The Theis Equation and Flow

Mathematics is remarkably effective in describing the physical world in part due to isomorphisms, relationships between concepts that reveal a similar underlying structure. In 1935 Charles Vernon Theis was working on groundwater flow, a subject with little mathematical treatment at the time. He thought that perhaps a well tapping a confined aquifer could be described using the same mathematics as the heat flow of a thin wire drawing heat from a large plate, as this work was better established. With a little bit of help from C. I. Lubin and considering how parameters describing underground water flow could be compared to those describing heat flow in solid materials, he developed the Theis equation which is used to this day to model the response of a confined aquifer to pumping over time.

I developed a small program which allows visualization of the potentiometric surface of a confined aquifer subject to pumping using Processing. This particular example uses aquifer and pumping parameters from a Geo-Slope whitepaper.

The source code may be downloaded here. All values including aquifer, pumping, visualization, and numerical parameters may be varied to apply to a wide variety of situations. The exponential integral (or “well function”) is calculated using a numerical approximation accurate to at least 1 part in 10,000,000 .

Convergence in the Lorenz Attractor

Most visualizations of the Lorenz attractor are of a long history of a single point after convergence to the attractor has occurred. I was interested in what the surrounding space looked like, so I randomly selected 20,000 starting points from a three dimensional Gaussian distribution with a standard deviation of 100. Each point was iterated, and a short history displayed as a trail.

Interestingly enough the points do not simply fall in from arbitrary directions like a gravity field, but display structure by instead swirling along a clear path up the z axis.

Our Modular Minds

I believe that ultimately human consciousness can be described by a program. Now this doesn’t mean we’re all in the Matrix, simply that our mind is a giant seething logical machine with values that are manipulated by rules. There is no strange new science in the sense of new specialties that must be discovered in order for the mind to be understood, but a progression in a new kind of science as Wolfram dubbed it – the study of how complexity arises.

A List of Rules

When I first heard that you could program a computer as a child I was amazed. A strange wonder that I could only spend a few shared minutes with at school, something that could draw, add, and write far faster than I could ever dream of – and I could tell it what to do? I wasn’t quite sure how to inform it to bend to my every wish, so I started with Turtle (actually called LOGO I later found) upon my teacher’s recommendation. I fed Turtle long lists of instructions – move forward, draw a line, turn left, repeated in all and any ways I could think of. He would draw glowing green shapes across my screen, and never tired.

The Need for Modularity

The only problem was that while the Turtle seemed infallible, I certainly could not say the same. I was making the classic beginner’s mistake – I would write one giant chunk of code that was supposed to cause my turtle to dance in precisely the way I wanted. Any little mistake would send it widely off course and I would end up with a mess that barely looked like the original design at the best of times. I later learned that using a programming concept called “modules” could help me isolate these errors and make code more efficient and reusable. Just like a company could have a manufacturing and engineering division which could communicate with standardized blueprints, a program could have different modules that would exchange data in a standardized manner. A modular program is more stable since mistakes are typically limited in influence to the module they’re contained in, and each module can be modified by separate influences with only the understanding that they are supposed to behave and communicate in a certain manner.

Damage as Evidence

So is our mind modular? Well, if it wasn’t, we could assume that a brain injury affecting a certain part of the brain would have a consistent and general impact across all of our consciousness. The only problem is that we generally only see a nonspecific mental decline like this from a nonspecific trauma, say impact blows to the head over a long period of time. Injuries in specific areas seem to be correlated with deficits in certain mental abilities – while leaving others totally intact.

A stroke can basically be thought of as an incident where blood flow is drastically affected in a certain specific area of the brain. This subregion of the brain is unable to function due to lack of blood flow, and very strange things can occur.

Howard Engel is a Canadian novelist who had a stroke. Upon waking one morning, he found that the morning paper seemed to be written in some strange script, an alphabet he could not understand. Everything else appeared normal, except his visual cortex had been damaged in a specific area which prevented him from visually parsing letters and words. As a writer, he despaired – it seems that his livelihood had been lost. Soon he realized a critical distinction which gave him hope – he may be unable to read visually, but could he write? Howard sat down and traced these strange looking symbols, his pen gliding over the bizarre shapes over and over. And eventually, the concepts came back to him. In a strange sense, he could now read again. Years and years of writing had associated certain movements of his hand with letters and concepts. Instead of words in his head put to paper by hand movements and a pen, he had to move concepts in the opposite direction – moving his hand over shapes written previously by others, the concepts echoed back up his motor cortex.

And it worked. There was irreparable damage to his visual cortex, a critical module malfunctioning. So he hacked his brain, redistributing resources from his motor cortex which had been trained to recognized these same symbols and concepts necessary for reading by his constant writing. Howard now traces the shapes he sees on the inside of his front teeth with his tongue. His speed has steadily increased, and he says he can now read about half of the subtitles in a foreign film before they flash off the screen.

It doesn’t seem too strange to suggest that there are different localized modules in the brain for motor control, visual interpretation, and other concepts easily identifiable with different aspects of the physical world. But are there modules with finer distinctions, working on different parts of our mental experience rather than different parts of the physical world?

The Wason Selection Task

The Wason selection task is a very interesting experiment in the field of psychological reasoning. Before I spoil it for you by talking too much, let’s just do it right now. Look at the following cards.

Assume the cards have a number on one side, and a color on the other. What cards need to be flipped over to make sure that all even numbers have red backs? Make sure you’ve picked a card or cards.

Got it? Now a bit of unsettling but ego-salvaging news. When this experiment is done with undergraduates, only 10 to 20 percent get it right. The correct answer is to flip over two cards: the number 4 to make sure it has a red back, and the blue card to make sure that it doesn’t have an even number on the other side. Most people suggest flipping over the 4 and the red card – this is wrong, as it doesn’t matter if the red card has an odd number on the other side.

Now let’s mix it up a bit. Instead of numbers and colors, let’s try people and social activities. Assume the cards have a drink on one side, and a person of a certain age on the other. What cards need to be flipped over to make sure everyone drinking beer is old enough to do so?

Now the answer flows quickly and easily, and almost everyone gets it correct. We need to flip over the beer card to make sure that the person on the other side is old enough, and we need to flip over the card showing the underage drinker to make sure they’re playing by the rules.

The weird thing is that both examples are logically equivalent. Instead of numbers and colors, we’ve just used people and drinks. But something very important has occurred, and it happens time and time again as these tests are administered. It seems that people are fast and accurate at solving this task only if it is described as a test of social obligations. They both can be described identically with logic – but that doesn’t appear to matter to our mind. We appear to have a module dedicated to social reasoning and conflict, and can only solve these problems quickly if it involves determining if someone is cheating or breaking social conventions. This ancient module would hold significant survival value – a general logic verification module not quite so much.

Modules Upon Modules

There appears to be significant evidence for a modular mind, not just in terms of divisions between senses such as sight or hearing and other actions like movement but also more abstract modules that deal with concepts such as social rules. Stroke victims can literally rewire their brains, passing concepts upward into their consciousness through paths never intended to be used in such a strange manner, duplicating the work of other modules lost to injury. These modules live in a strange world of physical interaction and abstract mental space, a huge interconnected mass with no clear outline behind it. The big question now becomes: is a sufficiently complex system able to understand itself, and are we that system?

Elementary Cellular Automata

Simple rules can often give rise to very complex behaviour.

applet

Applet built using Processing, Wolfram CA code example used as base. Source code available upon request. If this applet fails to load or screws up in any other way, please leave a comment with your browser version and operating system – thanks!

What is this?

Imagine we have a line running left to right made up of squares, and these squares can be either white or black. We then want to draw another line directly below this – but how do we do it? What rules should we use?

Well, we could use a very simple rule and just copy it directly.

eca-duplicate

But that gets pretty boring. Maybe we could use a slightly more complicated rule and say if a cell used to be black, it’s now white, and vice versa.

eca-invert

Well, it’s a bit more interesting, but not by much. What is a simple set of rules we can use that will produce more interesting behaviour? Perhaps instead of having a square rely just on the previous one, let’s have the rules depend on the previous square and its neighbors.

eca-concept

This means we’re looking at three squares that can either be black or white. There are eight total possible combinations, shown below.

eca-generic_rules

And we can decide what we want to happen when any of these situations occur. Lets try this set of rules:

eca-example_ruleset

And see what happens when we start with a single black cell.

eca-example

Could be interesting! It’d be a lot easier if we had more lines and squares for us to see a bigger picture of the structure produced by these rules though…

So that’s exactly what the application at the top of the post does. You can select whatever rules you want for the eight possible states, and toggle between a single point and random data to start by clicking on the circle in the top right.

Some Interesting Rule Sets

Since there are eight possible combinations that we can choose to either result in a black or a white square, there are a total of 28 = 256 possible rule combinations. A lot of these rules end up producing patterns that become very similar, and it was found that there are 88 unique rules, depending on your definition of unique. Here are a few interesting ones.

Rule 110

random initial conditions
rule-1
How complex can the behaviour be from these simple rules? Well, you can use this ruleset to simulate a computer. You’ve got to set up the initial line of black and white squares very carefully, and it’s equally hard to read the results, but the logic ends up being exactly the same. If you had a lot of time, you could do some interesting things…

Rule 90

single point initial condition
rule-2
This ruleset produces a fractal – the Sierpinski Triangle – when a single point is used to start.

Rule 30

single point initial condition
rule-3
If you’re wondering if these things have any real world application, this rule set displays chaotic properties and is used as the random number generator for large integers in Mathematica, used by millions worldwide.

Rule 169

random initial conditions
rule-5
I couldn’t find any information online about this, but it certainly stands out visually.

Rule 184

random initial conditions
rule-6
This rule set can be used to model traffic flow, or deposition of particles onto an irregular surface.