It's a tricky concept linked to chaos, but the Feigenbaum Constant is a special number which appears everywhere in nature. More symbols at www.sixtysymbol... With Philip Moriarty
In his book "Chaos: Making a New Science", James Gleick tells the story of Feigenbaum's Number's discovery. As he describes the moments where the number unveiled itself, my blood ran cold. It was that thrilling.
agreed, thats what I think happens with radioactive decay. They say its random, but decays at a predicable rate. There has to be some hidden variables.
The constant pi is the ratio between a circle's perimeter and its radius, yet pi is irrational (goes on forever). As far as I have been able to dig up, no one has yet proven whether the Feigenbaum constants are rational or not, yet I have seen a paper in which they assume they are irrational, and they are on the list of "suspected irrational numbers" on Wikipedia :)
A better explanation as to what the "forks" actually represent would be nice. I get that in the case of the helium cell that the forks represent the point at which when heated to a certain point the period oscillation doubled, but what that means for the population of fish? I haven't the foggiest.
This was in Scientific American almost 30 years ago. The iterations represent generations in the case of population. I wrote a program to graph it back then. See my other comment for the pseudocode.
Awesome video. It's great to see the professors get so excited about Feigenbaum's number. They both looked like little kids trying to explain something cool.
This was in Scientific American almost 30 years ago. The iterations represent generations in the case of population. I wrote a program to graph it back then. After removing the compiler-specific graphics stuff, it boils down to this pseudocode. Try these values first. You will want to set DONT_PLOT a little higher after seeing what it does. ITERATION_COUNT = 20 ITERATION_COUNT_DONT_PLOT = 0 INITIAL_VALUE = 0.3 for x = -2 to 4 y = INITIAL_VALUE for ITERATION_COUNT_DONT_PLOT iterations First do some number of iterations and don't plot them, to hide the noise of the initial values. y = x * y * (1-y) ) for ITERATION_COUNT iterations y = x * y * (1-y) plot_color = ITERATION_COUNT (The colors are nicer when DONT_PLOT is 0.) plot point at (x,y) ) )
Added to this is the fact that the mere observation of variables tends to change them, meaning that even if a system did have a relatively small number of variables it would still be impossible to exactly predict what would happen.
That is extremely interesting! The structure seem very much like fractal so maybe fractal matematics can be used on chaos study. In the future when tech goes to nano scales and very complex the term chaos will become more important. Also study of fractals is getting more important in communications etc.. This constant just might be the new Pi of the future!
That graph shows the amplitude of the function (ie a straight line shows a constant amplitude) when the line splits is shows two separate amplitudes. When the line splits is shows a point where the number of semi-stable populations has doubled. If you are looking at a population of fish the height of the line shows the number of fish and the splitting of the line shows that there are two value of the population that the model moves between.
A thing happens or not. A choice of 2 paths to follow. The graph is a map of all possible states inside the bounded condition. It is best to think of it as a pendulum with another attached to it. The biggest circle drawn by the combined pendula is the bound condition. The simplest oscillation is back and forth together. The first split is P2 does not swing at the same direction as P1 but has the same rate. Eventually you cant predict what the pendulum will do, but it will be inside the circle.
@HerrCaZini very quickly, think of this formula. X_{n+1} = R X_n (1-X_n) where underscore means subscript solving this equation and X_{n+1} = X_{n} give us a certain value(s) of X_n that solves it for certain value R. now if we have a graph with horizontal axis R and vertical X, it gives us this v. weird graph. plug in R=3 and find X then R=3.4 and find X. you'll notice for R=3 theres only one value of X, for 3.4 theres suddenly 2. thats what the bifurcation is basically,
I still don't understand what is splitting apart all the time. Population of fish? What is the splitting apart? Non-linear capacitors? What is splitting then? Pendulum, how can the frequency split apart?
When it splits, the population doubles. Each split is one fish producing two offspring. (Clearly this is not how it really works, but it is close enough as a model)
I believe when he's talking about the pendulum he is referring to a double pendulum system; where you attach one pendulum to another and give it a push, rather than a single pendulum.
Ronald de Rooij Capacitors are linear elements, he's speaking about capacitors put in a circuit with non linear elements (like diodes or transistors). Probably in that particular case he is talking about the resonant frecuency of the circuit, that "splits" into two different resonant frecuencies, this actually happens, he's to vague about the concept, but basically you will have a paralell resonance fecuency and a series resonance frecuency, The circuit model varies with the frecuency applied, so, for instance, at low frecuencies you do the calculations and you get a certain resonance frec, but as frecuency increases some parameters vary and add up to the calculations, and you have to modify your circuit model. That will yield a predominant series or paralell model(another two new frecs), and the ratio in wich the model has to be "re thinked" as a frecuency function behaves like the systems in the video. Let's say that above 30MHz each time the frecuency is multiplied by 4.66 you have to re calculate your constants (resistance, inductance and capacitance vary with frecuency, capacitors become inductors, inductors become capacitors, resistors may bacome the two of them, the attenuation changes, everything starts to become a CHAOS!). So you have a circuit model for 30MHz, other for 140MHz, other for 600MHz and so on...I worked with high frecuency circuits and this is certainly true, in fact, if these effect hadn't been taken into account probably we wouldn't have modern communication systems, like cellulars, satellites, etc....
I usually really like how Sixty Symbols presents the core of scientific field to scientifically inclined people, without dumbing it down, without relying on having knowledge in that field.. But here, I get totally lost. What do these diagrams represents? What are the junctions? What is it with the box and the waves? What have fish to do with it? I guess I need to get some knowledge on chaos theory before hand, in order to understand that video. Keep up the good work, but this one is subpar.
+House The y axis represents your "x value" i.e. the number of fish (for example). The x axis represents some parameter (usually called r), which alters the characteristics of the system. The lines represent recurring states of the system. (For fish it is a bit confusing, because fish population don't jump between two numbers instantly.) See my other comment for the splitting. ;)
I had some trouble as well, the split is simply a change, with fish a split could be when more predators move into the area, for the pendulum its when a force different from the force its currently under is applied.
@TonyMach01 Simply put, what they discovered was that in any natural system changes occur at set intervals related to the constant 4.66 - It is similar to the Fibonacci sequence, how natural design follows the sequence in general. It is just one of those fascinating 'rhythms' of nature that create a constant that can be used to describe when and at what intervals a dynamic system branches or changes. Does that help?
@HerrCaZini Kind of like the point where you have to change gears when driving a car up hill, you're changing the ratio of the gears against the force times the distance, or "period doubling" the force in relation to time. A dual pendulum setup does the same thing with force applied where it will double its occillations at a certain point in time.
@Moriarty2112 i wasn't saying they are the same, just that they sounded similar. i just though that what karma is: The belief that every action you make will affect the whole world around you. sounded a bit similar to what was described as chaos theory. might have misheard though.
If you check the graphic shown at 7:13 (shown multiple time in the video, I know), how comes it's not symmetrical? Is it only "this instance of a simulation", or are the "branches" always tilted upward? Well, now that I ask this, I wonder what does the Y axis represents on that graphics. I guess I need to do some research of my own to get any closer to any kind of comprehension!
I thought for sure he was going to start writing on the wall. I've seen it done. I saw a Physics Prof. start writing on a light colored, painted door; and he just went on and on, writing on this door, just because he happened to be standing there when he was seized with the need to explain something. Building maintenance and was not amused; and the Admin. produced a note asking him to "please not write on the doors".
AFAIR it was Lorenz' print function that rounded off the intermediate result. If Lorenz had written a better print function, and subsequently entered the new start value to the machine's full precision, he wouldn't have made his discovery, at least not at that point in time.
Prof Susskind holographic 2d math at high energies. One way to interpret is as geometric distortions at high accelerations.. ie. back-facing surfaces wrap around, infinity as an edge. If higher dimension/info-scapes have geometric representations in such functions.. then, like a quasi crystal, it is a geometric function/gradient through possible states. numbers as geometric structure. IMO, Moriarty, Susskind & underlying wave function, seem to be different interpretations of complex rotations.
The cause of the split is the driving force. If increase the driving force (in a population of fish you would increase the amount of space, food, mates etc) at a constant rate then the population would increase as shown in the bifurcation graph.
@HerrCaZini the easiest way I can think to explain it, imagine you have a really complex equation you want to solve -- for simplicity's sake I'll say Ax = B, but know that this equation doesn't actualy work for this. So you know the values of A and B over time and you want to solve for x. It turns out that for many natural phenomena, when you solve for x, over time you find that there can be more and more solutions, like x=5 and x=8 are both solutions and later there are 4 then 8 etc
@scottvalentine808 Say the environment can manage 100 fish. The graph has population growth factor on the X-axis, and stable population matching that value on the Y-axis. When fish multiply less than 2.5-fold each year, the population stabilises to 100 fish after several generations. If the growth is more intense, the population never stabilises, but swings between lets say 105 and 95. At 2,8 the population stabilises on four levels. And so on, all the way to chaos.
I understand that the actual constant comes from a ratio, but I don't understand the initial graph. it's clearly not a function, right? so what does the graph represent? can anyone clarify?
+Imfingbob What you do with these kinds of function is that you iterate them rather than calculate them like a 'normal' function which takes a value for a given input x. As you vary some control variable k (in the logistic case this represents the natural proportional increase in population from one generation to the next) you run each value for a considerable length of time, i.e. a large number of successive values, and you find that the behaviour initially settles in toward one value and just stays there. As you increase k eventually you get a first 'bifurcation' and instead of the function settling to a single value it starts to alternate between two values. The it cycles through four values etc. The graph is of the steady-state values the function takes after running for a while with a given value of k. That's why it looks so weird compared to a normal graph. Hope that makes sense.
This was in Scientific American almost 30 years ago. The iterations represent generations in the case of population. I wrote a program to graph it back then. See my other comment for the pseudocode.
this doesn't help you determine which particular atom will decay at what time. IE if you have 3 examples of a radioactive isotope with a decay rate that's been determined to be approximately 10 minutes. you wait for the first one to decay and start the clock there. The remaining two have an equal probability of being the next to decay and any given second has an equal probability of being the next one in which the decay occurs. (continued)
@Moriarty2112 I agree that they aren't alike. i just like finding out rational explainations for old superstitions, e.g. mis interpretation of ball lightning. but i understand the karma is completely unrelated to chaos, and that there similarity is purely coincidental.
Pretty much, yeah. But I wouldn't call it "descend". Chaos is the reason life formed (chemicals was mixed and mixed by energy from volcanoes, meteor impacts, the sun, etc, to the point one molecule became so complex it gained the ability to make copies of itself by using energy from heat, light, pressure changes, Ph level changes, etc).
Think of rate of decay as an average You can observe an atom decay. You mark that as your starting point, then you record the time it took until you observed the next atom to decay, and how long the one after that takes etc. You average out your results and the more intervals your average represents the more accurate the average is. (continued)
Yes but I'm only interested in it from a number perspective and want to improve the collection of irrational numbers I know about. So this number joins the ranks of pi, e, and the golden ratio by being common throughout nature?
I wish the video went into more detail on what a bifurcation diagram actually shows... I've been trying to type it out into a comment but it's not working so well =/. I just learned about this stuff at school, and the biggest missing piece for me what how the heck you draw a bifurcation diagram. In any case, awesome video!
People, people please. These are not meant to be tutorials, they are an introduction to ideas so that people who otherwise wouldn't know about these concepts or interesting tid bits about physics or maths etc... have something to "springboard" off of, in order to do more reading and learning... They aren't purposely leaving out certain things, they just have too because of time constraints etc..
fiegenbaum constant, fish follow the comfortably available path once the fish is hungry it goes to where the split is for the food is, once the fish eats a copy of the fish is created in the air then both fed fish swim off into their environment of dwelling. Until they hunger again then they find the split of water and air to feed and it happens until for the life span of the fishes or until the pen runs out of ink. chaos theory is when the fish decides it is hungry enough to find a split to eat
Rotations can map the information representing an evolving universe. very complex Perlin Noise functions. underlying mathematical structure this video describes.. is an indication of underlying matrix/rotation geometries. At high very energies, physics becomes 'simpler', forces converge. From 'unity', rotations can precess.. etc, into seemingly chaotic modes (a field represented by other rotations). string temp.. as position. translate.. everywhere is same universe rotated. not random ones
How to make a universe from rotations. Rotate spinor on 2 axis, returning to original state after 10^48 rotations, ie mapping out a proton shell in 'pseudo-planck' units. Multiply by 4 more axis rotations.. each representing a length element of a quasi crystal rombi.. so it iterates pseudo-protons through all possible states. Add more rotations for complexity. Would observers made of/ in such a system experience similar mathematical, geometric reals & determinants, complexity & chaos?
Really? I understood it. If you measure a pendulum's time from side to side, if it is 1 at first, then 2, then 4, then the time it takes for it to go from 1 to 2, divided by the time it takes to go from 2 to 4, is the feigenbaum ratio. Eventually the pendulum stops swinging in a regular pattern and just moves around due to wind and so forth (that's when it becomes chaotic).
i don't understand, If reality is quantized (planks constant) wouldn't that mean that Feigenbaums constant be a finite number in ratio with planks constant in the real world? I mean i understand why feigenbaums constant is infinitely long when it comes to just pure mathematics, but in experiment, all thing can literally be traced back to the movement of something at the plank length, which would give the Feigenbaum constant a finite number right?
Up until now, I thought that the Divine Proportion discovered by Luca Pacioli - 1.618 was the same thing as the Feigenbaum constant. Does the Divine Proportion have any applications in physics - experimental or theoretical?
The scary thing is, this applies to the rate at which the human population is making new advancements... The time between major advancements is exponentially decreasing, and mathematicians calculated that the point of chaos will be reached in.... december 21, 2012. I'm scared :(
The blank areas are just like the area before the first bifurcation (fork) , they represent nothing interesting. Stability I guess. Then chaos starts taking over, then later youll see periods of stability show up again, then right back to chaos and further bifurcation.
I'm afraid I agree. They both talk about CHANGES in the patterns, but they don't explain the BIFURCATIONS which are crucial. If it were only change the line would just go in another direction, but not split. As I understand it at the point of the bifurcation there suddenly are two states instead of one, and the system oscillates between those two. CMIIW
2:40 doesn't that make any measurement of the weather fundamentally wrong? If each decimal place varies the result so different, how do we know which one is 'right'?
lolzomgz1337 Yes and no. What usually happens is that meteorologists create multiple models with slight variations and arrive at an estimate through several steps of statistical analysis. That is basically taking the average of the different models and arriving at the statistically most likely outcome for the weather system. Not exactly though, since this is a youtube comment and not a lecture in meteorology.
lolzomgz1337 as he mentioned, they are about the same for a little while and then go crazy; this is why our weather forecasts only forecast a few days ahead, and aren't completely accurate
the constant was shown at the end and seem'd like it was going on forever. But it is derived from a ratio. So is it just long or am I missing something and its irrational?
experimental results have error. you would in that case have a perfectly finite rational answer that was close to delta but ever so slightly off. maybe millions or billions of decimal places into it, but it would be slightly off. just like you could draw a circle and then get some string and measure how long the string is to go all around it. you can derive pi as a finite number that way, but small errors in measurement would make it a (albeit close) incorrect number.
In his book "Chaos: Making a New Science", James Gleick tells the story of Feigenbaum's Number's discovery. As he describes the moments where the number unveiled itself, my blood ran cold. It was that thrilling.
I love how excited these guys get, it keeps me focused on my goal of getting into uni.
+S4R1N Keep going :)
Please remember us mental midgets! We're counting on new blood to try and give us a glimpse of this beautiful universe.
Have you gotten in yet ?
and me into getting out of it haha
I wonder if he got into / graduated uni
love these people/professors, they are so passionate and explain the concept really well
I don't understand. I need James and brown paper.
We're all looking forward to our lectures with this guy this year! :)
Thank you for this amazing content! 13 years later, still strong.
Someone please tell this very clever man to get a PEN for his whiteboard/wall :)
agreed, thats what I think happens with radioactive decay. They say its random, but decays at a predicable rate. There has to be some hidden variables.
The constant pi is the ratio between a circle's perimeter and its radius, yet pi is irrational (goes on forever). As far as I have been able to dig up, no one has yet proven whether the Feigenbaum constants are rational or not, yet I have seen a paper in which they assume they are irrational, and they are on the list of "suspected irrational numbers" on Wikipedia :)
A better explanation as to what the "forks" actually represent would be nice. I get that in the case of the helium cell that the forks represent the point at which when heated to a certain point the period oscillation doubled, but what that means for the population of fish? I haven't the foggiest.
I am glad there is a wikipedia article about Feigenbaum's constants to understand.
Please, if you enjoy Wikipedia as much as me (lots), contribute once a year, even $2 helps them.
This was in Scientific American almost 30 years ago. The iterations represent generations in the case of population. I wrote a program to graph it back then. See my other comment for the pseudocode.
I find the speech from 5:33 - 5:43 really inspiring and deep, gotta love it :3
Awesome video. It's great to see the professors get so excited about Feigenbaum's number. They both looked like little kids trying to explain something cool.
This was in Scientific American almost 30 years ago. The iterations represent generations in the case of population.
I wrote a program to graph it back then. After removing the compiler-specific graphics stuff, it boils down to this pseudocode.
Try these values first. You will want to set DONT_PLOT a little higher after seeing what it does.
ITERATION_COUNT = 20
ITERATION_COUNT_DONT_PLOT = 0
INITIAL_VALUE = 0.3
for x = -2 to 4
y = INITIAL_VALUE
for ITERATION_COUNT_DONT_PLOT iterations
First do some number of iterations and don't plot them, to hide the noise of the initial values.
y = x * y * (1-y)
)
for ITERATION_COUNT iterations
y = x * y * (1-y)
plot_color = ITERATION_COUNT (The colors are nicer when DONT_PLOT is 0.)
plot point at (x,y)
)
)
This one went over my head I'm afraid.
haha, I was thinking the same here
7:00 I love the geeky passion in these guys XD
Added to this is the fact that the mere observation of variables tends to change them, meaning that even if a system did have a relatively small number of variables it would still be impossible to exactly predict what would happen.
Stunning pedagogical breakthrough at 5:07: hands against the wall drawing a diagram in white on white.
Totally invisible. What an astonishing way of conveying the ineffability of the Whole Thing!
That is extremely interesting! The structure seem very much like fractal so maybe fractal matematics can be used on chaos study. In the future when tech goes to nano scales and very complex the term chaos will become more important. Also study of fractals is getting more important in communications etc.. This constant just might be the new Pi of the future!
I admire Prof. Moriarty's skills with Lab View. It's like he has a program for everything he want's to talk about.
That graph shows the amplitude of the function (ie a straight line shows a constant amplitude) when the line splits is shows two separate amplitudes. When the line splits is shows a point where the number of semi-stable populations has doubled. If you are looking at a population of fish the height of the line shows the number of fish and the splitting of the line shows that there are two value of the population that the model moves between.
A thing happens or not. A choice of 2 paths to follow. The graph is a map of all possible states inside the bounded condition. It is best to think of it as a pendulum with another attached to it. The biggest circle drawn by the combined pendula is the bound condition. The simplest oscillation is back and forth together. The first split is P2 does not swing at the same direction as P1 but has the same rate. Eventually you cant predict what the pendulum will do, but it will be inside the circle.
@HerrCaZini very quickly, think of this formula.
X_{n+1} = R X_n (1-X_n) where underscore means subscript
solving this equation and X_{n+1} = X_{n} give us a certain value(s) of X_n that solves it for certain value R.
now if we have a graph with horizontal axis R and vertical X, it gives us this v. weird graph. plug in R=3 and find X then R=3.4 and find X.
you'll notice for R=3 theres only one value of X, for 3.4 theres suddenly 2.
thats what the bifurcation is basically,
I still don't understand what is splitting apart all the time. Population of fish? What is the splitting apart? Non-linear capacitors? What is splitting then? Pendulum, how can the frequency split apart?
***** I have to look into this, because in my logic, an end population is a constant. It does not split, by definition.
When it splits, the population doubles. Each split is one fish producing two offspring.
(Clearly this is not how it really works, but it is close enough as a model)
I believe when he's talking about the pendulum he is referring to a double pendulum system; where you attach one pendulum to another and give it a push, rather than a single pendulum.
Ronald de Rooij Capacitors are linear elements, he's speaking about capacitors put in a circuit with non linear elements (like diodes or transistors). Probably in that particular case he is talking about the resonant frecuency of the circuit, that "splits" into two different resonant frecuencies, this actually happens, he's to vague about the concept, but basically you will have a paralell resonance fecuency and a series resonance frecuency, The circuit model varies with the frecuency applied, so, for instance, at low frecuencies you do the calculations and you get a certain resonance frec, but as frecuency increases some parameters vary and add up to the calculations, and you have to modify your circuit model. That will yield a predominant series or paralell model(another two new frecs), and the ratio in wich the model has to be "re thinked" as a frecuency function behaves like the systems in the video. Let's say that above 30MHz each time the frecuency is multiplied by 4.66 you have to re calculate your constants (resistance, inductance and capacitance vary with frecuency, capacitors become inductors, inductors become capacitors, resistors may bacome the two of them, the attenuation changes, everything starts to become a CHAOS!). So you have a circuit model for 30MHz, other for 140MHz, other for 600MHz and so on...I worked with high frecuency circuits and this is certainly true, in fact, if these effect hadn't been taken into account probably we wouldn't have modern communication systems, like cellulars, satellites, etc....
Ronald de Rooij Ah thanks all, I need to study this more, so much more...
I usually really like how Sixty Symbols presents the core of scientific field to scientifically inclined people, without dumbing it down, without relying on having knowledge in that field.. But here, I get totally lost. What do these diagrams represents? What are the junctions? What is it with the box and the waves? What have fish to do with it? I guess I need to get some knowledge on chaos theory before hand, in order to understand that video.
Keep up the good work, but this one is subpar.
In the case of the convection loops (and I suppose all of the other scenarios) what does the y axis represent and why does the line split?
+House The y axis represents your "x value" i.e. the number of fish (for example). The x axis represents some parameter (usually called r), which alters the characteristics of the system. The lines represent recurring states of the system. (For fish it is a bit confusing, because fish population don't jump between two numbers instantly.) See my other comment for the splitting. ;)
Nice too see another video about this number!
If you liked this video, I recommend James Gleik's introdoution and overview of the chaos theory, "Chaos: making a new science". I loved it.
This is absolutely amazing.
Can I suggest using a pen when drawing bifurcation diagrams? Or has someone already suggested that?
I was wondering when they were going to address the random fish clips.
Very good video!
Why does the line on the graph break into two? what is that showing?
My belief is that nothing is random just that there are so many variables we can't take into account completely
@TheCarnun There's one episode about waves where he plays on it!
I dont get it. What is it that happens when the pitchfork splits? And what is it thats required to make it split? Energy?
This video is listed as a source in the Wikipedia article on Feigenbaum constants.
Well that said... nothing.
How does a graph of population numbers split? How does the period of a pendulum split?
I had some trouble as well, the split is simply a change, with fish a split could be when more predators move into the area, for the pendulum its when a force different from the force its currently under is applied.
@TonyMach01 Simply put, what they discovered was that in any natural system changes occur at set intervals related to the constant 4.66 - It is similar to the Fibonacci sequence, how natural design follows the sequence in general. It is just one of those fascinating 'rhythms' of nature that create a constant that can be used to describe when and at what intervals a dynamic system branches or changes. Does that help?
Near the end of the video, there are 2 constants, the one discussed in the video, and another one. What is that other one?
@HerrCaZini Kind of like the point where you have to change gears when driving a car up hill, you're changing the ratio of the gears against the force times the distance, or "period doubling" the force in relation to time. A dual pendulum setup does the same thing with force applied where it will double its occillations at a certain point in time.
@Moriarty2112 i wasn't saying they are the same, just that they sounded similar.
i just though that what karma is: The belief that every action you make will affect the whole world around you.
sounded a bit similar to what was described as chaos theory.
might have misheard though.
If you check the graphic shown at 7:13 (shown multiple time in the video, I know), how comes it's not symmetrical? Is it only "this instance of a simulation", or are the "branches" always tilted upward? Well, now that I ask this, I wonder what does the Y axis represents on that graphics. I guess I need to do some research of my own to get any closer to any kind of comprehension!
my thesis for high school is about this , and this video sure helped out a lot , thx guys
These bifurcations/splitting phenomena, do they have any relation to (for example) degeneracy and splitting of modes of vibration in crystals?
The size of my eyes as I was watching this !
Very fascinating stuff. =)
I thought for sure he was going to start writing on the wall. I've seen it done. I saw a Physics Prof. start writing on a light colored, painted door; and he just went on and on, writing on this door, just because he happened to be standing there when he was seized with the need to explain something. Building maintenance and was not amused; and the Admin. produced a note asking him to "please not write on the doors".
AFAIR it was Lorenz' print function that rounded off the intermediate result. If Lorenz had written a better print function, and subsequently entered the new start value to the machine's full precision, he wouldn't have made his discovery, at least not at that point in time.
@HerrCaZini Yea Brady it would be great if you could do more on this subject... it is hard to understand without knowing what the graphs mean.
I thought he was going to talk about the magic numbers of nuclear physics, but this is much more interesting.
Prof Susskind holographic 2d math at high energies.
One way to interpret is as geometric distortions at high accelerations.. ie. back-facing surfaces wrap around, infinity as an edge.
If higher dimension/info-scapes have geometric representations in such functions.. then, like a quasi crystal, it is a geometric function/gradient through possible states.
numbers as geometric structure.
IMO, Moriarty, Susskind & underlying wave function, seem to be different interpretations of complex rotations.
The story of how Feigenbaum discovered his number is very interesting; it's worth looking up; a human interest story.
Lyapunov exponents would be a good compliment to this video (and the eigenvalue one).
Anyways, great job and 5 stars as always.
The cause of the split is the driving force. If increase the driving force (in a population of fish you would increase the amount of space, food, mates etc) at a constant rate then the population would increase as shown in the bifurcation graph.
@HerrCaZini the easiest way I can think to explain it, imagine you have a really complex equation you want to solve -- for simplicity's sake I'll say Ax = B, but know that this equation doesn't actualy work for this. So you know the values of A and B over time and you want to solve for x. It turns out that for many natural phenomena, when you solve for x, over time you find that there can be more and more solutions, like x=5 and x=8 are both solutions and later there are 4 then 8 etc
It's Feigenbaum's constant alpha, the scaling factor between x values at
bifurcations.
@CheezeFis But did it flap its wings?
@Wilder Riley shut the f up. No body wants your advertising
@scottvalentine808 Say the environment can manage 100 fish. The graph has population growth factor on the X-axis, and stable population matching that value on the Y-axis. When fish multiply less than 2.5-fold each year, the population stabilises to 100 fish after several generations. If the growth is more intense, the population never stabilises, but swings between lets say 105 and 95. At 2,8 the population stabilises on four levels. And so on, all the way to chaos.
I understand that the actual constant comes from a ratio, but I don't understand the initial graph. it's clearly not a function, right? so what does the graph represent? can anyone clarify?
+Imfingbob What you do with these kinds of function is that you iterate them rather than calculate them like a 'normal' function which takes a value for a given input x. As you vary some control variable k (in the logistic case this represents the natural proportional increase in population from one generation to the next) you run each value for a considerable length of time, i.e. a large number of successive values, and you find that the behaviour initially settles in toward one value and just stays there. As you increase k eventually you get a first 'bifurcation' and instead of the function settling to a single value it starts to alternate between two values. The it cycles through four values etc. The graph is of the steady-state values the function takes after running for a while with a given value of k. That's why it looks so weird compared to a normal graph. Hope that makes sense.
This was in Scientific American almost 30 years ago. The iterations represent generations in the case of population. I wrote a program to graph it back then. See my other comment for the pseudocode.
this doesn't help you determine which particular atom will decay at what time.
IE if you have 3 examples of a radioactive isotope with a decay rate that's been determined to be approximately 10 minutes. you wait for the first one to decay and start the clock there. The remaining two have an equal probability of being the next to decay and any given second has an equal probability of being the next one in which the decay occurs.
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@Moriarty2112 I agree that they aren't alike.
i just like finding out rational explainations for old superstitions, e.g. mis interpretation of ball lightning.
but i understand the karma is completely unrelated to chaos, and that there similarity is purely coincidental.
Professor! you look so young!
Could this explain the start of the universe to the unimaginable vastness of stars & planets
A constant in chaos. How fascinating.
Pretty much, yeah. But I wouldn't call it "descend". Chaos is the reason life formed (chemicals was mixed and mixed by energy from volcanoes, meteor impacts, the sun, etc, to the point one molecule became so complex it gained the ability to make copies of itself by using energy from heat, light, pressure changes, Ph level changes, etc).
Just brilliant.
Pi times what equals the Feigenbaum constant? I'm calling it the Dan constant. Checkmate!
Think of rate of decay as an average
You can observe an atom decay. You mark that as your starting point, then you record the time it took until you observed the next atom to decay, and how long the one after that takes etc. You average out your results and the more intervals your average represents the more accurate the average is.
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The world needs more Physicists and mathematicians!!!
Order in chaos? Fascinating.
I'm a chemistry guy myself, but these videos are really interesting.
Yes but I'm only interested in it from a number perspective and want to improve the collection of irrational numbers I know about. So this number joins the ranks of pi, e, and the golden ratio by being common throughout nature?
I wish the video went into more detail on what a bifurcation diagram actually shows... I've been trying to type it out into a comment but it's not working so well =/. I just learned about this stuff at school, and the biggest missing piece for me what how the heck you draw a bifurcation diagram.
In any case, awesome video!
Its interesting how this is related to Zeno's paradox.
People, people please. These are not meant to be tutorials, they are an introduction to ideas so that people who otherwise wouldn't know about these concepts or interesting tid bits about physics or maths etc... have something to "springboard" off of, in order to do more reading and learning...
They aren't purposely leaving out certain things, they just have too because of time constraints etc..
fiegenbaum constant, fish follow the comfortably available path once the fish is hungry it goes to where the split is for the food is, once the fish eats a copy of the fish is created in the air then both fed fish swim off into their environment of dwelling. Until they hunger again then they find the split of water and air to feed and it happens until for the life span of the fishes or until the pen runs out of ink. chaos theory is when the fish decides it is hungry enough to find a split to eat
Great work! I liked this video a lot.
no brown paper :O??? BRADY HAS BEEN HACKED AND THIS IS NOT THE REAL PROFESSOR MORIARTY.
Rotations can map the information representing an evolving universe.
very complex Perlin Noise functions.
underlying mathematical structure this video describes.. is an indication of underlying matrix/rotation geometries.
At high very energies, physics becomes 'simpler', forces converge.
From 'unity', rotations can precess.. etc, into seemingly chaotic modes (a field represented by other rotations).
string temp.. as position.
translate.. everywhere is same universe rotated.
not random ones
How to make a universe from rotations.
Rotate spinor on 2 axis, returning to original state after 10^48 rotations, ie mapping out a proton shell in 'pseudo-planck' units.
Multiply by 4 more axis rotations.. each representing a length element of a quasi crystal rombi.. so it iterates pseudo-protons through all possible states.
Add more rotations for complexity.
Would observers made of/ in such a system experience similar mathematical, geometric reals & determinants, complexity & chaos?
Really? I understood it. If you measure a pendulum's time from side to side, if it is 1 at first, then 2, then 4, then the time it takes for it to go from 1 to 2, divided by the time it takes to go from 2 to 4, is the feigenbaum ratio. Eventually the pendulum stops swinging in a regular pattern and just moves around due to wind and so forth (that's when it becomes chaotic).
i don't understand, If reality is quantized (planks constant) wouldn't that mean that Feigenbaums constant be a finite number in ratio with planks constant in the real world? I mean i understand why feigenbaums constant is infinitely long when it comes to just pure mathematics, but in experiment, all thing can literally be traced back to the movement of something at the plank length, which would give the Feigenbaum constant a finite number right?
Up until now, I thought that the Divine Proportion discovered by Luca Pacioli - 1.618 was the same thing as the Feigenbaum constant. Does the Divine Proportion have any applications in physics - experimental or theoretical?
I, I, I really enjoyed this!
Surprising you haven't put this on numberphile :p
I don't think numberphile existed at the time of upload.
They finally did. lol
The scary thing is, this applies to the rate at which the human population is making new advancements... The time between major advancements is exponentially decreasing, and mathematicians calculated that the point of chaos will be reached in.... december 21, 2012.
I'm scared :(
damn, this is a nice sub :)
thx for those vids
Hello from 2020. I like your comment. :D
Or cells. one cell divides into two, then those 2 cells divide, then those cells divide, etc
Can you explain the blank areas on the chart at 7:16? What have they got to do with the constant? Are they important?
The blank areas are just like the area before the first bifurcation (fork) , they represent nothing interesting. Stability I guess. Then chaos starts taking over, then later youll see periods of stability show up again, then right back to chaos and further bifurcation.
@TixTipx Your last statement is correct. Read about "Logistic map" on Wikipedia.
7:47 I understood ''feces'' at first -.-
Funny how chaos makes physicist wave their arms around a lot.
I'm afraid I agree. They both talk about CHANGES in the patterns, but they don't explain the BIFURCATIONS which are crucial. If it were only change the line would just go in another direction, but not split.
As I understand it at the point of the bifurcation there suddenly are two states instead of one, and the system oscillates between those two. CMIIW
very nice, I loved it!
2:40 doesn't that make any measurement of the weather fundamentally wrong? If each decimal place varies the result so different, how do we know which one is 'right'?
lolzomgz1337 Yes and no. What usually happens is that meteorologists create multiple models with slight variations and arrive at an estimate through several steps of statistical analysis. That is basically taking the average of the different models and arriving at the statistically most likely outcome for the weather system. Not exactly though, since this is a youtube comment and not a lecture in meteorology.
lolzomgz1337 as he mentioned, they are about the same for a little while and then go crazy; this is why our weather forecasts only forecast a few days ahead, and aren't completely accurate
lolzomgz1337 but not completely jagged
lolzomgz1337 jagged as in screwd up
@Maladath Very well put!
Put a grain of rice on the first square of the chess board, two on the next, double that on the next and so on.
the constant was shown at the end and seem'd like it was going on forever. But it is derived from a ratio. So is it just long or am I missing something and its irrational?
experimental results have error. you would in that case have a perfectly finite rational answer that was close to delta but ever so slightly off. maybe millions or billions of decimal places into it, but it would be slightly off. just like you could draw a circle and then get some string and measure how long the string is to go all around it. you can derive pi as a finite number that way, but small errors in measurement would make it a (albeit close) incorrect number.
6:47 There is a Marshall amp in the bottom left corner...
i want Professor Moriarty to draw me that graph and let me get it as a tattoo to go with my other math tattoos. a nerd can dream...lol