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Alan Norton
United States
Приєднався 10 сер 2010
This channel consists of videos of fractals that are calculated by iterating complex functions. Fractals, named by B. Mandelbrot, are shapes that have repeated detail at all scales of magnification. These videos illustrate evolving Julia sets and Mandelbrot sets. The videos look better if you set playback quality as high as possible. More information about these shapes can be found in the article "Visualization and Animation of Herman Rings of Rational Functions of Degree 3" at: docs.google.com/document/d/1qhtyoq9t6qXxnnkxHMmFpq9zfYH2PCIe/edit?usp=sharing&ouid=111487949632689085985&rtpof=true&sd=true .
Doubled Rings with Long Branches
This video animates Julia sets of the functions
f(z) = z^2(z+b)(z+c)/(bz+1)(20z+1), with values of b and c chosen so that these Julia sets usually have two Herman rings, one inside the other. These Julia sets are similar to those in the video “Evolving Inner and Outer Rings” on this channel, but these values of c result in long narrow branches containing the inner Herman ring and its preimages. The values of b are near the circle of radius 1, with argument decreasing during the animation from about 95 degrees to about 52 degrees. The values of c are near the circle of radius 20 with argument increasing from about 142 degrees to 168 degrees. The winding numbers of the inner ring decrease from 2/3 to 0.63 during the animation, while the winding numbers of the outer ring increase from 0.19 to 1/3. The values of b and c in these images are calculated using the Mandelbrot sets shown in the video “Rotating Double Rings”. That video illustrates two curves of Herman rings for the function f(z) when |b| = 1. Increasing or decreasing the value of |b| causes the two curves to expand or contract relative to one another, which makes it possible to find a value of b for which the two curves of Herman rings intersect. For each frame of this animation (i.e., each value of arg(b)), a value of |b| is calculated so that the two curves intersect in the c-plane. Where those two curves intersect, the corresponding Julia set has two different Herman rings. The points in the image are colored pink for points iterating to zero, and dark green for points iterating to infinity. Points that iterate to the inner Herman ring (and nearby cycles) are colored brown and points converging to the outer ring (and nearby cycles) are colored turquoise.
f(z) = z^2(z+b)(z+c)/(bz+1)(20z+1), with values of b and c chosen so that these Julia sets usually have two Herman rings, one inside the other. These Julia sets are similar to those in the video “Evolving Inner and Outer Rings” on this channel, but these values of c result in long narrow branches containing the inner Herman ring and its preimages. The values of b are near the circle of radius 1, with argument decreasing during the animation from about 95 degrees to about 52 degrees. The values of c are near the circle of radius 20 with argument increasing from about 142 degrees to 168 degrees. The winding numbers of the inner ring decrease from 2/3 to 0.63 during the animation, while the winding numbers of the outer ring increase from 0.19 to 1/3. The values of b and c in these images are calculated using the Mandelbrot sets shown in the video “Rotating Double Rings”. That video illustrates two curves of Herman rings for the function f(z) when |b| = 1. Increasing or decreasing the value of |b| causes the two curves to expand or contract relative to one another, which makes it possible to find a value of b for which the two curves of Herman rings intersect. For each frame of this animation (i.e., each value of arg(b)), a value of |b| is calculated so that the two curves intersect in the c-plane. Where those two curves intersect, the corresponding Julia set has two different Herman rings. The points in the image are colored pink for points iterating to zero, and dark green for points iterating to infinity. Points that iterate to the inner Herman ring (and nearby cycles) are colored brown and points converging to the outer ring (and nearby cycles) are colored turquoise.
Переглядів: 326
Відео
Flagella
Переглядів 2,6 тис.2 місяці тому
This video illustrates Julia sets of the functions f(z) = z^2(z a)/(1 conj(a)z), emphasizing the progression of winding numbers 1/N as N goes from 2 to 40, and arg(a) goes from 90 degrees to 19.9 degrees During the animation |a| starts at 4 and gradually decreases to 3 by the end. For most winding numbers, there is a Herman ring containing the unit circle |z|=1. Whenever the winding number is r...
Collapse of Herman Rings Between Opposing 2-Spirals
Переглядів 1,4 тис.2 місяці тому
This video illustrates Julia sets of the functions f(z) = z^2(az b)/(cz a), with |a| = 1, c real between 4 and 12, and b varying along a curve of Herman rings. Initially c = 4, a = 1, and b is chosen to yield a Herman ring with winding number the golden ratio, .61803… During the animation, c increases to 12, arg(a) increases to 45 degrees, and the winding number approaches 0.5. When the winding...
Pinwheels beside a Mandelbrot set
Переглядів 4632 місяці тому
This video illustrates a planar slice of a portion of the Mandelbrot set of the function f(z) = m z^2(z b)/(z d), with |b-d| small (0.0001), a set which was also seen in the video “Visualization of an Embedded Mandelbrot set” on this channel. This video illustrates the dynamics in 3 complex dimensions of the Mandelbrot set of f(z), using planar images near and parallel to the plane of the class...
Visualization of an Embedded Mandelbrot Set
Переглядів 1,2 тис.3 місяці тому
This video shows how the classic Mandelbrot set fits inside the Mandelbrot set of third-degree rational functions of the form f(z) = m z^2(z b)/(z d). This illustrates the structure of a narrow sliver of the larger Mandelbrot set, nearby the classic Mandelbrot set. Note that when b=d, f(z) is just the standard quadratic polynomial m z^2, which is where the video starts. The video shows what hap...
Periodic Herman Ring Pairs
Переглядів 6514 місяці тому
The music in this video is “Lil Song” by Jack Lamos. This video animates Julia sets of fourth degree rational functions f(z) = x^2(x a)(x c)/(x b)(x d) m. Complex values of a, b, c, d, and m were chosen so that the Julia sets of f(z) have two different Herman rings, and each ring has a cycle of period 2. The values of a, b, c, d and m were chosen in the intersection of two curves of Herman ring...
Evolving Inner and Outer Rings
Переглядів 1,4 тис.6 місяців тому
The music in this video is “Whiny Funk in E Major” by Jack Lamos. This video animates Julia sets of the fourth degree rational functions f(z) = x^2(z b)(z c)/(bz 1)(20z 1), following a curve through the space of complex pairs (b,c). Most of the Julia sets in the video have two different Herman rings: one ring is inside the unit circle and a larger ring encircles the inner ring. Points that iter...
Rotating Double Rings
Переглядів 2,8 тис.10 місяців тому
This video illustrates the Mandelbrot set of the fourth degree rational function f(z) = x^2(z b)(z c)/(bz 1)(20z 1), intersected with the c-plane. This Mandelbrot set shows two different curves of Herman rings, that rotate in opposite directions during the video. Wherever the two curves cross, the corresponding Julia set of f(z) has two Herman rings, one inside the other. The parameter b follow...
Lambda Map for Herman Rings
Переглядів 2,4 тис.10 місяців тому
The lambda map is a variant of the classical Mandelbrot set defined by the quadratic polynomial λz(1-z) This video illustrates the Mandelbrot set of the mapping f(z) = z^2(z b)/(cz 1) to show how the lambda map is approximated by the Mandelbrot set of f(z) when |c| is large. The video shows the Mandelbrot set of f(z) in the b-plane, calculated by iterating the two critical points of f(z). Point...
Wiggly Rings
Переглядів 766Рік тому
This video animates Herman rings of the functions f(z) = z^2(z b)/(1-z conj(b)) During the video, the complex number b follows a curve in the complex plane, following the edge of the Mandelbrot set associated with this function. The winding number of f(z) about the origin varies from about 5/11 to about 1/3. Points in the plane are colored blue if a point iterates to infinity or green if it ite...
Squirmy Critter
Переглядів 3,2 тис.Рік тому
This video animates Herman rings of the functions f(z) = z(z b)(z d)/(1 cz)(1 dz), illustrating the diversity of shapes associated with Herman rings in this space. Here d=0.3, and the complex numbers b and c follow curves of Herman rings in the Mandelbrot set of these functions, with |b| and |c| near 3. Initially the argument of b is about -36 degrees and the argument of c is about -20 degrees....
Millipede Evolution
Переглядів 3,3 тис.Рік тому
This video animates Herman rings of the functions f(z) = z(z a)(z b)/(-1 conj(a)z)(-1 bz), where b=0.2, and the complex number a follows a curve that goes halfway around the origin in the complex plane, with distance from the origin varying between 2.5 to 5. The argument of a varies from 0 to pi during the video, as the winding number about the origin varies from 1/2 to 1, then from 0 to 1/2. P...
Rings Rolling Alongside a Mandelbrot Set
Переглядів 1,6 тис.Рік тому
This video illustrates the Mandelbrot set of the 3rd degree rational functions f(z) = z^2(z a)/(1 conj(a)z) m where the complex number a follows the circle of radius 3.1 about the origin. The video follows a curve of Herman rings that emerges from the cusp of the classic Mandelbrot set and follows alongside the main body of the Mandelbrot set as the winding numbers of the Herman rings increase ...
Rings around a Five-cycle
Переглядів 4,9 тис.Рік тому
The music on this video is "Disco LOL" by Jack Lamos. The video is based on the 4th degree rational function: f(z) = z^2(z b)(z c)/( z 1)(1 conj(c)z) where the complex number c has magnitude 20 and argument about 72 degrees. The value of b varies during the video, following a curve of Herman rings around the inside of a bulb of period-5 points in the Mandelbrot set of these functions. The Mande...
Doubled Herman Rings at a Crossing in a Mandelbrot Set
Переглядів 2,4 тис.Рік тому
This video is based on the 4th degree rational function: f(z) = z^2(z b)(z 22)/( cz 1)(1 22z) where the complex number c has magnitude 1 and argument -39 degrees. The value of b varies during the video, starting and ending with b=conj(c). The Mandelbrot set of these functions is like the Mandelbrot set illustrated in the video “Bubble Crossing” on this channel, with two curves (following Herman...
A Three to Nine Cycle Herman Ring Transition
Переглядів 1,4 тис.Рік тому
A Three to Nine Cycle Herman Ring Transition
Mandelbrot sets of rolling rings in 3-cycles
Переглядів 1,4 тис.2 роки тому
Mandelbrot sets of rolling rings in 3-cycles
Make a herman ring with the function z^3/(z + b) + c Both b and c are complex constants
What is the polynomial for the critical points?
This is a bit complicated. The function is of the form f(x)=x^2(x+a)(x+b)/(cx+1)(dx+1), where a,b,c,and d are complex numbers. The numerator is N(x)=x^2(x+a)(x+b) and the denominator is D(x)=(cx+1)(dx+1). The polynomial you need to solve is the numerator of the derivative of f(x), since the derivative of f(x) vanishes where its numerator vanishes. The numerator of the derivative of the quotient N(x)/D(x) is N'(x)D(x)-D'(x)N(x) where the prime(') indicates a derivative. The polynomial N'(x)D(x)-D'(x)N(x) is the one you want. It is of degree 5, but there is no constant term so you only have to solve a 4th degree polynomial (quartic). I use Ferrari's formula in my program to find the roots.
How do you generate those overlapping fractals?
It's actually just one image. Each pixel is colored based on the result of iterating 4 critical points. If the critical points converge to different cycles the pixel is colored according the longest cycle length. That way you see the (smaller) bulbs with longer cycles appearing to be superimposed over the (larger) bulbs with shorter cycles. Does that make sense?
4 critical points? You mean 4 starting points corresponding to each complex extremum of the function?
The critical points are the complex numbers where the derivative is zero. They are the roots of a 4th degree polynomial. It is solved at each pixel in the image and those 4 roots are then used as the starting points for 4 iterations.
@@VAlanNortonThat's what I said. The "complex extremum" I dubbed is a point in the complex plane which has a complex derivative of zero. Anyways you make sense.
i wanna touch it
This is an accurate depiction of worldbuilding is like
I thought it was a Minecraft map
Trippy
Is that true?
Mathematicians when they discover the Paisley pattern
Muy bueno
mandelbrot but minecraft version??
I don't wanna touch that
glad I'm not the only one
one only ever has to once
Yeah it's pricky
Too bad
You will touch is or you will die
I'm enjoying playing count-the-flagella on this one
I read thwt as "collapse of human rights"
idk how i got here but this video is both cool and made me feel nauseous
I have no idea what I just watched but I am so here for it
It’s the collapse of Herman rings between opposing 2-spirals 👍
this is really cool. does the fractal’s equation change through the video to achieve this?
Yes, the values of a and c change continuously during the video and the value of b is chosen each frame to get the desired rotation. This is discussed a bit in the text.
Thank you!
Watery and dreamlike...
Is this the same Alan Norton that taught the SIGGRAPH 1991 Fractal Modeling course? My friend and I just purchased a Pixar Image Computer off ebay and I remember that Alan Norton presented some visualization of the cubic connectedness locus rendered on a PIC during that course.
I think so, but I don’t remember anything about the cubic connectedness locus. Was that the course John Hart organized?
Mandelava Lamp
0:35
Mandelbrot?
Not a Mandelbrot set, these are Julia sets, but I used their Mandelbrot set to specify how they change over time.
Julia sets are the best kind
It looks like the blue is infecting green
The rest of it is a Siegiel disc
Yes, actually a period-5 cycle of Siegel discs.
Landba
It's actually the Mandelbrot set based on the Herman ring.
Are they part of the madelbot or julia set, my understanding is the julia set contains the mandlebrot set?
This is the Mandelbrot set . I want to animate the Julia sets of these functions but that will take some more work.
@@VAlanNorton Its great. Can't wait
Math? More like meth.
your playing with. the fractal called "lambda"
Right. Maybe I can do something to exploit that similarity.
looks like a cell 0:39
mandelbirds trying to kill the Mandelbrot
Julia set.
Long julia set😂😂😂😂😂
sqworm
I think I will name it swiggy
what if it touches itself?
it dies
it dies
it dies
The universe collapses on itself
we die
Cool critter!
lmao bro
Beautiful,this will be popular very soon
first
Hmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm
Infinity god created universe
Mandelbrot inside mandelbrot inside mandelbrot inside mandelbrot inside the mandelbrot set
We would like a parasite
I WILL QUIT VIDEOS OF REST OF MY LIFEEEEEEEEEEEEE!!!!!!!!!!!!!!!
😢
Promo-SM 😱
Lol
So cool! What's the music? Is it copyright free?
Hey, Wes! I made this song and it’s not copyrighted- you’re free to use it if you’d like, just give me a shoutout if you don’t mind 🤘
.rs..slow
Formula: julia set pbs + c^2 x.y relastic break; + c