--- The Highline Canal: Complexity ---


Here are several sources of information about the Highline Canal.

https://www.denverwater.org/sites/default/files/LargeMapFINAL.pdf

https://www.denverwater.org/recreation/high-line-canal/guide-to-the-high-line-canal-trail

https://www.denverwater.org/recreation/high-line-canal

https://highlinecanal.org

One thing you will notice right off is how the canal meanders across the countryside for 71 miles.

Actually, that is not exactly correct. Rivers meander - the canal does...something else.

The reason a stream meanders is that various factors pull it in different directions - the same kind of factors that makes a thin stream of water meander across a gently sloping pane of glass. The glass isn't perfectly smooth so the stream has to flow around the tiny imperfections. Also, the earth is rotating out from under the water as it flows, creating the Coriolis effect.

The South Platte River shows this kind of meandering as it flows through it's valley near Denver and, even more as it flows out onto the plains.

https://www.google.com/maps/place/South+Platte+River/@41.056048,-101.5642861,12z/data=!4m5!3m4!1s0x8776f2ebc5765be1:0xbe93203c9541fa6c!8m2!3d41.1143877!4d-101.4849727

The Highline Canal is very convoluted for another reason. It follows the topographic contours of the eastern rim of the valley cut by the South Platte River. It flows by gravity so it's gradient must always be down and the Highline flows downhill at a gradual 2 feet every mile. If you've ever looked at a topographical map, you know that it's rare that a contour ever follows a straight line.

By the way, THE Highline Canal is not the only highline canal in the Denver area. There is also the Farmer's Highline Canal that flows through Golden to Westminster and Thornton. There are also several other highline canals in Colorado and, certainly, many in the world. A highline canal is simply a canal the follows a topographical high line (contour).

The highline canal, at a stretch, could be called fractal. That would not be completely accurate because fractility implies repetition at different scales. If you magnify a design and it looks the same at each magnification, that would be fractility. The classic example is the Mandelbrot set.

In physics, scale is capped by the size of the universe at the top and quantum "graininess" at the bottom. There are no such limits in mathematics. You can always come up with a smaller number. So when I say that the Mandelbrot set has infinite detail, I mean it. You can keep magnifying the Mandelbrot set for ever and you will still discover more detail.

Check out this series of magnifications of the Mandelbrot set.

                                                                  [Mandelbrot set]

That last one....is that a tiny Mandelbrot set that I see? Why, yes...I believe it is!

There are, in fact, an infinite number of Mandelbrot sets embedded in any Mandelbrot set. There are also other designs like this one.

                                                          [Detail from the Mandelbrot set]

A shocking detail is how the Mandelbrot set is generated. It grows from this simple equation.

                                                       [Equation for the Mandelbrot set]

This equation tells you to set c to some complex value and then calculate the result of the equation when z is 0+0i. Next place the result back into the equation as the new value of z and keep doing that.

Some values of c, for instance c=0+0i, just sits there quietly. In the case of 0+0i, each step just returns 0+0i. For some values, like 0.804608667883013+0.820834819347395i, the equation "explodes" on the 11th step and my spreadsheet can't even calculate the 12th step because the result is too large. On the other hand, 0.107533656386414+0.479408179270122i shows no sign of getting out of hand through 25 steps.

To create the Mandelbrot set, you just do that for every value of the complex plane and, if a point "explodes", graph it. You can use different colors to indicate how quickly it explodes. The points that don't explode are members of the Mandelbrot set.

If you look at a map of the Highline Canal, a mile long section doesn't look like a 10 mile long section and there doesn't seem to be a copy of the whole canal hidden within a shorter segment, so it would be a stretch to call it "fractal". But is complexity conserved at different scales? In other words, is a 5 mile segment just as wiggly as a 20 mile segment?

Obviously, this could only go so far. A 5 foot section of the canal is not complex at all - it would look straight! But what about "reasonable" lengths - something you could see on a map.

There are measures of complexity that require a computer and a very complex equation to calculate - I wanted something a little simpler. I decided to use a measure that compares the "hiking distance" of a section with a "as the crow flies" section. The Highline Canal trail has been measured - I'm going to assume that the trail mileposts were set up according to hikers with pedometers, but I don't know. I can determine straight line distances using the Google Maps measuring utility.

Now for the comparisons. It would be easy enough to just subtract the straight line distance from the much longer hiker's distance, but then I couldn't compare a 5 mile section with a 10 mile section. I needed a standardized measure. A ratio would do the trick so I used the ratio of the hiker's distance to the straight line distance.

Ratios are dimensionless. As a fraction, if the numerator and denominator units of a ratio are the same, they cancel out. That's good for my purposes because, a ratio of two measurements will be the same whether the units are feet, miles, or kilometers. They aren't dependent on scale.

My measurements are in miles, to the nearest mile. I'm taking measurements off maps that show the positions of trail mileposts so I can't be more accurate than a half mile; therefore, I round map distances to the nearest mile and I can't do any better than that in calculations. That's okay. It's good enough for my purposes.

I have a ratio for the entire trail, four for the quarters that I'll be walking this year, and 15 for the roughly 5 mile segments shown in the Highland Canal Conservancy maps. I've also calculated the arithmetic and geometric means and standard deviations for the quarters and segments, and I have a histogram for the ratios for the segments.

                                                                         [histogram]

The reason for the histogram is that, if the distribution of the ratios looks close to normal (which it probably is not), I can put some stock in the statistics I've calculated. Luckily, the histogram shows a curve that could be mistaken for normal - it's probably binomial but I'll ignore that.

The Highline Canal is 71 miles long, walking distance. As the crow flies, it's only 30 miles long. The ratio is about 2. How similar are segments of the trail at different scales?

The quarter ratios range from 2 to 3 for an average of 2 and a very similar geometric average. The calculated standard deviation is about 0.4. Smaller standard deviations mean less variation. If you select a 20 mile segment of the canal trail, there is a 68% chance that it's ratio will be within 0.4 of the average: somewhere between 1.9 and 2.7.

The 5 mile segments look very similar in terms of complexity. The average and geometric average ratio is close to 2 which is well within the expected complexity of the quarter sections and not far off from the ratio for the whole trail. The standard deviation is about 0.7 - more variation than what is seen in the quarter sections, but still not a lot.

What all these numbers mean is that, in terms of how much the canal and it's trail meanders, 5 mile segments look a lot like 20 mile sections, and the 20 mile sections look a lot like the whole trail.

If you walk the trail, you'll see what I mean. You can see that it's very "wiggly" from the maps. There are places that you can walk 10 minutes and you find yourself very close to where you were 10 minutes ago.

Some parks have very convoluted walking trails simply to pack longer trails in smaller areas. Complexity can be useful. The Highline canal is convoluted so that it can follow the terrain with a constant, very shallow down-hill gradient.

On a surface more complexity provides more surface area. Your small intestine membranes are very complex, packed with tiny fingers jutting into the inner space of your gut. They increase the surface area of the membrane that absorbs nutrients from your food. Chemical engineers who design catalysts for things like catalytic converters and chemical reactors want the catalysts to have as much surface area as possible so that as much of the reactants can get to the catalyst as possible.

If the Mandelbrot set could be translated into a physical object, it would have infinite surface area and would make a great catalytic surface or absorber. Of course, it can't, but you can see why fractal surfaces can be useful.

If you would like to explore fractals, there are many Mandelbrot viewers on line such as the one at:

http://math.hws.edu/eck/js/mandelbrot/MB.html

or you can download the Mandelbrot explorer at

https://www.mandel.org.uk

There are many examples of fractals in nature. The way limbs split off a larger limb looks a lot like the way the larger limbs split off even larger limbs, and the way vein in leaves divide. Smaller sections of the swirl of sunflower seeds look like the whole swirl. The next time you walk, look for fractals on the trail. You might be surprised at how common these intricate patterns are.

--- Adventuring philosophies ---

True enough, my final home is still out there, but this is most certainly my home range and I love it. I love every rock I fall off and tree I trip over. Even when I am close to dying from exhaustion, a beautiful sunset doesn't lose it's power to refresh and inspire me and that, in itself, is enough to save me sometimes.

W. Ross Ashby came up with the Law of Requisite Variety that states that, for a system to be able to adapt effectively to its environment, the internal complexity has to at least match the complexity of its environments. That certainly makes sense. For a system to survive an assault from outside - temperature fluctuations, stuff falling from the sky, squirrel attack, plantar blisters - the system has to respond with a defense specifically designed for that attack. Acorn bombing - umbrella. 60 degrees to 20 degrees frozen rain (remember, I live in Colorado) - packable rain gear.

Mostly, where I grew up, the environment was manageable. The exception was the third weekend in March. That was the weekend of the SEHowl. We had only two Howls with fine weather throughout. More typically, the first part of the week was bitterly cold (we had one howl where water would freeze as we tapped it out of the water carrier) and the last part was warm enough to cause you to sweat if you were active. With a change like that, you can probably figure out what the middle of the week was - torrential storms. Flash floods and tornadoes were a fairly common activity in northeastern Alabama.

But in Colorado, there are not many buffers in the environment. The Army Corp of Engineers seem to have gotten the flash floods under control on the Denver side of the Front Range but they haven't got a handle on the weather. It can still jump 40 degrees in a couple of hours. Bear Creek still rises and falls in the matter of minutes. In Alabama, the meteorologists do a pretty good job of predicting weather five days ahead. Here, weather forecasts change hourly the day before. I'm pretty sure that, if a dog pees on the ground here, the pH of the soil will go nuts.

Alabama sorta protects people against the environment. If a person falls, they're likely falling on something they can eat. Here, there are a lot of berries and such that are really good; unfortunately, there are a lot of plants that look like those edible berries but they're likely to make you quite ill.

Here, requisite variety comes into play much more strongly than in Alabama. Colorado inspires an adventurous spirit.

I wouldn't say that I live randomly. Age has forced me to be more regimented. I take three handfuls of pills everyday and two eye drops. Certain pills, if I miss the dose, I know as soon as I hit the trail. But I have ways of injecting randomness into my life. I use the roll of the dice (see my ToolBook http://www.theriantimeline.com/ToolBook.ods for a convenient randomizer) to decide a lot of my activities. I often take the "wrong way" to get to where I'm going. And it has been a long tradition of mine to just get lost. I can afford to do that with impunity here since, in Denver, it's really hard to actually get lost. Find the mountains - that's west.

I like science kits. I always have. That's what I always got for Christmas. But once you hit the trail, it's a different story. In the lab, you practically know what's going to happen when you start doing something. The satisfaction is in seeing (as Walter Lewin says on his videos) that science works. On the trail, you get surprises.

In the lab, things are clear cut. On the trail there is a phenomenon called sensitivity to initial conditions. I've recently watched a lecture series on weather and the lecturer often made the point that, not only can't we predict the weather far in advance, but we'll never be able to predict the weather far in advance. In fact, the father of chaos theory (that's the short name for sensitivity to initial conditions) was a weatherman named Edward Lorenz (that might be a little unfair since Henri Poincaré was working with chaos theory almost a century before.)

In the laboratory, if you do an experiment and then you redo it under the same conditions, as accurately as you can, you're going to get pretty much the same outcome. Outside the lab, you would have to repeat the experiment under exactly (and I do mean exactly) the same conditions to have the same outcome and it is literally impossible to be that precise. And a tiny deviation in starting conditions can lead to wildly different outcomes. That's why studies in the social sciences, ecology, and other field sciences can be so "imprecise". In the lab, you can control most of the irrelevant conditions that might contaminate an experiment. In nature, there are just way too many confounding variables.

In nature, you need enough requisite variety to deal with sensitivity to initial conditions and complexity. There are always things coming at you and, if you want to record all the important stuff, you really have to stay alert. Conservation of energy doesn't go away in chaos. Nature will balance it's forces.

I've said before that you can't force nature and expect to come up with predictable outcomes. The way to handle nature is to be part of it and to influence it from inside the system. Forcing nature is like trying to patch up an old dam. Patching one leak causes pressures at another spot to create another leak somewhere else.

That's the problem of walking on black ice. If you ever start slipping, the sudden motion of trying to catch yourself throws you off balance in a new direction and you're going to go down.

They used to talk about preventing tornadic storms in the southeast. It sounds like a good idea on the surface but that is where the southeast gets it's water. I suspect that preventing tornadic storms in the southeast would turn places like Alabama and Georgia into deserts.

People keep bringing plants and animals into the US to deal with environmental problems. They figured that kudzu would deal with erosion and made it worse. Providence Canyon in Georgia is the product of erosion caused by kudzu. If you drive down there, you'd better have deer whistles on your car. Deer don't obey pedestrian laws. Game management people thought it would be nice to stock the area with white tails. Without the natural predator (wolves), the deer population went crazy. And I'd like to know which brilliant hunter thought it would be a good idea to stock Alabama with wild hogs.

So, why should the answer be "get inside and influence?"

Well, let's say you wanted a friend to go to a concert with you. You force him into a car at gunpoint and force him to drive to the concert. You get what you want, but you also get several things you don't want. You're friend pouts through the whole concert and neither of you enjoys it. Also, your friends wonders how you could do such a thing and decides that he never wants to see you again.

Here's a different solution. You know your friend well enough to know that he likes the band and will go if you are willing to pay for the tickets, so you offer to carry him to the concert and get the tickets. He agrees and you both have a great time.

The "superpower" that allowed you to find a better solution is called "theory of  mind". You're able to develop a model of people you know in your mind that allows you to predict what an other will do in response to things that you do. Part of theory of  mind is what people call empathy. If you're good at it, you can read others so well as to practically read their minds.

Does nature have a mind? If you're an animist like me, then you would say, "yes". but that's not necessary for you to be able to develop a theory of mind of nature. A theory of mind is just that - a theory. It's a conceptual model that allows you to predict what an other will do in response to what you do. It doesn't really matter whether the other mind is a real  mind or an imaginary as long as the model works.

What does matter is that you have enough information - that you "know" the other. The reason people can develop theories of mind is that people have brains that are not logical  machines like computers. Brains are pattern processors. They gather all the patterns in the world around them that are relevant to the issue at hand and they merge them into a master pattern. The master pattern behaves as a person would and if you're good at problem solving or empathy, you can read that master pattern - because that's what brains are good at.

You might think I'm talking about intuition and, yes, you would be right, but I am specifically talking about a well-trained, knowledgeable intuition.

In one of my many jobs, after my initial interview, I was talking to one of the people that would be a coworker. I mentioned that I'm a good judge of character. He asked what I thought of the person who had just interviewed me and, without pause, I said, "You mean that he's completely self-serving and that he'll do anything and surround himself with whoever he thinks will get him ahead?"

When I looked up, the guy was agape. He was used to everyone's first impression being, "Wow! What a great guy!"

You can understand nature by paying attention to how she responds to things. You can develop a cooperative relationship with her in the same way you develop cooperative relationships with anyone.

Relationships are like gardens. In order to get the results you want, you have to cultivate them. If you just let a garden develop on it's own steam, you might get the plants you want but they'll be sickly, and there will be lots of plants you don't want. A healthy garden requires nurturing and weeding. The same goes for a relationship.

People are mercurial; so is nature. Working with either requires one to be as changeable as they are. Adaptability is important in survival or evolution - it is just as important in adventuring. Surprises just happen and you have to be ready for them, therefore, you have to be able to fend off distractions.

My favorite poet is Robert Service, the "poet laureate of the Yukon." His poetry is eminently lyrical. It begs to be put to music and, many years ago, I did just that. If you want to hear the result, here's a link:

http://theriantimeline.com/Robert%20Service's%20Canada.mp3

One of the songs, "The Reckoning" has the following stanza:

"Time has got a little bill - get wise while yet you may,
For the debit side's increasing in a most alarming way;
The things you had no right to do, the things you should have done,
They're all put down: it's up to you to pay for every one.
So eat, drink, and be merry, have a good time if you will,
But God help you when the time comes, and you
Foot the bill."

Actually, pain and inconvenience are payment that you make to live in this world. You're going to get hurt and that hurt is going to leave a mark. If you place yourself in harms way, as you do when you adventure, you will most certainly end up with a collection of dints and dings. The good news is that most of the pain is not an issue.

At it's best, pain is a signal that something is wrong in the body and that something needs to be done to restore a healthy equilibrium. More often, the body thinks that something is wrong but is either mistaken or is noticing something that will resolve on it's own or will never resolve but isn't going to interfere with anything you want to do.

The trick is to pay attention to your body until you can discern which is which. I know that my feet could be bleeding with blisters and it will come to nought in three days so I can ignore them with impunity. You  may need to realize how far you can push your blisters before you really need to do something about them - and you might need to know what to do about them.

Regardless, pain isn't the issue. Injury might be an issue, cramps that immobilize you  may be an issue, but pain is just signal or noise. Pay attention and then respond appropriately.

Pain can immobilize you if you let it; so can philosophy. Philosophy is as dangerous an adventure as just about anything.

Consider Rene Descartes. He was so popular that he was invited by Queen Christina of Sweden to organize a scientific academy and to tutor her. Evidently, they did not like each other much and Descartes died there of a respiratory ailment, far from his beloved France. His claim to fame was his dictum, "I think, therefore I am." but almost as soon as it was said people started finding holes in it and Descartes had to start defending it. There wasn't much defense available. It's easy to knock it down. For instance, your thinking certainly indicates that something exists, but not necessarily you. You might just be a character in someone elses imagined story.

One of my favorites was Rudolf Carnap, a giant of symbolic logic and a member of the Vienna Circle, a group of people who believed that, if it couldn't be talked about scientifically or logically, it shouldn't be talked about at all. That philosophy, called logical positivism, like many nineteenth and twentieth century philosophies, didn't last long, quickly being superseded by Post-Modernism, which was largely a reaction to philosophies like logical positivism. Honestly, trying to dump four fifths of the Dewey Decimal System was doomed to failure.

But I did like the depth to which he developed logical systems, and I very much sympathized with his disdain of many of the central questions of philosophy; although I think he went too far. It is important to establish some understanding of the nature of existence, moral issues, and whether there is anything beyond the material world.  But a lot of the classical philosopher's conclusions seem a bit sophomoric. For instance, Kant said that the central tenant of ethics is that humans should never be a means but always an end. But that eliminates a large part of the basis of relationships. There is a such thing as good faith in use. Much of friendships and love relationships are based on common, honest use. People are the means of cultivating relationships.

Jean-Paul Sartre has always seemed to me to be a miserable individual - much more so than is called for. He said, "Hell is other people" (from his play No Exit). He taught that self-consciousness was a curse. Methinks the philosopher doth protest too much.

And B. F. Skinner, just don't get me started. I came into psychology when Skinerian behaviorism was big. I had a textbook that expounded the glories of behaviorism with the fervor of a Marxist. The authors stated flat out that all other schools of psychology would (not "should") be relegated to historical studies. Twenty years later, behaviorism had morphed into cognitive behavioral psychology and emphasis has definitely shifted to what is going on inside peoples' heads. I hope the authors of that text book are embarrassed.

I think that one big problem with philosophical thought has been that philosophers just think too much. I liken it to a treasure hunter that digs a little too far to the right and completely missed the treasure, coming up with only dirt - interesting dirt, to be sure, but dirt nevertheless.

There has been lifetimes wasted on consideration of the puzzle, "Can God create a stone too heavy for Him to lift?" The sentence looks like it makes sense purely because it is a reasonably constructed sentence; otherwise, it makes no sense at all. Many philosophical statements are perfectly good statements. They have good grammar; the words mean something; but they just don't make sense.

I was in a speech team in the 70s and it was around that time that debate judges started disallowing nihilistic arguments. It's perfectly reasonable to argue that, since you can't even argue that the things around you are really real, you certainly can't argue anything else. The problem is that, if you start with the assumption that there is no reality (or no provable reality), then you can't go any farther. Debate is dead in the water before it even starts.

But debate is important. It is the way people come to conclusions about issues that can't be solidly proven, some of the most important issues there are - issues of policy, responsibility, value.....

Even if you lock yourself in your house and never go out, there are dangers and the biggest danger is nonparticipation. Life is participation and survival is not just living. In the same way, thinking is dangerous, but not thinking is the worse choice by far and the key is to finely hone that incredible thinking tool, your brain, through learning and experience.