, which he calls mass, from massiveness, giving $F=m$ This fits with a contempories paper on balance weighing scales, that noted how not only could a weight put a balance beam into equilibrium, but ones arm could as well, by pushing downwards. Since a force can also cause an object to move, such as the case of a ball being thrown, an objects intrinsic force downwards should also cause it to move, towards the ground. To shorten our story, lets assume he has a concept of friction, and is already considering idealized thought experiments without it, leading to the concept that displacement is a function of velocity and time, $\Delta d=v \Delta t$ Since we are, at this point, already considering that velocity is fixed, we have Newton's first law. As stated by Newton himself: \begin{quote} Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed. \end{quote} The next leap is to consider how does the force impressed change the objects state? He already thinks that the downward force due to an objects mass is constant, and for an object at rest, it is only an upward force applied by the surface the object is on that balances out the downward force. Lets assume he is also refering to the change in velocity over time as acelleration. He could say that ths downward force accellerates the object, because obviously its velocity continues to grow during its fall. If that growth is fixed, then $v=kt$, with k being some sort of constant factor. Graphed, that would yield a triangle, with the height of the triangle being the velocity at the end of the time period. Now, mathematicians studying geometry already know how to calculate the area of a triangle, so using that we can determine how far the object will fall, in a given amount of time, $d=\frac{1}{2}kt^2$ and solving that for $t$ gives us, $t=\sqrt{\frac{2d}{k}}$ Ah-ha!'' says our theoretician, That equation looks just like what my collegue studying falling fruit came up with!'' Now he just has to figure out what is this $k$ constant anyway? Jumping ahead again, lets say he has realised that accelleration is porportional to the force, divided by the mass of the object, $a=\frac{F}{m}$ He also knows that the downward force, which he is now calling gravity, is porportional to the mass of the object $F_g=mg$, where $g$ is the accelleration due to gravity. Substituting that into his accelleration equation gives $a=\frac{mg}{m}$, the mass terms cancel out, meaning that an object, no matter what its mass is, will accellerate towards the ground at the same rate, $a=g$ Of course, actual observations don't show this to be true, objects like a feather fall far slower than a cannon ball, but at this point our theoretician has come up with the idea of friction due to moving through air, drag, and may even understand its exponential growth as $v$ grows, $F_d=kv^2$, causing the two forces to balance at some given velocity, the terminal velocity, where $kv^2=mg$ All of the above chain of logic is fairly easy to check against reality with tools available to Newton, and in reality Newton acted both as an experimentalist, and as a theoretician, although the exact line of logic presented above (and continued below) is purely the authors imagination. In any case we are still left with determining just what is $g$? Suppose that all objects with mass attract \emph{each other}? We could say that $Fg=k m_a m_b$, although it seems that more likely would be that the distance between the objects comes into play somehow, $Fg=k \frac{m_a m_b}{d}$, otherwise one would be as equally tugged by a rock very far away, as a rock nearby.\footnote{Indeed, many consider that exact sort of infinit reach of effects to be exactly why software, which runs on Von Neuman machines, is so unreliable: one can't simplify the reasoning of interactions, if everything interacts equally with each other.} To make a long story short, we're nearly at Newton's law of Universal Gravitation, $Fg=G\frac{m_a m_b}{r^2}$, although it turns out that the force drops off by the square of the distance. This equation \emph{is} consistant with our previous force due to gravity equation, if $m_a$ is very large compared to $m_b$ It also tells us that $G$, our gravitational constant'' must be a very small number, as the Earth is very large, which would also explain why no one has been able to observe this force between two objects.\footnote{As it turns out the value of $G$ is $6.673 \times 10^{-11} N m^{2} kg^{2}$, a value so tiny it was only until 1798 that it was measured by Henry Cavendish, 111 years after Newton hypothesized the inverse-square law of universal gravitation in Principia.} Remember though, we can't test this theory, at least not with the tools available to us. What we have come up with is in fact, theoretical physics. \subsection{The elegancy of equations} Physicist Murray Gell-Mann states that in fundemental physics there is an idea that a beautiful or elegant theory is more likely to be right than a theory that is inelegant''\cite{murray_ted_talk} This concept is taken surprisingly far, Gell-Mann relates how Einstein frequency brushed off criticism of his work, even from experimentors who had experiments that seemed to contradict his theories, on the basis that the equations of his theory were far simplier than the more complex equations of his detractors. Ultimately, Einstein was right, and the experiments flawed. Between objects fall'' and Newton's Law of Universal Gravitation, we've shown a thought process that goes between basic observation, to what at the time was theoretical physics. Of course, theoretical physics often doesn't stay theoretical for long, and universal gravitation soon turned out to describe the motions of the planets quite accurately. What we haven't yet shown, is similarities. For instance the force between two charged particles is described by Coulomb's Law, $F=k_e\frac{q_1 q_2}{r^2}$, which is identical to the law of universal gravitation. Similarly Einstein's famous $e=mc^2$ is identical in structure to whole hosts of other relations, like the above mentioned $d=\frac{1}{2}kt^2$, which in turn is identical\footnote{Don't be fooled by the $\frac{1}{2}$; the \emph{structure} is equivilent as we could have simply defined mass to be half of what we happen to define it as, and simply add a $2$ to any $m$ in other equations.} to the energy of a moving object, $E_k=\frac{1}{2}mv^2$ Gell-Mann explains these strange simularities by describing physics first as the description of a logical system, and then, from that, as an onion skin, where each layer resembles the previous one to some extent. He states: \begin{quote} The result is that newly encountered phenomena are described rather simply, and therefore elegantly, in terms of mathematics close to what was already developed for phenomena studied earlier. That is a property of the basic law, not of human observers. The manifestations of the law at different scales exhibit approximate self-similarity. Newton called it "Nature Conformable to Herself." \end{quote} Essentially math studies self-consistant logical systems, shifting at times between discovering new systems, and designing new systems, the balance between the two paradigms being fuzzy, and ultimately a matter of taste. Physics then comes along and studies reality, and finds the logical systems in math that best describe reality. Usually those logical systems are simple, and if they are simple, there just aren't that many to choose from, so the same systems appear over and over again. Of course, that is not to say that the \emph{results} of applying those logical systems have to be simple. The well known Mandlebrot fractal is defined simply as the set of complex values of $c$ for which the orbit of 0 under iteration of the complex quadratic polynomial $z_n+1 = {z_n}^2 + c$ remains bounded; Chaos Theory shows that iterating seemingly simple rules, may not produce simple results. \section{Application} \begin{quote} When you go out to paint, try to forget what object you have before you - a tree, a house, a field, or whatever. Merely think, here is a little square of blue, here an oblong of pink, here a streak of yellow, and paint it just as it looks to you, the exact color and shape, until it gives you your own naive impression of the scene before you. - Claude Monet \end{quote} \subsection{Representations of reality} \bibliographystyle{plain-annote} \bibliography{report} \end{document}