The Mandelbrot Set - The only video you need to see!

[Music] Mandelbrot's fascination with the visual side of math began when he was a student his only in January 44 that suddenly I fell in love with mathematics and not mathematics in general with the geometry most complete central form that that part of geometry which in which mathematics and the eye meet the professor was talking about algebra but I began to see my mind geometric pictures which fitted this algebra and once you his pictures the answer become obvious so I discovered something which had no clue before that I knew how to transform in my mind instantly the formulas into pictures it was 1958 the giant American corporation was pioneering a technology that would soon revolutionize the way we all live the computer IBM was looking for creative thinkers nonconformists even rebels people like Benoit Mandelbrot in fact we had cornered the market for a certain type of oddball we never had the slightest feeling of being establishment Mandelbrot's colleagues told a young mathematician about a problem of great concern to the company IBM engineers were transmitting computer data over phone lines but sometimes the information was not getting through they realized that every soften the lines became extra nosey error occurred in large numbers it was indeed extremely a messy situation Mandelbrot graphed the noise data and what he saw surprised him regardless of the time scale the graph looked similar one day one hour one second it didn't matter it looked about the same it turned out to be safe similarly the Vengeance mandelbrot was amazed the strange pattern reminded him of something that had intrigued him as a young man a mathematical mystery that dated back nearly a hundred years the mystery of the monsters the story really begins in the late 19th century mathematicians had written down a formal description of what a curve must be but within that description through these other things things that satisfied the formal definition of what a curve is but were so weird that you could never draw them we couldn't even imagine drawing them they were just regarded as monsters or things beyond the realm they're not lies there nothing white lines they're not circles they were like really really weird the German mathematician Georg Cantor created the first of the monsters in 1883 he just took a straight line and he said I'm gonna break this line into thirds and the middle third I'm gonna erase so you're left with two lines at each end and now I'm going to take those two lines take out the middle third and we'll do it again so he does that over and over again most people would think well if I've thrown everything away eventually there's nothing left not the case there's not just one point left there's not just two points left there's infinitely many points left as you zoom in on the counter set the pattern stays the same much like the noise patterns that Mandelbrot had seen at IBM another strange shape was put forward by the Swedish mathematician Hale Gavin coke said was where you start with an equilateral triangle one of the classical euclidean geometric figures and on each side I take a piece that I substitute two pieces that are now longer than the original piece and for each of those pieces I substitute two pieces that are each longer than the original piece over and over again you get the same shape but now each line has that little triangular bump on it and I break it again and I break it again and I break it again each time I break it the line gets longer every iteration every cycle he's adding on another little triangle imagine iterating that process of adding little bits infinitely many times what you end up with is something that's infinitely long the Koch curve was a paradox to the eye the curve appears to be perfectly finite but mathematically it is infinite which means it cannot be measured at the time they call it a pathological curve because they're made no sense according to the way people were thinking about measurement and Euclidian geometry and so on but the Koch curve turned out to be crucial to a nagging measurement problem the length of a coastline in the 1940s British scientist Louis Richardson had observed that there can be great variation between different measurements of a coastline it depends on how long a yardstick is how much patience you have if you measure the coastline of Britain with a one mile yardstick you get so many yard sticks which gives you so many miles if you measure with a one foot yardstick it turns out that it's longer and every time you use a shorter yardstick you get a longer a number because you can always find finer indentations Mandelbrot saw that the finer and finer indentations in the Koch curve were precisely what was needed to model coastlines he wrote a very famous article in science magazine called how long is the coastline of Britain a coastline in geometric terms said Mandelbrot is a fractal and though he knew he couldn't measure its length he suspected he could measure something else its roughness to do that required rethinking one of the basic concepts in math dimension what we would think of as normal geometry one dimension is the straight line two dimensions is say the box that has surface area and three dimensions is a cube but could something have a dimension somewhere in between say 2 & 3 Mandelbrot said yes fractals do and the rougher they are the higher their fractal dimension though all of these technical terms like fractal dimension and self similarity but those are the nuts and bolts of the mathematics itself what that fractal geometry does is give us a way of looking at in a way that's extremely precise the world in which we live in particular the living world Mandelbrot's fresh ways of thinking were made possible by his enthusiastic embrace of new technology computers made it easy for Mandelbrot to do iteration the endlessly repeating cycles of calculation that were demanded by the mathematical monsters the computer is totally essential otherwise has taken a very big long effort Mandelbrot decided to zero in on yet another of the monsters a problem introduced during World War one by a young French mathematician named Gaston Julie Gaston Julia he was actually looking at what happens when you take a simple equation and you iterate it through a feedback loop that means you take a number you plug it into the formula you'll get a number out you take that number back to the beginning and you feed it into the same formula get another number out and you keep iterating that over and over again then the question is what happens when you iterate a lots of times the series of numbers you get is called a set the Julia set but working by hand you could never really know what the complete set looked like there were attempts to draw it doing a bunch of rithmetic by hand and putting a point on graph paper you would have to feed it back hundreds thousands millions of times the developments of that new kind of mathematics had to wait until first computers were invented at IBM Mandelbrot did something Julia could never do use a computer to run the equations millions of times he then turned the numbers from his Julia sets into points on a graph my first step was to just a throw mindlessly a large number of Julia sets not one picture hundreds of pictures those images led Mandelbrot to a breakthrough in 1980 he created an equation of his own one that combined all of the Julia sets into a single image when Mandelbrot iterated his equation he got his own set of numbers graphed on a computer it was a kind of road map of all the Julia sets and quickly became famous as the emblem of fractal geometry the Mandelbrot set is not easy to describe the Mandelbrot set visually it looks like a man it looks like a cat it looks like a cactus it looks like a cockroach it's got little bits and pieces that remind us of almost anything that you can see out in the real world that particular living things so it has a character that reminds us of lot of things and yet it itself is unique and anew the Mandelbrot set is real an absolute thing no question whatsoever any mathematician or any computer scientist or student in a school could study it and find the same describe the same thing it's a common experience and so such things that can be magnified forever that have infinite precision do exist but they're not touchable it's a geometrical shape and an icon if you wish which somehow embodies as an example a very important aspect of how the world works somebody recently actually called this set the thumbprint of God with this mysterious image Mandelbrot was issuing a bold challenge to long-standing ideas about the limits of mathematics the blinders came off and people could see forums that were always there but formerly were invisible the Mandelbrot set was a great example of what you could do in fact or geometry just as the archetypical example of classical geometry is the circle when you zoom in you see them coming up again so you see self-similarity you see by zooming in you zoom zoom zoom you're doing it usually in the pop suddenly it seems like you're exactly where you were before but you're not it's just that way down there it has the same kind of structure is way up here and the sameness can be dropped now we'll begin our serious exploration of the Mandelbrot set a voyage which in fact could last forever and ever much longer than the lifetime of the universe I have here the full set about six inches across now if I blow this up now link she's the magnification now thirteen times and you see more and more detail disappearing an interesting thing is you see mini Mandelbrot's replicas almost identical yet that's subtly different of the original set and I can go on doing this here is a magnification of more than three thousand times so the original picture about six inches across is now half a mile across and no matter how much we magnified it million times a billion times until the original set was bigger than the entire universe we would still see new patterns new images emerging because of the frontier the EM set is infinitely complex and when I say infinitely I really mean that most people and they say infinitely in the only will rather baby this is great infinity [Music] [Applause] [Music] [Applause] [Music] what is so remarkable in fact astounding about the Mandelbrot set is it although it's infinitely complex it's based on incredibly simple principles and like almost everything in modern mathematics in fact anybody who can add and multiply can understand the principles on which it's based you don't even have to subtract or divide still less use logarithms or trigonometric functions to comprehend how the Mandelbrot set is created in fact in principle it could have been discovered any time in human history and not merely in 1980 but the problem is this although it's only based on adding and multiplying you have to carry out those operations millions billions of times to create a complete set and that's why he was not discovered until the era of modern computers it's an interesting parallel with the equation that almost everybody is familiar with the only equation on what everybody is familiar with in equals mc-squared albert einstein's equation that says matter and energy are equivalent to each other that was a very simple equation with very far-reaching consequences and the equation for the Mandelbrot set is equally simple Z equals Z squared plus C the letters in the manual product creation stand for numbers unlike those in Einstein's equation where they stand for physical quantities mass lost the energy the Mandelbrot numbers are coordinates positions on the plane defining the location of a spot another difference from Einstein's equation and a very important one is this double arrow it's a kind of two-way traffic sign the numbers flow in both directions constantly feeding back on themselves this process of going round and round a new is called iteration installer like a dog chasing its own tail the output of one operation becomes the input of the other and so on and and on when the Mandelbrot equation is given a number representing a point and that number is iterated through the equation one or two things happen either the number gets bigger and bigger and shoots off to infinity or it shrinks to zero depending on which happens the computer then knows where to draw a boundary line so what we get from this basic iteration is a kind of map dividing this world into two distinct territories outside of it are all the numbers that have the freedom of infinity inside it numbers of the prisoners trapped and doom throughout them at extinction and it was happening at the tip of each hair it splits into two others and so on each is splitting going on indefinitely this splitting up the spy for occasion going off into a pond in random directions quite abruptly is typical of a class of mathematical entities called fractals the Mandelbrot set is the most famous fractal the word fractal means any geometrical structure that has detail on all scales of magnification no matter how big you make it you still see extra new detergency before and the name was actually vented by Mandelbrot himself he thought he had to have a name for this area he realized he was working in and so he coined the term fractal because it conveys this feeling of fragmented broken fractional irregular a common characteristic of fractal systems is branching trees circulatory systems and rivers for example all display branching fractal patterns there are lots of fractal systems within the body and my work on the heart the heart there's actually full of fractals the most famous example of it is the blood vessels coming into the heart the coronary circulation is a pattern of a branching network of blood vessels that's typically fractal where the branching structures look very similar at different scales even snowflakes known for the dissimilarity from each other display a fractal pattern internally fractals are shapes which we are extraordinary used to in how to say our subconscious ill organized life for example everybody knows that if you take a map of Britain on a small school globe you see a very simplified shape Cornwall is just the kind of triangle and waste perhaps a little rectangle you can't put the details on the bit on a small map if you look at in a larger map you add more detail the closer you come in certain since imagine yourself like somebody coming in on a rocket from far away she very little closer you come the more detail you see if we come very very close you believe she rocks and finally the idea of coastline disappears because when there's no longer where as where it's landing with water so and he was always in my mind to put together a geometry based upon many known facts in mathematics that advanced mathematics many scatter facts in in our experience many scattered facts in the results what scientists had done of various kinds many all kinds of of putting together all these things and using them as breaks if you will of a new building which is a new geometry living creatures seem to be complicated structures produced from simple rules simple laws of physics and chemistry and a lot of the structure that you see in living creatures is organic but pattern structure leaves on trees ferns particularly things like that have the same feature that the man what set has of you look at little pieces of them and they have lots and lots of detail and in fact the little pieces look very similar sometimes to the whole thing it's very tempting to compare the way a simple formula produces a complicated Mandelbrot set with the way very tiny things in nature produce complicated organisms and there were certainly some similarities in there is the same kind of unfolding of a process the instructions are there but not an actual description of the object you

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Keywords: Mandelbrot, fractals, fractal geometry, math, electronica, psychedelic, Fibinacci, The BitK, BitK, Bros in the know, Mandelbrot set, Amazing, Science, Nature, Earth, Intelligent design, Geometry, Fractals, infinity, Infinite, spiral, spirals

Id: 56gzV0od6DU

Length: 21min 18sec (1278 seconds)

Published: Sun Jun 12 2016

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Fractals everywhere! check out Bitcoin booms and busts over the years and zoom in and out