A Magic Number - Sixty Symbols
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- เผยแพร่เมื่อ 29 พ.ค. 2009
- 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.sixtysymbols.com/
With Philip Moriarty - วิทยาศาสตร์และเทคโนโลยี
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.
love these people/professors, they are so passionate and explain the concept really well
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
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 :)
I don't understand. I need James and brown paper.
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.
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 find the speech from 5:33 - 5:43 really inspiring and deep, gotta love it :3
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.
This is absolutely amazing.
Nice too see another video about this number!
Great work! I liked this video a lot.
7:00 I love the geeky passion in these guys XD
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.
This one went over my head I'm afraid.
haha, I was thinking the same here
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 :)
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...
very nice, I loved it!
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.
I admire Prof. Moriarty's skills with Lab View. It's like he has a program for everything he want's to talk about.
The size of my eyes as I was watching this !
Very fascinating stuff. =)
I, I, I really enjoyed this!
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)
)
)
Just brilliant.
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.
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.
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!
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.
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. ;)
Near the end of the video, there are 2 constants, the one discussed in the video, and another one. What is that other one?
These bifurcations/splitting phenomena, do they have any relation to (for example) degeneracy and splitting of modes of vibration in crystals?
@Maladath Very well put!
Lyapunov exponents would be a good compliment to this video (and the eigenvalue one).
Anyways, great job and 5 stars as always.
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.
I was wondering when they were going to address the random fish clips.
Very good video!
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!
@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,
Phenomenal!
my thesis for high school is about this , and this video sure helped out a lot , thx guys
@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.
This video is listed as a source in the Wikipedia article on Feigenbaum constants.
I dont get it. What is it that happens when the pitchfork splits? And what is it thats required to make it split? Energy?
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.
My belief is that nothing is random just that there are so many variables we can't take into account completely
Could this explain the start of the universe to the unimaginable vastness of stars & planets
@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?
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..
I thought he was going to talk about the magic numbers of nuclear physics, but this is much more interesting.
@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.
@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.
@TheCarnun There's one episode about waves where he plays on it!
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.
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.
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?
Why does the line on the graph break into two? what is that showing?
The world needs more Physicists and mathematicians!!!
Can I suggest using a pen when drawing bifurcation diagrams? Or has someone already suggested that?
@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.
damn, this is a nice sub :)
thx for those vids
Hello from 2020. I like your comment. :D
@tomaskvapil Very briefly... Suppose you're trying to model a population. The size of this population depends on the growth rate. Now, for certain growth rates, your population will tend to hover around a fixed size. This size is what the y axis on the graph represents. The x axis corresponds to the growth rate. When the growth rate becomes too big, you get multiple "steady" sizes (your population will tend to oscillate between them.) After a certain point, you get chaos... That help?
Order in chaos? Fascinating.
@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.
A constant in chaos. How fascinating.
@CheezeFis But did it flap its wings?
@Wilder Riley shut the f up. No body wants your advertising
What is that program running on your laptop that plots the line and the bifurcation points?
Thanks
It's Feigenbaum's constant alpha, the scaling factor between x values at
bifurcations.
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.
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".
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).
@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
hi brady i have a question out of curiosity (hopefully it isnt toooo late to ask) . is there a relationship between this constant and the golden ratio (phi)?
thank you
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!
how does the splitting graph show the population of fish or whatever it the subject is??
Does this overcome Phi?
well put
Maby its... understanding the control numbers in order to understand infinite fractle posibilitys. in nature like how he said a small change leads you to a layer in which small changes equal a multiplyed or magnifyed result path... explaining "diversity in the infinite structure". Anyone have more light to shed on the subject?
Professor! you look so young!
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?
The story of how Feigenbaum discovered his number is very interesting; it's worth looking up; a human interest story.
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)
Q.Is the Feigenbaum Constant applied to the results of the LHC?
You guys rock :D
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.
Might be a stupid question, these periods or breaks depending on what you apply the constant too.
Are there always the same amount of splittings or do they vary ?
The Feigenbaum Constant is the same for any chaotic system that shows period-doubling. This is because chaos is a result of nonlinear differential equations of sufficient complexity, not the result of any particular field of physics.
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).
Or cells. one cell divides into two, then those 2 cells divide, then those cells divide, etc
So you can tell if something will descend into chaos by the presence of feigenbaum ratio?
I'm a chemistry guy myself, but these videos are really interesting.
Me too. We have so much in common.
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
does this repeated pattern count as a fractal (having to do with fractal geometry)?
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 only don't understand the "splitting" of the graph... anyone more info?
@Aviatorsmith
I see. Thanks :)
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?
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?
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)
Are these the numbers that appeared in the movie π ?