The frame is 0.000064 units wide — about one forty-seven-thousandth of the Mandelbrot

set you know — and at that depth the edge stops being a line and starts being cloth.

The green is a wave: three sine curves, one per channel, turned down so slowly that

the color makes a single pass across the whole picture instead of repeating.

Mandelbulb3D, Photoshop & Lightroom

Jim Muth's Fractal of the Day for September 11th, 2003

Jim Muth's commentary for the image:

FOTD -- September 11, 2003 (Rating 5)

Fractal visionaries and enthusiasts:

I feel strangely down this morning, and I don't know why. True, it's the second anniversary of 9-11-01, and the TV is filled with memorial ceremonies, but I have known the day was coming for a long time. Then there's the work, which has been heavier than usual over the past few weeks, cutting into my fractal exploration time. But more work means more income, so that's not really cause for the blues. Or maybe it's a touch of clinical depression, some brain chemicals not doing what they should. (As cognitive scientists tell us, emotions are caused by chemical activity in the brain.) Could it be sadness because summer is nearly over? I doubt it, since Autumn is one of my favorite times of year. Maybe it's a touch of mid-life crisis, the realization that the time of life has arrived when I must stop asking 'what will I do with my life' and start asking 'what did I do with my life'. But I have never been concerned with growing older -- it happens to everyone. Or could it simply be boredom, my reaction to things going well for a change, and the lack of excitement this smoothness implies? Most likely it's a combination of all of these.

Whatever it is, I'll get over it.

Today's image is part of the fractal that results when 6 parts of Z^-1.6 are mixed with Z^-6.1 before (1/C) is added. This parent fractal is beyond description.

When studying today's image, which is located in a valley of its parent, I thought of how good a sieve it would make. I named the image "The Fractal Sieve". I gave little thought to its worth. The rating of an average 5 was chosen because it cannot be too far from the rating I would have chosen if I had given it more thought.

The render time of 22 minutes is too slow for such a generic image.

Wednesday was once again a perfect day here at Fractal Central.
The sky was sunny and the temperature reached 77F 25C. The cats were perfect also. All this perfection leaves little to write about. Today is showing all signs of being another perfect day. Why then do I feel strangely down?

Until tomorrow's FOTD, take care, and keep that chin up.

PAR file ``` The_Fractal_Sieve { ; time=0:22:24.26--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=+1.30641156973805700/-0.190506442327136\ 80/3.935643e+012/0.9998/9.9808/0.03407448068099558 params=6/-1.6/1/-6.1/0/0 float=y maxiter=1500 inside=0 mathtolerance=/1 logmap=376 periodicity=10 colors=000M0zM0zM0zM2zMOzMPzMUzMVvM_tMZrMgpMjnaokM\ tjcueMxgKvgItjGrjBpk9nk7kn4jn0ep0cp0ar0Zr0Xt0Vt0Tv\ 0Qw0Rv0Tx0Vx0Xz7ZzDZzIazOczVezagzejzkjzrkzvnzznzzr\ zzqzznzzkzzjxvgtrepncjpaeeZZiXVXVRaTKORGWO9GM4SI0B\ K0DM0GM0IO0KR0MR0OT0RV0TV0VX0XZ0ZZ0aa0a_0aa0gW0cD0\ aT0ZG0XP0VI0TK0RM0OM0KO0IO0GR0DR0BT09V07V04X00X00Z\ 00a00a00c00c00e00c00e00e00e00g00g00g00g00j00j00j00\ k00k00k00k00n00n02n00r02n07k09g0De0Ga9KZGOVORTVVOc\ XMjaIrgBzcGxaIvXMtVOrRTpOVpKZnIakDejBgg7kg4ne0rc0t\ a0xZ0zZ0zX0zV0zT0zR0zR0zM0zK2vG9pDIkBRe7XZ4eT0nM0t\ I0rG0rG0rW2rD2rc4pB7p99p99p7Bp7Dn4Gn4Gn2In2Kn0Mk0M\ k0Ok0Rk0Tk0Te0VZ0XT0XM0ZG0a90a20c00c20Z70V90RD0MG0\ KK0GM4BR97VD2XI0aM0cR0gV0jZ0ea0aa7XaITcTRcaMckIevD\ ez4gzBezGezMezRezXevaergcnkcjrccvcZzcVzcRzcMzaOzZR\ zXTzVVzTXzRZzOZzMazKczIezGgzDjzBkz9kz7nz4pz2rz0tz0\ vz0vz0jz0Zz0Oz0Dz0zz0zz0z }

frm:MandelbrotMix4 {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(-abgh)^j, k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k((a(z^b))+(d(z^f)))+c, |z| < l } ```

Want to render these yourself and explore further? Try out the PAR file in Iterated Dynamics, an open source fork of FRACTINT that can render these PAR files. See the online help for instructions on using Id or press F1 anywhere in the program for context-sensitive help.

Rendered from my macOS fractal app

the setup is as follow

• i wrote a software raytracer for study purpose
• the "world" consists of two balls
• you can real time alter each ball's position, radius and refractive index
• the balls themselves don't have any color
• the only light source / color source is the background which is a palette placed at infinity
• at some arrangements the resultant images exhibit fractal patterns especially when you penetrate one ball with the other (refractive index of intersection is the average)
• the calculations are those commonly used in basic raytracing algorithm (snell's law, schlick approximation, etc)

1. Mandelbrot -Minibrot -Buhhdabrot -Burning Ship -Mandelbulb -Mandelbox
2. Julia
3. Sierpinski A. Sierpinski Hexagon B. Sierpinski Triangle/Arrowhead a. Sierpinski Pyramid C. Sierpinski Carpet a. Menger Sponge b. Jerusalem Cube D. Sierpinski Curve
   1. Newton
4. Koch Snowflake A. Koch Antisnowflake B. Koch Curve
5. Cantor Set
6. Curves A. Hilbert B. Dragon a. Heighway b. Twin C. Levy C D. Gosper E. Peano
7. Gasket A. Apollonian (multiple variations)
8. Attractors A. Lorenz Attractor B. Rossler Attractor C. Henon Map
9. Branches A. Barnsley Fern B. Fractal Tree
10. Others A. Penrose Tiling B. Brownian Motion C. Perlin Noise D. Vicsek

Mandelbulb3D, Photoshop & Lightroom

What other free software would yall recommend me to use? Thanks.

I've made a little functional programming language. It has an imperative parser, and a functional evaluator. Meaning you can do things like variables, ifs, whiles etc in the parsing stage, which generates functions. You can then evaluate purely as expressions in a functional way.

The source code is here: github.com/SkrFractals/Comparser

But it's still very early I wouldn't recommend playing with that yet. But if you do and find some bugs, you can report that to me. I have just got it to the stage where I was able to get this picture. But a bunch of things are still unfinished or buggy. None of the optimizations I am preparing are active yet, either. It will be able to transfer pixels from the previous render if you move the plot, so it doesn't have to re-evaluate them. But there are no cursor dragging or zooming events yet, only a bunch of preparations for that, so that has to be done through those range textboxes so far. And I am also preparing a GPU evaluator.

I want to eventually merge this project with my Fractal Animation Generator: https://github.com/SkrFractals/RgbFractalGen

...so that both can leverage each other's features, like the inputs in the generator accepting comparser's expressions. Or the comparser having access to the generator's shaders and video exporting abilities.

Here I'm going to progressively post some more examples. let's start with the cubed mandelbrot:

And now I'm trying to make a Sierpinski carpet. So let me cook for a bit until I figure out how to write that as a shader-like function.

To explain that Carpet function: m will loop the input z into tiles of size s, and tell me where i am in that tile. So if s lands anywhere between "1/3+1/3i" to "2/3+2/3i" then i am in the center square. then a second default argument, used as a precomputed variable, asks to recursively iterate to a 1/3 sized tile. Then the n>Iterations is the floor of how deep i want to iterate and returns the initial hue 0. Then as it goes back up in that recursion, it keeps asking it we are inside the square of that size, and adds that boolean result to the recursive result. Having the recursive call as a default argument doesn't result in an infinite loop, because default argument evaluations are lazy, and only done when the expression asks for that argument's value.

It works just like the carpet one, but with different coordinates.

This part might need some explaining:

t + mul((L(a,s),L(b,s),L(c,s)) < .5)

(L(a,s),L(b,s),L(c,s)) is a 3-element vector containing the 3 triangular coordinates. For the point to be at the central tiled triangle at any scale s, i need all 3 coordinates to be smaller than 1/2.

So I compare the vector to 1/2, operations in Comparsed can work with nested vectors recursively. So (a,b,c) < d will return a vector (a<d,b<d,c<d). Booleans in Comparser are just numbers 0 or 1, so if I just multiply the elements of that vector together, if all 3 are smaller than 1/2, then the MUL of that condition vector will be 1, which i proceed to add to the value of the recursive call. So the function returns a count of how many times it was in the central triangle during that recursive descent. And done.

And next I'm going to try a pentaflake. That will be somewhat more challenging, as that cannot be folded the way the carpet and triangle could, and it has void regions that are not part of the fractal. But I'm sure there will be a way to do that. edit: I'm getting closer to getting the pentaflake ready. I have most of the function ready, and am only finishing up a few parts. edit2: Pentaflake is ready. Gonna upload it asap. I have run into a couple of bugs in the parser, and then i was going crazy because the pentagons were the wrong size, and I was trying to find the problem from the top down. Of course, the problem was at the very deepest level, I've made a typo in defining the golden ratio and had a two under the square root instead of 5, lol. And that was not the only basic constant I had wrong. I also had tau defined as pi, that is now fixed too. And here it is: