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Curious, this may be more of a philosophical question: what is the reason why these organic-seeming movements can be modeled mathematically, specifically with t

Curious, this may be more of a philosophical question: what is the reason why these organic-seeming movements can be modeled mathematically, specifically with trigonometric functions? Why do biological processes seem to obey trig like so?

xfs 14y ago
Organic-seeming movements can be modeled with a sum of a set of sines and cosines. Are you talking about Fourier series?
dietrichepp 14y ago
Biological processes are composed of chemical processes, and chemical processes are often modeled with differential equations. The trig functions (sine and cosine) and the exponential function are all solutions to the simplest second and first order linear differential equations, respectively. You might equally ask why the planets follow the trig functions so closely when they orbit the sun, or why the shape of the Earth is close to a sphere, or why the first digit of the amount of money in your pocket has a roughly 30% chance of being 1, but only an 18% chance of being 2.
- rodh 14y ago
  Would second order differentials imply linear acceleration? Would we then always get better approximations the higher the order? EDIT: I ask this because linear acceleration doesn't seem like something which might readily occur in nature.
  - dietrichepp 14y ago
    You can get constant (not linear) acceleration with a second order differential equation. Here is an example: d^2 x ----- = -g dt^2 Solving this gives us the parabola we are all familiar with, when an object falls under constant acceleration: x = -1/2 g t^2 + v_0 t x + x_0 This is a second order differential equation, but it is too trivial an example. Here is a more interesting differential equation: d^2 x ----- = -x dt^2 All solutions to this equation take the form: x = a sin t + b cos t There are a lot of physical materials in nature which produce this kind of relationship between force and position. Basically, anything that is kind of like a spring. Reach over and flick a glass on your desk with your fingernail. Hear it ring? The sine wave that it's ringing with is predicted by the relationship between force and position that we understand. Every tuned musical instrument in the world, with one exception, generates notes that can be modeled by some differential equation. Pianos, violins, and horns are relatively easy; drums are much harder but still tractable. The chemical and physical systems in living tissue are much more complicated. Our best models for them often incorporate nonlinear elements (the sine wave example is linear) or a large number of variables. But periodic behavior is still quite common, and the sine wave is in some senses the simplest function that is smooth and periodic. Note: The one musical instrument for which differential equations are not the best model is the digital synthesizer.
tolmasky 14y ago
> what is the reason why these organic-seeming movements can be modeled mathematically, specifically with trigonometric functions? Any motion can be modeled mathematically. There are probably infinite equations that provide "organic looking motion". I could make one that made really good looking motion but was really long and complex and you could have a parallel person asking "why is nature so messy and not giving us simple motion?". The insight here is to not attach any special significance just because "it looks organic". In reality what we are probably saying is that it "looks smooth". (In fact you could look at Bezier curves which are often used for animations to see non-trigonemtric examples, which while easy to define are relatively complex to actually run through http://en.wikipedia.org/wiki/Bézier_curve http://en.wikipedia.org/wiki/Bézier_curve . All of iOS's built-in famous smooth animations use these.) So let's break these down. All of these examples are periodic. So for starters we need a good periodic function. We could have started with y = |(x%2) - 1| ( http://tolmasky.com/letmeshowyou/notsmooth.png http://tolmasky.com/letmeshowyou/notsmooth.png ). That would give us the back and forth motion we want, but would have not looked smooth, since we linearly increase, then linearly decrease, and in the middle abruptly change. Your eye would catch that abruptness and that's what would give it the "inorganic feel" (despite the fact that there are plenty of things in biology that are super abrupt). So what we want instead is the same kind of back and forth, but rounding off the edges. You could imagine forming a function by taking a normal parabola (y=x^2) and an inverted one (y=-x^2), and putting them together to get the rounded bottom and top corners ( http://tolmasky.com/letmeshowyou/twoparabolas.png http://tolmasky.com/letmeshowyou/twoparabolas.png ). Luckily for us, we've already discovered a class of functions that give us something similar in the form of sine and cosine. Now, the reason why these are smooth and periodic is simply because the values of these functions are derived from tracing along a circle ( http://www.galaxygoo.org/math/sineCurve.html http://www.galaxygoo.org/math/sineCurve.html ). Now that you know this, you have a "tool" for your mathematical bag of tricks: every time you want to do something smooth and periodic, you are probably going to reach for sine or cosine. Similarly, if you want to now tweak this and have things ramp up in speed or down, you could try using exponents, etc. Hopefully this sheds a little light on what's going on here.
arethuza 14y ago
"Why do biological processes seem to obey trig like so?" I'm not sure I would say that biological processes obey anything - merely that we can find it useful to describe real world behaviour in terms of these abstract mathematical functions. Of course, it's interesting to speculate if there is a link between the level of maths required to usefully model the real world and the level of maths we are happy working with - i.e. if the real world was much more complex would we also have developed a much more richer natural ability for mathematics.
andrewcooke 14y ago
are you asking a reasonable question? do you have a clear idea of some regular change in size that would not "appear organic"? because i suspect that almost any smooth, repeating change would be acceptable... and if that's so, then your question is largely meaningless.
VMG 14y ago
They are not organic, they are just smooth. Any other smooth motion probably would do.

xfs 14y ago

Organic-seeming movements can be modeled with a sum of a set of sines and cosines. Are you talking about Fourier series?

dietrichepp 14y ago

Biological processes are composed of chemical processes, and chemical processes are often modeled with differential equations. The trig functions (sine and cosine) and the exponential function are all solutions to the simplest second and first order linear differential equations, respectively. You might equally ask why the planets follow the trig functions so closely when they orbit the sun, or why the shape of the Earth is close to a sphere, or why the first digit of the amount of money in your pocket has a roughly 30% chance of being 1, but only an 18% chance of being 2.

rodh 14y ago
Would second order differentials imply linear acceleration? Would we then always get better approximations the higher the order? EDIT: I ask this because linear acceleration doesn't seem like something which might readily occur in nature.
- dietrichepp 14y ago
  You can get constant (not linear) acceleration with a second order differential equation. Here is an example: d^2 x ----- = -g dt^2 Solving this gives us the parabola we are all familiar with, when an object falls under constant acceleration: x = -1/2 g t^2 + v_0 t x + x_0 This is a second order differential equation, but it is too trivial an example. Here is a more interesting differential equation: d^2 x ----- = -x dt^2 All solutions to this equation take the form: x = a sin t + b cos t There are a lot of physical materials in nature which produce this kind of relationship between force and position. Basically, anything that is kind of like a spring. Reach over and flick a glass on your desk with your fingernail. Hear it ring? The sine wave that it's ringing with is predicted by the relationship between force and position that we understand. Every tuned musical instrument in the world, with one exception, generates notes that can be modeled by some differential equation. Pianos, violins, and horns are relatively easy; drums are much harder but still tractable. The chemical and physical systems in living tissue are much more complicated. Our best models for them often incorporate nonlinear elements (the sine wave example is linear) or a large number of variables. But periodic behavior is still quite common, and the sine wave is in some senses the simplest function that is smooth and periodic. Note: The one musical instrument for which differential equations are not the best model is the digital synthesizer.

tolmasky 14y ago

> what is the reason why these organic-seeming movements can be modeled mathematically, specifically with trigonometric functions? Any motion can be modeled mathematically. There are probably infinite equations that provide "organic looking motion". I could make one that made really good looking motion but was really long and complex and you could have a parallel person asking "why is nature so messy and not giving us simple motion?". The insight here is to not attach any special significance just because "it looks organic". In reality what we are probably saying is that it "looks smooth". (In fact you could look at Bezier curves which are often used for animations to see non-trigonemtric examples, which while easy to define are relatively complex to actually run through http://en.wikipedia.org/wiki/Bézier_curve http://en.wikipedia.org/wiki/Bézier_curve . All of iOS's built-in famous smooth animations use these.) So let's break these down. All of these examples are periodic. So for starters we need a good periodic function. We could have started with y = |(x%2) - 1| ( http://tolmasky.com/letmeshowyou/notsmooth.png http://tolmasky.com/letmeshowyou/notsmooth.png ). That would give us the back and forth motion we want, but would have not looked smooth, since we linearly increase, then linearly decrease, and in the middle abruptly change. Your eye would catch that abruptness and that's what would give it the "inorganic feel" (despite the fact that there are plenty of things in biology that are super abrupt). So what we want instead is the same kind of back and forth, but rounding off the edges. You could imagine forming a function by taking a normal parabola (y=x^2) and an inverted one (y=-x^2), and putting them together to get the rounded bottom and top corners ( http://tolmasky.com/letmeshowyou/twoparabolas.png http://tolmasky.com/letmeshowyou/twoparabolas.png ). Luckily for us, we've already discovered a class of functions that give us something similar in the form of sine and cosine. Now, the reason why these are smooth and periodic is simply because the values of these functions are derived from tracing along a circle ( http://www.galaxygoo.org/math/sineCurve.html http://www.galaxygoo.org/math/sineCurve.html ). Now that you know this, you have a "tool" for your mathematical bag of tricks: every time you want to do something smooth and periodic, you are probably going to reach for sine or cosine. Similarly, if you want to now tweak this and have things ramp up in speed or down, you could try using exponents, etc. Hopefully this sheds a little light on what's going on here.

arethuza 14y ago

"Why do biological processes seem to obey trig like so?" I'm not sure I would say that biological processes obey anything - merely that we can find it useful to describe real world behaviour in terms of these abstract mathematical functions. Of course, it's interesting to speculate if there is a link between the level of maths required to usefully model the real world and the level of maths we are happy working with - i.e. if the real world was much more complex would we also have developed a much more richer natural ability for mathematics.

andrewcooke 14y ago

are you asking a reasonable question? do you have a clear idea of some regular change in size that would not "appear organic"? because i suspect that almost any smooth, repeating change would be acceptable... and if that's so, then your question is largely meaningless.

VMG 14y ago

They are not organic, they are just smooth. Any other smooth motion probably would do.