Dimensions | Chapter 6

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Dimensions | Chapter 6
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École Normale Supérieure de Lyon (ENS-Lyon)

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nations transforming a lot well if you don't mind we're going to transform 1 portrayed caps a let's begin as something simple the transformations Edgar's is added to each point on the photo corresponds to a complex numbers this divided by 2 we get another point intimidated by the transformation into new picture less surprised by shrank a half the size since each is being divided by 2 this transformation is called a harmless back John they let's go to the multiplication by Carl him easy we know that multiplied by eyes just a quarter turn also noted the modulus does not change the beyond image increased 90 degrees by indeed this is quite a sophisticated way of saying that we just rotated the picture at sea back well but a bit more complicated not vacation by 1 plus sign a look at a complex number 1 plus sign it corresponds to the point with a 1 and warden it on its argument is 45 degrees at its modulus the square root 2 Bush using Pythagoras Iran hints a modification by 1 plus on announced Pfister multiplying but the square meters to and they adding 45 degrees to the argument it's simple 1 has to combine a homer 50 and a rotation this is called a similarity as a rule Croat up Is that a more interest we're going to transform the points state into their squad the state multiplied by saying that a let's begin by placing the photo a suitable place flush against the cold taxis that is man I assume little bit since the squaring process will change the size of things and I need space to show you took change now we can transform a photo progressively notice that the argument is square is twice the argument is so that the right angle of the left of the doubled on the transformation after now it is turned into a 180 degree angle let me place the photos somewhere else she lets to look again at the same transformations it square elects to gain the same argument toppling a incidents the combined extinct pull the transformation its argument is about 45 degrees and after the transformation points at 90 degrees she did you could also observe the marginalized way tho a the big not go-between new transformation sending the point Z 2 minus 1 bring man don't forget with complex numbers on had multiplied but also not not play 0 schools does doesn't this image remind you of the Sistine Chapel bring back large complex numbers With a large margin becomes more money and takes maybe and comments here's a similar transformation look at the formula the value of k changes slowly some parts are expanding others a contracting if 1 looks close to shake presented even dialects on not Air better where a circle remains circular even in so it might cross my hand grew my face became smaller still recognized me yeah at an and 1 more transformation more involved well this 1 is not really a weight-loss program me but now wants more that evangelical because I shapes small plots did not change for if you look at that but on my show he keeps a secular Shiite on these transformations are calm formal Fulham Norfolk rather complicated laughing in Greek words saying that 1 presenters shapes indeed with complex numbers what could do quite a lot 1 could even take the exponential if you know what this means but even if you don't have to
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explain somebody objects or been studying closely she here you see a number of points some opted inside these days a it's all Council measures that's a former transformations it squares ponds and let's look at the resort you could see that point stains on the disk and the yellow points on the contrary escaped from the disk and even escape from screen laughter my mom 1 says that the big disk Is the fielding Jr. said of the transformation square points outside the issue said escaped to infinity with on repeats the transformation death we
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understand how these shapes change I will show you several things in parallel on the left-hand side the red sign you could see a point that Stockton this is the point on the right-hand side you see the corresponding Jr. said this is performing as he slowly changing but sometimes for some values and the Junious said seems to disappear but cannot see anything anymore on the screen what now for instance the truth is that Jr. said 3 walked into an infinite number of pieces so small that you didn't see anything on the screen they were mental approach who popularized practical steps suggested the study of this thorny red that describes the values of the which 1 concede that Julius clearly on the screen in other words that issue which they GSA did not into multiple beast Of course is red said his call the man Price said I spent quite a lot of sun studying it To I suggest that we look closely very closely at dismantle products said ends you lean side said you can appreciate how beautiful it let's go here it is that but
once I will not explain everything imagine the mantle bread said as bright colors surrounded by tropical seas and that you could see that often but really your looking at Troon microscope picked details if the mantle process With size a soccer field well we will be looking at details the size a single act a feel water million mm if if ahead if they have a right now they say him ahead if if they had a bad if if if if if a have room a head of maybe you want Dre while I he'd Tracy Ms. vegetables because it is beautiful and because understanding subjects gay much pleasure the main is reason enough to spend time on this question also because these transformations simple what can find these essence of chaos such a fundamental concept in modern science but simple things generating rich structure but To study complicated phenomena through their simplest incarnation this is off the role of the mathematician but a
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