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\title{Congestion - Mimesis of traffic flow}
\author{Peter Todd}
\date{April 8 2010}
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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.
\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:
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.
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
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,
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:
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."
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
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
\subsection{Representations of reality}
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