Hidden Structures of the Mandelbrot and Julia Sets

in my previous video I animated through all of the Julius heads for Z squared plus C along the real and complex axes question was asked about the transition point where the majority the points go from outside to inside and why it seems to become very chaotic I thought this might be a good start to discuss the relationship between the Mandelbrot set and the Julia sets for Z squared plus C but before we look at some cool pictures let's first take a quick refresher on the math the Mandelbrot sets iterative equation is commonly given as Z squared plus C so we have a two dimensional plane where we have X as the real number axis and y as complex number axis all the points represent a complex number we can take any single point or number and use it as C in our equation we start with a z0 plus 0i and run the equation works we then take that result and use that as our new z leaving C as our original number rinse repeat etc until a number does one of two things escapes off to infinity or it hangs around the origin those that hang around even after millions of iterations we call inside and those that escape we call outside now you may wonder why we start this process at zero instead of skipping that first iteration and load Z with just our point number well put a pin in that because we'll come back to that in a bit for now let's just take a look at the Julia set specifically Julie is set with the equation C square plus C Julius it's comprised any type of equations go look it up if you want more in fact for the rest of this video when I think Julia said I will mean specifically the Julia set for Z square plus C so I don't have to say Z squared plus C over and over them the very astute among you may have notice of the Julia equation is exactly the same as the men apart equation the main difference however is the value that we use as our C which in the Mandelbrot set we use our complex point them but now we use an arbitrary value so for any point our complex number plane we start the first iteration with disease loaded with our point number and some arbitrary value for C at this point the sequence is the same as before we iterate through until the result escapes were getting around you as you can see changing a few value commuted practically different results now comes the fun part the two equations are the same we can draw a direct connection from a point on the Mandelbrot set to a specific point on a specific Yuliya set to consider our first metal rod example where we chose the point zero point five plus zero point five I we then started our iteration sequence with zero squared plus zero point five plus zero point five I the same sequence happens as the Julia set when we use zero point five plus zero point five I and our arbitrary C value and we run the point at the origin the result of this is that for any point on the Mandelbrot set inside or outside there's a direct correspondence to the origin of the Julia set with a see of that point in effect this means that the picture of the Mandelbrot set is indexed to all of the Julia sets for the point zero plus zero I let's illustrate this by comparing the two on the left the close-up of a subsection of the land alongside with a target around a specific point the right is the tuna set with a semen were under the target in the middle on set put a target around the fortune as we move up the nanobot check you've seen at the course of engineers that changing those who quit guys have seen that the point under both targets changes at the same time but don't worry I'll slow it down a bit in a second get a little closer to the edge little bit more female [Music] [Music] [Music] [Music] [Music] as you can see here inside on the mandelbrot left side we will also be inside on the Julius that side and we get closer to the main body of the metal offset virtues as much like ice clothes come together to form a more solid shape [Music] here we all melt side even with crank up the iteration of home from school back alley River the course modules have become solid finally reaching a perfect circle in origin okay we've accounted for a single point on each of the Jew is that but what about the rest of them do they connect to the Mandelbrot set and any full way the short answer is yes they absolutely correspond to points on the medibot set their suit up for the more complete answer we have to work backwards it's from Julius Meno bottom remember I said in the beginning to put a pin in the fact that we start at zero when we run our iteration to the nanobot site even though we can easily skip that step well what happens we don't start at zero instead what if we offset the start point mathematically this becomes the same is if we choose a point not at the origin in our Julia set let's look again at our comparison except this time we'll leave the target on the metal oxide at a single point and we'll move the target on the Julia side which is the point value from the Julia set as our starting offset value for the middle oxide as we increase in the positive complex direction we can see that the Mandelbrot set doesn't preach interesting gymnastics however the value under each of the targets remains the same from both sides here we travel in the negative complex direction which produces the same result as the positive side did this makes sense to some squaring the number in the first math operation that starts the whole sequence in fact any direction we choose for our offset will be able to the opposite offset when both the mental bite enjoy eternal this is why all the German is said to reverse mirror images of themselves and any line that passes through the origin genitals in a metal box at asthma who are off set out to addition to the one and travel around in the circle oh you the Julius that receives cirrhosis as I said the metal rod sent us pretty interesting gymnastics depending on the offset we use the closer we are the edge of the metal outside police start off setting give us a higher chance of writing into the elastics and this explains why the Julia set gets extremely chaotic right before our sea becomes completely inside and in effect this makes a Julia set the value of seeing a component index or the label offset at the point corresponding to that c4 all starting concept items here's another way to look at this let's start with our good old men of offset now let's overlay the Julius that preserve or furohai at its corresponding point and tilt it up we'll remove all the Jewett points except to those that ride the complex axis we can now do the same for the other points down Randall Boggs relaxes [Music] when we do this for all the points down the real axis will end up Buddhist funny-looking triangle shape and we have a picture of all the points along the metal rods real axis with all all the complex only offsets now if we scan in the complex direction we get a cat scan of the full Mandelbrot set with a complex offset [Music] the we could do the same with a real offset as well we'll tilt the set sideways to get a more complete picture [Music] [Music] [Music] now two dimensional slices are pretty informative but to see the full beauty of these mathematical structures we need to go 3d here we have a tree dividual slice of the complete plan apartments and prostitutes from complications [Music] the better box set with the sweep along to real [Music] you [Music] Julie five relapses combined more construction [Music] all the jewels that's the right order [Music] lightly a close elders the viewer to the temptation the whole structure of the metal rod set with every offset this will effectively show every possible Julius appointed Percy square plus see I've been their home records thank you for watching [Music] [Music] [Music] [Music] [Music] you

Channel: Damnimp

Views: 224,578

Rating: 4.9261913 out of 5

Keywords: Mandelbrot Set, Julia Set, 3d math

Id: vfteiiTfE0c

Length: 18min 23sec (1103 seconds)

Published: Fri Mar 24 2017

Reddit Comments

[u/XDFreakLP]

Ooooh this is a good one

[u/leadnpotatoes]

The music is mixed way too loud, you can barely hear what the author is trying to say over all that 90's synth.