Saturday, July 17, 2010

Thinking in Straight Lines

Let’s face it, we, the civilized, educated, enlightened part of humanity like things to be straight. Let primitive tribesmen live in picturesque and practical round huts—we require abstract boxes of steel and concrete clad in plate glass, with plenty of nice straight lines, true vertical and horizontal planar surfaces and lots of ninety-degree angles to please the eye. Let these tribesmen spend their days meandering up and down picturesque winding paths laid down by grazing animals—when we build a road, we take a map and apply a ruler to it, and anything that’s in the way of that ruler, picturesque or not, must be dynamited and bulldozed because everyone knows that traveling in straight lines is more efficient.

This is good enough for most of us, and so we have come to regard straight lines as natural. In fact, in our world there are just two types of natural phenomena that give rise to straight lines: objects drop or hang down in straight vertical lines, and light beams travel in straight lines; beyond plumb lines and lines of sight everything is either a curve or a squiggle. But since most of our environment is artificial—and crammed full of straight lines and flat horizontal and vertical surfaces—we hardly ever have to confront this fact. Of course, the more scientifically astute among us know that straight lines are but a convenient fiction. We start with a conceptual framework of space that consists of x, y and z axes, and proceed to coerce our observations to fit this framework until the mismatch becomes too obvious to ignore, as with objects dropped from orbit, or with light from far-away galaxies that’s so warped by nearby galaxies that the image looks like a smear.

But the fiction is indeed very convenient. To start with, all straight lines are interchangeable and compatible. When we build, we tend to put things either on top of or next to other things, and if they involve straight lines, then no intricate fitting is involved—we can just slap it together any which way and efficiently move on to our next box-building exercise. When we go to a lumberyard, what we buy is not so much wood as straight lines cut through wood. Trees know a lot more than we do about constructing maximally efficient structures out of wood, but we like straight lines, and so we cut through the strongest part of the tree—the concentric rings of wood that make up the trunk—for the sake of making a perfectly straight stick. We could build beautiful, strong, long-lasting structures using round timbers grown to order (as some of us do) but generally we don’t because we are mentally lazy, always in too much of a hurry, and have made a fetish of straight lines.

Quite unsurprisingly, our preference for straight lines carries over into the way we think about relationships between things—the mental models we construct of our world. For instance, we consider it a matter of moral rectitude and straight dealing that the price be linearly proportional to the amount of stuff we get: if you pay twice as much, you should get twice as many potatoes. Quantity discounts are acceptable and sometimes expected, but pricing on a curve is generally seen as underhanded. We mistrust curves. Stepwise functions are fine, though, because they are made up of straight line segments. We can put up with having tax brackets, but try taxing people based on a nonlinear formula, and there is sure to be a tax revolt. Were the potato market a product of biological evolution rather than of human artifice, it would perhaps work like this: the price would be some nonlinear function that’s directly proportional to the customer’s net worth, and the number of potatoes dispensed would be some nonlinear function that’s inversely proportional to his net girth. Place your moneybags on one sliding scale, your flab-bags on the other, and some potatoes come out. Such a natural regulatory mechanism would prevent fat, rich gluttons from out-eating the rest of us, but it cannot be, for we have a very strong cultural preference for a simple linear relationship between price and quantity.

Straight lines are popular with grocers and their customers, but nobody loves a straight-edge more than the technocrat. Real-world data generally look like a collection of unique artifacts described by a multitude of qualitatively dissimilar properties and inferred relationships, all fluctuating unpredictably over time in a way that resists the direct application of the straight-edge. Therefore, the first step is to quantify the properties and, if at all possible, ignore the relationships. The next step is to choose just two parameters and to plot these artifacts as points on a piece of graph paper. Then, finally, a technocrat can grab a straight-edge, slap it down on the piece of paper, move it around a bit to find what looks like a good fit, and draw a straight line. Voilà: a linear relationship between two complex phenomena has been found, which can now be treated as real and objective—something that can be shared with one’s colleagues and be used as a basis for setting policy—because it involves a straight line which tells that one thing is proportional to some other thing, so that we know what result to expect when we perturb one or the other.

Straight lines are popular with engineers as well. Engineers work hard to design linear, time-invariant systems in which the output is directly proportional to the input any time you like. To them, deviations from linear behavior are defects. They are to us as well: we can hear it if the audio amplifier has nonlinear effects because it distorts the sound, and we can see it if the optics distorts the image. We can tell a straight line from a crooked one without any tools. But the mathematical tools which engineers use when they design these linear time-invariant systems are particularly good, as mathematical tools go. Mathematics can be quite fun as a sort of advanced parlor game for philosophers, but most math is rather problematic from an engineer’s point of view. You can describe just about anything using a set of differential equations, but most of the interesting phenomena—the behavior of an airfoil in an airstream, for instance, or the behavior of high-temperature gases in a combustion chamber—produce equations that can’t be solved analytically, and can only be approached using numerical methods, using a computer. A mathematical model is constructed, and random numbers are thrown at it to see what comes out. But linear time-invariant systems are described using a singularly well-behaved class of differential equations which do have closed-form, analytical solutions that directly provide answers to design questions, and so engineering students are drilled in them ad naseam and then go on to design and build all kinds of machinery that behaves as linearly as possible, from humble volume knobs to complex aircraft control surfaces. In turn, this well-behaved, predictable machinery allows us to achieve linear effects within the economy: build more stuff—get proportionally more money; spend more money—get proportionally more stuff. But, just as one might suspect, this only works up to a point.

Let us recall: straight lines are but a convenient fiction. There is no physical analogue of a mathematical straight line that goes from minus infinity to plus infinity. The best we can do is use all of our artifice to create relatively short straight line segments. Truth be told, the engineers can’t create linear systems; they can only create systems that exhibit linear behavior in their linear region. Outside of that region, nature does what it always does: make crazy curves and squiggles and generally behave in random and unpredictable ways. An example of what happens when we exceed the limits of the linear region from our everyday experience is the phenomenon of overloading an audio amplifier. The resulting effect is called clipping, and it sounds like a particularly unpleasant, harsh, grating noise. There are only two solutions: turn down the volume (return to the linear region), or get a more powerful amplifier.

In the economic realm, the effects of exceeding the limits of the linear region can be even more unpleasant. While within that region, building more houses generates more wealth, but just outside of that region strange things begin to happen rather quickly: house prices crash, mortgages go bad, and building any more houses becomes a singularly bad idea. In the linear region, having more money makes you richer, in the sense of being able to buy more stuff, but outside of that region one is forced to realize that since most money has been loaned into existence, it is in fact composed of debt, and once this debt goes bad, no matter how good your net worth looks on paper you are still facing destitution, greatly exacerbated by the fact that you are out of practice when it comes to being poor. In the linear region, investing more money in energy production produces more energy, but just outside that region it produces less energy, and may also inadvertently destroy entire industries and ecosystems.

If linearity is a fiction that is only useful up to a point, then what about time-invariance? Clearly, it too must have its limits. Stepping on the accelerator may produce the same acceleration every time, but the amount of fuel in the tank decreases monotonically until there is none left. When it comes to more complex, dynamic systems—industries, economies, societies—they may continue to respond to external stimuli in a linear and time-invariant manner up to a point, but behind this stable façade their capabilities erode, their resources dwindle, their complexity increases, and beyond a certain point an entirely different process begins: the process of collapse. Such systems generally do not become smaller, spontaneously become less complex or reduce their resource use while continuing to respond to external stimuli in a controlled, linear manner.

But so strong and so deeply ingrained is our habit of thinking in straight lines that often we cannot imagine that we can ever leave the linear region, or, once we do, that we have done so, even when the evidence is staring us in the face. Forensic analyses of airplane crashes have revealed that sometimes, as his last act, the pilot ripped the control console off the cockpit floor—an act that requires superhuman strength—so hard was he pulling back on the yoke to bring up the airplane’s nose. I am sure that there are plenty of pilots—in all walks of life—who will prefer to crash, gripping the controls with all their might, gaze fixed on the distant, irrelevant or fictional horizon, than to push the eject button. Their entire life’s experience has been confined to the linear region, and so they cannot imagine that it can ever end.

One particularly significant example of this thinking is the belief in Peak Oil, generally expressed as the idea that global oil production already has or will soon reach an all-time peak, and will then gradually decrease over a time span of several decades. Oil depletion is being modeled as a linear function of oil production: a few percent a year, holding more or less steady from one year to the next. At the same time, the use of oil by industrialized societies is often quite usefully characterized as an addiction. Let us exercise this metaphor a bit and see where it takes us. Suppose you have a junkie who has an ever-increasing heroin habit and who has to go out and hustle harder and harder to score his next fix. Now, suppose global heroin production peaks, prices go up, supply dwindles, and our junkie has to start cutting the dose. Not too far along what you then have is a sick junkie, in withdrawal, who cannot go out and hustle for his next fix. And very soon after that you have a collapse of the heroin market because the junkies have all been forced to kick the habit to one extent or another. This disruption of the heroin market, even if temporary, causes heroin production to decrease even faster, production costs and associated risks to go up, and so forth. Beyond a certain point, the heroin market would no longer be characterized as a linear, time-invariant system where the more you pay the more of it you get any time you like, because there would be so little of it around.

Similarly with oil. Right after Hurricane Katrina there was some disruption of gasoline supplies in some of the southern US states. People have written to me to tell me that the result was instant mayhem: society at all levels swiftly stopped functioning. The shortage was temporary and was quickly forgotten, but were it a long-term, systemic shortage, we would no doubt observe all the usual effects: much extra fuel evaporated from topping off fuel tanks and burned from driving around with a full tank and full jerrycans in the trunk, much fuel wasted from driving around looking for gasoline and from idling in long lines at filling stations, a lot of siphoning of gas from tanks and motorists left stranded as a result, a lot of people unable to get to work, and, shortly after that, hoarding, looting and rioting, commerce at a standstill, use of federal troops to restore public order, curfews and limitations on all travel, bank holidays and a balance of payments crisis, and, finally, the general inability to pay for further oil production or imports. All of these disruptions cause oil production to fall even faster, along with all other economic activity, until there is simply not that much demand for the stuff. As much of the global oil industry is idled, drilling rigs, refineries and pipelines fall into disuse and become inoperable. Instead of a nice few-percent-a-year gradual decline, we would have what Douglas Adams would have described as a “spontaneous existence failure.”

I am sure that some people would like me to whip out my straight-edge, plot some straight lines and make some projections: What is my price forecast? What production numbers are we talking about, ten or twenty years out? Well, that to me feels like a complete waste of time. I’d rather spend time learning how to train trees for round timber construction. The future is certain to be nonlinear, but I am quite sure that there will be trees in it. The reason I bring this up is that there are a few of pilots out there who I hope will have the presence of mind to push the eject button instead of clutching at the controls with their eyes locked on the artificial horizon.


Sixbears said...

My oil use has already fallen off a cliff. Don't use any gasoline. Currently have enough wood in my driveway to heat the house for the winter. Got rid of one car. 95% of the motor fuel I do burn is waste vegetable oil. However, looking for a replacement for that.

The easiest solution is to just stay home. No job to go to anyway.

The future is here. Many of us are picking and choosing what tools from the old life will be useful in the new. It's a sparse tool kit, but big enough.

It isn't a straight line from here to our new lives, but that's cool. The view is interesting.

I must admit to being amused by the fact that I live in a dome.

mgmquincy said...

Well writen, thought provoking and eye opening as usual. I often think that I have thought of everything until I read your next article and realize that once again a new and valuable perspective has been posed. Thanks again, I await your next thoughts. Mike

RebelFarmer said...

This is a really fascinating concept. Not being an engineer, being female, and more "artsy fartsy" in my thinking, I had a tough time getting through the first half of this article. I really connected when you used the junkie metaphor.

May years ago, maybe 40, I remember some research being done with aboriginal tribes that had never seen or experienced straight lines. Everything in their experience had curves. They literally could not imagine, draw, or visualize straight lines or 90 degree angle.

I guess my point is that individual perception depends on how closely we observe and experience the natural world in our every day lives. It seems to me it is a lot easier and more productive to work within the "rules" of the natural world than to impose our linear preconceptions on a non-linear world.

Unknown said...

When I was one year old this guy was telling me the same thing...

pretty fresh

Anonymous said...

OK, I did rather ask you to whip out your straight edge, (with no consultation fee offered either).
Others have thought along similar lines about the often tragic straigtness of the human mind, like Alan Watts here, if you are interested.

Ronald Langereis said...

Brilliant! mgmquincy already worded what I would have said, if he hadn't been first. Ronald

Anonymous said...

Watt's talk was also a year or so before the birth of fractal geometry, and lo there are indications that all those straight lines support a naturalistic looking order after all. One of many examples on web....

Plus, there are a couple of places in nature where straight lines are found - crystals, and here...

Lady says structures supporting cells take the form of straight lines. If cities are like cells...

Is the oil depeletion curve following a fractal pattern? Academic question perhaps.

steven taylor said...

numbers are good and useful, but no one can predict all the outcomes of life with math and numbers, some times their is the unexplainable , and the higher power to take into account ..which can act in what appears to be an irrational way at times.
I wouldn't be shocked though to see another major economic dislocation in the US in the next 2 years as the financial crisis is about 5 years away from conclusion.

snuffy said...

Dimitri,your analysis is spot on as usual,and gives me a edge in discussions I have had with my brother.He is also a engineer,and has a lineal mindset when it comes to looking to how the collapse will transpire.This has been a ongoing conversation for many years now,as we watch current events.{At least those that are public knowledge.}
His belief is in a lineal transition to a lower energy state,I am more a "fast track collapse" type,as your example of the "fast" effects of Katrina.

When that pipeline shut down,the country went from "0" to "Man Battle Stations" In a heartbeat of time,so to speak.Weeks not months...some places,days not weeks.[I had visions of it might just keep getting worse..and made appropriate expenditures]
A international "incident",such as might occur at any time with Iran in the gun-sights of the US military,could well be the type of event which could spiral out of control of all parties,and result in non-lineal,fractals-type acceleration leading to.....Lets not go there..please.
I can look to the future and quite easily spot the signs of our sliding into the same chaos which lead to the disintegration of the soviet state.How it plays out here is still to be determined.I hope it will not involve the use of those nuclear toys we have had sitting around for so many years...

Bee good,or
Bee careful


Tony said...

I basically agree with you, but there's one problem with your metaphor, which is that you equate all junkies with one another. Some junkies are bigger, or smaller, or more or less violent, or have more or less access to their drug of choice. The world ain't made equal and we have gross geographic disparity in oil reserves, not to mention violence reserves and bloody-mindedness reserves. I think expecting all industrial society to collapse (non-linearly, if you like) at the same time is, itself, an example of linear thinking.

However, I'll grant you that, from where I sit, in a violent society without much domestic supply, things will probably continue to accelerate in the not-happy direction.

Charles Edward Frith said...

This thinking isn't so far away from my own dislike binary options or outputs. We tend to think of one or the other when often it's neither.

DeVaul said...

This is a very thoughtful article, especially coming from an engineer. I don't recall hearing an engineer speak like this before.

I remember getting a D in calculus in college and leaving Economics 101 in Germany after just 10 minutes. None of this stuff made any sense to me. I was unable to see how it could fit into real life. Today, I guess my response would be: "does not compute!"

I had already decided that there would be no gradual descent based on my observation of those around me. Humans do not react to things that may happen years or months from now. We are designed to react immediately to an immediate danger.

Some of us can see approaching catastrophes and try to prepare for them, but our preparations are inhibited by those around us who do not wish to react until the last possible moment. I believe this is genetic.

If humans reacted years ahead of time to all possible dangers and catastrophes, the constant adrenaline rush would kill off most us at a young age, assuming we did not go insane first. Look at all the cult groups who were convinced the world would end on a certain day. How many are still alive?

I believe the end will be sudden, and that most of us will die in the ensuing conflagration. But, I also believe some of us will survive and go on living, possibly reverting to a previous dormant genetic code that helps us to survive in the present.

bryant said...

As a geologist who is forced to work with engineers regularly, this essay was deeply satisfying. Trying to explain the fractal nature of geologic reality to mining engineers enamored of linear functions makes me crazy. Thank you.

Also, I am happy you are willing to turn off the comment feature when a critical mass of asshats descend. Frankly, not being able to comment makes them less likely to return. The relationship in probably nonlinear but... anything helps.

Charley said...

And of course, any object subject to a gravitational field is also moving in a curved path, since the field itself is moving (in a curved path, since it's subject to gravitational attraction). Light, too, is always curved, refracted by the field of which it is a part.

It's all one big field, everything moving (only nothing doesn't move), everything curving. There are NO straight lines anywhere.

Bruce Lewis said...

Crystals and crystalline solids are composed of atoms arranged in straight-line lattices.

Rate Crimes said...

Three or more lines and you have a corner. Ianto Evans' book, "The Hand-Sculpted House" was a revelation of the tyranny of corners.

Rate Crimes said...

Bruce Lewis said...
"Crystals and crystalline solids are composed of atoms arranged in straight-line lattices."

Yes, in static conception. But there they remain uninteresting. As soon as crystals become dynamic and interesting, their properties are subject to the curvelinear tyranny of probability.

Carolyn said...

Crystals were already mentioned above; honeycombs and spiderwebs also have straight lines in them. No 90-degree angles though, I can't think of any examples of those in nature...

AgentGenev said...

I have to tell you, Mr Orlov. You are one of my very favorite writers. I don't know if it's because I'm also from the ex-Soviet bloc and later moved to the USA or what, but I find everything you write so intelligent and humorous. You are also among the most perceptive writers there are as you grasp concept probably not even 0.5% of the population dwells upon. I have often thought about them, but you put them in words so well and always surprise me with some greater insight I hadn't been able to reach myself.

Keep it up!

Bill Chapko said...


Excellent post. What struck me most is that wandering out of straight thinking will lead to a quick collapse. Very sobering.

The United States has a very bad case of "straight-line-itis". For example, the maps of almost all American towns, big or small, have straight grid lines while those in Europe are wiggly. My suspicion is that American puritanism is at work here. As Watts says, wiggles are much more sensual.

I believe the preference for straight lines goes deeper. It indicates a fundamental project since the scientific enlightenment to separate man from nature, imagining we'll be safe and powerful there. Straight-line thinking also invades grammar, writing and speech and cannot tolerate dancing meanings, and even less paradoxes. This is why I think aphorisms may have a future as expressing one's spirit.

Nietzsche pointed out that math, science and straight thinking is merely a description of us humans, not the world.

"... Mathematics is merely the means for general and ultimate knowledge of man." Gay Science, section 246

"... But how could we possibly explain anything? We operate only with things that do not exist: lines, planes, bodies, atoms, divisible time spans, divisible spaces. How should explanations be at all possible when we first turn everything into an image, our image! It will do to consider science as an attempt to humanize things as faithfully as possible; as we describe things and their one-after-another, we learn how to describe ourselves more and more precisely." Gay Science, section 112

Anonymous said...

Great post, Dmitry.

And excellent comment from Bill Chapko... very well said, Bill.

Christine Johnson said...

Awww, c'mon Dmitri.

I think you sell engineers short. As an EE, I'm very familiar with nonlinearities and the difficulties of predicting behavior in non-linear regions.

I'm also very familiar with runaway behavior, and I see it happening in a number of areas. Once things get going, things can very quickly get out of hand.

If anything, my education in linear time-invariant systems has taught me that I can't apply that thinking to non-linear time variant systems. Life is so much more complex and changing, it would be sheer folly to suppose otherwise. Admittedly, I may have a minority view :-)

Phil said...

I wonder if a Polynesian Yuri Orlov thought there would always be trees on Easter Island? Jared Diamond in Collapse of Civilizations makes it clear that linear thinking did not hold true in that unhappy place where someone did indeed chop down the last tree.

With runaway climate change, sea level rise, desertification and massive over population it is possible the last tree will get chopped down in the next few centuries. Bad news for all boat dwellers and my money would be on stone rather than wood.

Yuri are you falling into linear thinking yourself on this one? I prefer to think that it shows you possess that quality we all need (amongst many others), hope.

To conclude, I look forward to your blogs and books more than any others, they are always insightful and thought provoking. I'm glad you got rid of the maniac comments and I hope this one, my first, gets through.

Anonymous said...

I had the same thoughts while reading yet another recent, and utterly absurd, set of comments from an oildrum comment poster who is somewhat obsessed with trying to construct some new mathematical formulas predicting oil production decline, to replace the Hubbert curves, to give more 'accuracy' to decline models, making up complex mathematics using premises just barely, if that, examined, what's his name? You know the one, if you follow tod.

Anyway, his attempts to fine tune mathematical analysis to handle something as ultimately real world and non mathematical, and utterly non linear, and most decidedly non modelable, as failures in global oil production/consumption patterns has repeatedly struck me as one of the most ridiculous things I come across in that website.

I've been tempted to point out the total waste of mental horsepower his efforts involve, but it seems to make him happy to make up his silly formulas and to design models based on unknowable circumstances. I have to suspect you've noticed this too, since now and then he takes over threads there with his pseudo-science/pseudo math drivel.

Loving non linearity you've been poking around in lately, you're raising the bar pretty high here though, tough act for the other guys out there to follow I think, but that's as it should be. Keep up the great work.

Unknown said...

When I was in my twenties, studying physics and linear systems theory, I harbored notions that I would soon be master my world. These were such elegant tools and yielded answers without those tedious approximations. This naturally spilled over into the rest of my life where I ignored all that didn't fit my rectilinear models.

Thankfully, after many years of frustration, I managed to discover (with the help of wiser voices like Prigogine and Bak) a new aesthetics grounded in the chaotic, nonlinear, unfolding universe. In exchange for my "mastery", it became a hauntingly beautiful place.

kaimiddleton said...

1) Dmitry: you are the cat's meow. I really appreciate the work you're doing. I feel weird that I'm half in and half out of our current modern world. I mean, if everyone is using a car/bus to get to work, I'd kind of be a fool not to, so I'm still functioning in a job situation ... but I'm also eyeing the future with a view towards collapse, and have been taking many steps to prepare. Hence
2) As a rather linear thinker, I'm working (maybe struggling) with the question of how to function in this society appropriately. I work as a programmer. So I try to be practical and not too foofy about my work (I have a temptation towards perfectionism in solving any particular problem, when a napkin sketch of a solution might be enough). Derrick Jensen asserts in "Endgame", that the only sustainable level of civilization/technology is stone age. Computer programs that help you edit your term paper aren't quite relevant in the milieu. But I feel I do have to believe in what I'm doing, and that it is appropriate to do so. So:
3) What's the balance? How much linear thinking should we apply to the world; i.e., how much should we try to simplify all the equations we see around us, so we can apply logic/procedure to those equations to figure things out ... and how much should we just throw out all the equations and try to go with our gut instinct?

Lukas Vangheluwe said...

Thank you Dimitri.
#kaimiddleton :

There is some evidence that mammals such as rats, mice and other rodents show adaptive behavior towards overpopulation. Symptoms include: heightened aggression, diminished parental care, non-reproductive sexual behavior, widening gap between the physical condition of stronger and weaker individuals.
Most of these 'symptoms' can be detected in our societies, which could point us to desperation. However, the fact that these may be adaptations, and that we are carrying the gene complexes for them, points to some sort of hope : our ancestors have been through this kind of thing before, so there is a chance some of us may wiggle through. If those traits are adaptations, collapse and overpopulation must have happened quite a few times, and so we may not come totally unprepared, even if we aren't.
It is important to remember that animals in overpopulation are brutal, but once equilibrium is established, our natural empathy should mount to the surface again.

Jerry McManus said...

Great post! Non-linear behaviors are indeed alien to the thinking of most people. The exponential function in particular is difficult for people to grasp, as Dr. Albert Bartlett is fond of reminding us.

Just as insidious, perhaps more so, is the idea of feedback delays.

This can be easily observed in video games where aspiring astronauts repeatedly crash their lunar landers in an environment where inertia responds slowly to applied thrust. Our doomed astronauts pound on the decelerator only to watch helplessly as their now runaway lander zooms to oblivion, dutifully responding to the equally exuberant pounding on the accelerator applied only moments before.

Now consider the Earth's oceans, ecosystems, and atmosphere which have feedback delays measured in decades. Applying the brakes now, several decades too late, on the runaway freight train that is industrial civilization will make little difference in our trajectory over the cliff edge.

The critical difference being that the Earth has no "play again" button.


cafesneinton said...

Nice article! In mathematics the image that lies behind Numbers is indeed the infinite line (made from points). And of course for most people maths is grounded in Numbers...However the true ground for maths is Sets, and the image that lies behind Sets is pretty much the yingyang symbol - one kind of of stuff surrounding another kind of stuff is a Set. If this was our popular image of maths no doubt we would have a more holistic deeper way of thinking...

manray said...

Bravo, Dmitry. Once again you have nailed it. Every time I read your words I am grateful for the Internet. Your lucid and spot-on perspective of life in these United States is refreshing and inspiring. Please keep writing. I look forward to every post.

russell1200 said...

Tension between two points causes a straight line. As a sailer you will of course recognize this from the rigging of the ship, where the tension pulls the rope and cables straight.

Of course items under tension can also snap.

Dmitry Orlov said...

A cable under tension forms a straight line in zero gravity and with zero angular momentum. With gravity it forms a catenary. With angular momentum things get complicated. But a straight line is a good enough approximation for rigging on a sailboat.

kyleinabox said...

Holy poop. I've been thinking about this for years now but never really got what was going into my head until I related it to mathematics (personally I described it in terms of people seeing life in plateau and infinity which has different implications besides the straight lines.)

Not sure which part is best, the tree part or the pilot part because they were so great.

Do you have any thoughts of this concerning art or religion?