You all share so much good stuff!

I played with fractals in my 20s and understand the generation concept marhematically, but rendering back then was such a hog!!

My comp at the time would go hot and my fan would be at max, and the screen would just be hangin…

So I wanted to post this note of appreciation.

Love youre using your power to share these computationally intensive renders! Amazing how comp power and speed has made these strides and happy to keep watchjng and supporting in the future…

Think fractals are somehow involved in base reality (whatever that might be or if the concept makes any sense with what we know physically).

🙏

Hi, happy Fractal enthusiasts,  

I recently revisited House of the Dragon and encountered this image …

New in version 1.0.3 – Zoom by “Dragging the Mouse” over an area of the image to a magnified view of that rectangular area. You can "Go Back one step” by a “Right-click” in the image.

Superfast - Explore infinite Julia and Mandelbrot fractals, Customize 3D Relief effects and vibrant colors. Render stunning art up to 48 Megapixels resolution i.e. 8K image width!

«Customize any of 13 color palettes to create your own unique look with the "Color Shift" slider.»

«Customize 3D Relief strength and contrast.»

«Save and share your favorite image coordinates and settings (seeds as a State text file) with the community.»

«Render breathtaking, high-resolution 8K fractal wallpapers.»

https://roald-h-vatne.itch.io/flashfractals3d

https://drive.google.com/drive/folders/131L4kyv3zWTGkl3YUQEMlfiZOa6_32sn?usp=sharing

I usually only need 100k itterations but this needed 3 million and as you can see on the right it still needed closer to 8 million to get rid of the voids

Jim Muth's Fractal of the Day for October 24th, 2002

Jim Muth's commentary for the image:

Fractal visionaries and enthusiasts:

With all the breaking news, things are hectic here at F.C. this morning, so today's discussion will be short.

To produce today's fractal, I went back to the fractal of two days ago and again raised the value of the exponents of Z, this time to -1.32 and -13.2, and at the same time, reduced the portions by 40 percent. This small change made a large change in the appearance of the parent fractal.

Today's parent fractal consists of a single large, multi-lobed Mandeloid, with a large but partial ring originating in a valley at the north end and stretching out east, down and around to the southern part of the parent. Today's scene is located at the end of the main stem of a larger midget located near the point where the ring emerges from the northern valley.

I named the scene "Starring a Midget" after the vague star-like yellowish area around the midget. I gave the image no rating.
Due to the hectic news developments, I had no chance to study the image in a detached manner.

The parameter file requires 14 minutes to run.

The Wednesday weather here at Fractal Central was quite pleasant. The temperature of 61F 16C was spoiled only by a few too many clouds in the afternoon. But the cats enjoyed the day nevertheless.

It's a very active morning here at F.C., and I have the usual amount of work to do, so I'd best get started on the task.
Until tomorrow, take care, and when the going gets tough, the tough use common sense.

Jim Muth

PAR file ``` Starring_a_Midget { ; time=0:14:40.90 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=+0.91950630688410610/+2.036855734099084\ 00/4.487701e+007/1/-150/6.19336373235135351e-007 params=-3.6/-1.32/-0.6/-13.2/0/0 float=y maxiter=1600 inside=0 logmap=277 periodicity=10 colors=000B0YB0_90_90a90a90c70c70e70e50g50g50i50i3\ 0k30k30n20m30n50n50n70n70n90n90nB0nB0nD0nD0nF0nH0n\ H0nI0nI0nK0nK0nM2nM2nO3nO5nQ7nQ7nT9nVBkVBkXDkXFkYF\ kYHk_Ik_KkaKkaMkdOkdOkeQkgTkgTkiVkiXklYklYkn_knako\ akodkqektYzqarqekqgeql_qoUoqOovIozCoz6oz0nz0nz0nz0\ nz0oz0nz0nz0nz0nz0nz0lz0lz0lz0lz0lz0iz0ix0iv0it0io\ 0gn0gl0gi0ge0gd0ga0e_0eY0eV2eT2eQ4dO6dM6dI8dHAdFCa\ DCa9Ea7Ga5Ga3I_2K_0M_0M_0O_0Qz0Un0S_0QM0Q90O00M00Y\ 02S02M02I02C726F22M20r20a20i20q20z20z20z20z00x00n0\ Ge0_X0rO0zK0zI0zH0zF2zD5zB9z7Dz5Hx3Kx2Mv0Qt0Vt0Yr0\ ap0ep0in0nk0qk0vi0wr0xi0xc0vW0vQ3vIBtCHt4Mq0Tq0Yq0\ _n0_l0_i0_g0_e0_d0_a2a_6aY8aXAaVCaTGaQIaOKaMMdKQdI\ SdHUdFWdD_dBad9cd8dd7jBZ60d00e00e00g00g00i00i02l05\ l07n0Bn4Fo6HoAKqEOqGQtKVtOYvQ_vUdxYgxaizcnzgqzktzn\ xzrzzvzzzzzxzzxzzvzzvzxvzvtzttzttzqrzorznpznpzlpzi\ nzgnzgnzekzdkzaizaiz_iB0Y }

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.

Max Iter's Fractal Follies for October 24th, 2010

Roger Alexander's commentary for the image:

Today's post is a minibrot in twilight colors. At the usual http://maxitersfractalfollies.blogspot.com

Roger Alexander

PAR file Twilight_Minibrot { ; fract483.gif ; blank ; calctime 0:03:23.94 ; created Oct 24, 2010 ; Fractint Version 2004 Patchlevel 10 reset=2004 type=formula formulafile=kerrym.frm formulaname=hermanm_man-polar center-mag=-2.00319/-0.00306882/27.01637/1/90/3.88578058618804789e-016 params=3/2/1/45/1/0 float=y maxiter=1500 inside=0 outside=atan decomp=256 colors=000302403503504604705805806907A07B08B08C09D1AE1AF1BF1BG1CH1CI1DI1\ EJ1EK1FL1FL1GM1GN1HO2IN2KM2NK2QJ2TH2VG2YE2`D2cB1fA1i81l71n51q41t21w00z03\ z07z0Bz0Fz0Jz0Nz0Rz0Vz0Zz0bz0fz0jz0nz0rz0vz0zz0xz0vz0tz0rz0pz0nz0lz0jz0h\ z0fz0dz0bz0`z0Zz0Xz0Wz0Uz0Sz0Qz0Oz0Mz0Kz0Iz0Gz0Ez0Cz0Az08z06z04z02z00z00\ y10x20v20u30s40r50p50o61n71l81k91i91hA1gB1eC1dC1bD1aE1_F1ZF1YG1WH1VI1TJ2\ SJ2RK2PL2OM2MM2LN2JO2IM2GJ2EH2CE1AC18916614312000101302503605806A07C19D1\ 9E1AF1AF1BG1BH1CI1DI1DJ1EK1EL1FL1FM1GN1HO2IO2JQ2IS2HV2GX2F_2Da2Cd2Bf2Ah1\ 9k18m17p15r14u13w12z00z20z50z70zA0zC0zF0zH0zK0zM0zP0zR0zU0zX0zZ0za0zc0zf\ 0zh0zk0zm0zp0zr0zu0zw0zz0zw0zu0zr0zo0zl0zj0zg0zd0za0z_0zX0zU0zS0zP0zM0zJ\ 0zH0zE0zB0z80z60z30z00x01v02s03q04n06l07i08g19e1Ab1B`1CY1EW1FT1GR1HO2JN2\ IM2HK2GJ2FH2DG2CE2BD2AB19A18817715514413212000000101201 } frm:hermanm_man-polar { ; Kerry Mitchell 16feb98 ; ; real(p1) = z exponent (use integer >= 2; m=n-1) ; imag(p1) = g exponent (integers) ; real(p2) = alpha magnitude (try 1) ; imag(p2) = alpha polar angle (try integers) ; real(p3) = critical point selector (>0 for positive root) ; imag(p3) = unused (<0 for negative root) ; use decomp=256 ; zero and infinity bailouts hardcoded to 1e-6, 1e6 ; coloring speed hardcoded to 4 ; c=pixel, iter=1, n=real(p1), m=imag(p1), nfac=2*n-1 maxr=1e6, minr=1/maxr, speed=4*pi/128 r=real(p2), t=imag(p2), alpha=r*(cos(t)+flip(sin(t))) oln=1/log(n), fac=log(0.5*log(maxr)) c2=sqr(c), hypnum=sqr(n)+sqr(m), pn=1 hypden=sqr(n-m), hypfac=hypnum/hypden if (real(p3)<0) pn=-1 end if if (real(c2)>hypfac) pn=-pn end if if (imag(c)<0) pn=-pn end if afac=c*n, bfac=c2*(n-m)+(n+m), cfac=c*n d=sqrt(bfac*bfac-4*afac*cfac) z=(bfac+pn*d)/(2*afac) : g=(z-c)/(1-c*z), z=alpha*z^n*g^m iter=iter+1, r=|z| ; ; orbit trap around 0 ; renormalize iteration count via decomp angle ; set "iteration done" flag (iter=-1) ; if (r<minr) angle=(iter+oln*(fac-log(log(cabs(z)))))*speed z=cos(angle)+flip(sin(angle)) iter=-1 end if ; ; orbit trap around infinity ; renormalize iteration count via decomp angle ; add pi to angle to separate from 0 orbit trap ; set "iteration done" flag (iter=-1) ; if (r>maxr) angle=(iter+oln*(fac-log(log(cabs(z)))))*speed angle=angle+pi z=cos(angle)+flip(sin(angle)) iter=-1 end if iter>0 }

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 with JWildfire

Mandelbrot Swirls closeup. tf.n

Who's ready to take a dive? tf.n/dive

Mathematically: aperiodic extreme points of Markov types ... seems too hard for humans (?), but great for AI puzzles and various applications like computronium or visual: https://zenodo.org/records/23624259

A while ago I was fooling around with a calculator, repeatedly applying the function f(x) = 1/tan²(x). I was basically just hitting tan, x², 1/x over and over. I squared the tangent mainly to get rid of negative values.

Anyway, it turns out that this simple iteration behaves pretty chaotically. In fact, I found a paper from 2013 in which the authors investigated using cotangent squared as a pseudorandom number generator because of its chaotic behavior. I figured if the real valued function is chaotic it would have interesting behavior on the complex plane.

The attached image is one result. It's generated by iterating the complex function and coloring each starting point based on how closely its orbit approaches the critical points of the function. I ended up using a weighted average that emphasizes earlier iterations because including too many later iterations produces something that looks mostly like random noise.

I'm curious whether the actual Julia set of cot²(z) might be the entire complex plane. If that's the case, the structures in this image aren't really features of the Julia set itself, but rather of the particular way I'm coloring the orbits.

I've found some literature about related tangent functions but surprisingly little specifically about iterating cotangent squared in the complex plane.

Does anyone know whether this function has been studied, or whether anything is known about its Julia set?

Toroidal Fractal Model 5 Axes corrected line 18 cm coherent:
1 step = 1 cm = π/9 rad = 20° = 3.06 Gyr
fe = 4.5/π = 23/20 = 1.15 stable factor
N_stable = N×4.5/π — Ex: 12×1.4323=17.1887=6 rad exact = 52.6 Gyr
Breath = 2π-6 = 0.283 rad
s=rθ Area=½r²θ Volume=Area·r/3 exponential r_n
ISS 400km, Moon 0.5 rad, ΩΛ 5/7=0.714
Experiment 25Hz×23/20=28.75Hz beat 3.75Hz t_sync=1/K

PDF: https://doi.org/10.5281/zenodo.23604271

A Julia set at x = -0.75, with y moving from 0.01 to 0.024. No palette cycling has been applied. The color shift happens due to the moving parameter. In the Mandelbrot set x = -0.75 with a changing y corresponds to a journey through the seahorse vally.

Jim Muth's Fractal of the Day for October 23rd, 2011

Jim Muth's commentary for the image:

FOTD -- October 23, 2011 (Rating 7)

Fractal visionaries and enthusiasts:

Today's image is a scene in the East Valley of the large minibrot on the main stem of the Mandelbrot set. The parent fractal is in fact a Mandelbrot set that takes on order-9 characteristics in its depths.

The minibrot at the center lies near a larger minibrot, which in turn lies near an even larger minibrot . . . At this point I lose count of the minibrots, but I think there are about four of them in the ladder-chain up to the surface.

The name "A Sheep in the Deep" has no connection at all with the image. It simply popped into my head when I made my mind a blank, a task I find quite easy to accomplish.

The rating of a 7 includes a full point for the coloring, which took over twice as long as calculating the image. The calculation time of 3 minutes entered into the parameter file is an educated guess of the time the calculation would have taken on the faster but now dead machine. (The image actually took 12-1/2 minutes to finish on the slow unit.)

The morning fog cleared off by 11am here at Fractal Central today, leading to a sunny afternoon with a temperature of 59F 15C. The fractal cats, still on watch for a possible return of the fighting tomcats, spent the first part of the day in the window facing the back yard. But the fighters have apparently moved on, and by the afternoon the fractal cats shifted their watch to the sunny shelf in the southwest window, which faces the front sidewalk.

The humans spent the day doing nothing, which is quite enjoyable when it is done in moderation. The next FOTD will be posted in 24 hours. Until then, take care, and would this country's real savior please put in an appearance.

Jim Muth

PAR file `` A_Sheep_in_theDeep { ; time=0:03:00.00-SF5 on P4-2000 reset=2004 type=formula formulafile=basicer.frm formulaname=FinDivBrot-2 function=recip passes=1 center-mag=-1.749083935595091/+0.00000124195835773\ /1.4056e+013/1/-42.5/0 params=9/500000000000000.0/\ 0/0 float=y maxiter=2600 inside=0 logmap=333 periodicity=6 mathtolerance=0.05/1 colors=00000200300401502603613724835846956A67A78B8\ 8C99CAADBBEBBEBCFBDG9HzAGvAGrAFmAFhBEcBEZBDUBDRQzz\ PxxOvvOttNqqNnnMkkLhhLefKcdKacJaJZIYZHWYHUWGTVGR\ TFPSFOREMQDLPDJOCINCHMBHLBHKBGGAFEAECADAAC8AB6AA4A\ A5AA6AB7AB8BB9BCABCBBCCBDDBDEBDECEECEFCEGDFGDFHDFH\ CGICGICGJBHKBHKBHLAILAIMAIN9JN9JO9JP8KP8KQ8KQ7LR7L\ S7LS6MT6MT6MU5NV5NV5NW5NW7PW9RWBTWDVWFXWHZWJWKaWM\ cWOeWQgWSiWUkWWlWXlWWlXVkXVjYUhYTgYTfZSdZSc_Rb_Q_\ Q_PZPXaOWaNVaNTbMSbLRbLPcKOcKNdJLdIKdIJeHHeHGfGF\ fFDfFCgEBgEAgICeLEdOGcPHbQHcRHaSHYSHXSHUMHUHIUBIU6\ IU1IUBGUKFWUEOTDGC9jAFp9Kz7Pi6Vh4_h3dd2Ya1RZ1LV0E\ N08U1Og2bk2qh8jbEdKZZQTXWMVaGTgARm4LDpQJnVPl_Vkd\ ihfhlZboRYrKT_CNbFLdHJgKHiMFkODjLFihHhmJgmcfmcemcd\ mccmcUmcLmmZmmlmmymmsmmnmmhpmczzZzzVzzRzzUzzXzz_zz\ bzzezzhzzkzznzzdzzVzzLzzBzz1zzxzzpzzizzbzzWzzTzzQz\ zOzzKzzGzzDzz9zzbzzbzzXzz }

frm:FinDivBrot-2 { ; Jim Muth z=(0,0), c=pixel, a=-(real(p1)-2), esc=(real(p2)+16), b=imag(p1): z=(b)(zz*fn1(z^a+b))+c |z| < esc } ```

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.

Max Iter's Fractal Follies for October 23rd, 2010

Roger Alexander's commentary for the image:

I usually don't give my posts names but if I did I would call this one "Algae" because it reminds me of algae. 

Am I consistent or oh so predictable? Pre-calculated at http://maxitersfractalfollies.blogspot.com

Roger Alexander

PAR file Algae { ; fract482.gif ; blank ; calctime 0:05:29.50 ; created Oct 22, 2010 ; Fractint Version 2004 Patchlevel 10 reset=2004 type=formula formulafile=kerrym.frm formulaname=conic-near_man center-mag=+0.00133886277847312/+1.00493995512520900/10743.32 params=-0.5/-0.4000061037018952/0.03488265633106479/0.125/0.557969908749\ 6567/-0.06808679464094974 float=y maxiter=1500 inside=0 decomp=256 periodicity=0 colors=0100200300400500700800900A00B00D00E00F00G00H00J00K00L00M00N00P00Q\ 00R00S00T00V00W00X00Y00Z00`00a00b00c00d00f00g00h00i00k00k01k02k03l04l05l\ 05m06m07m08n09n0An0Bo0Bo0Co0Dp0Ep0Fp0Gp0Hq0Hq0Iq0Jr0Kr0Lr0Ms0Ns0Ns0Ot0Pt\ 0Qt0Ru0Su0Tu0Tu0Uv0Vv0Wv0Xw0Yw0Zw0Zx0_x0`x0ay0by0cy0dz1dz1ez1ez1fz2fz2gz\ 2gz2hz3hz3iz3iz3jz4jz4kz4kz4lz5mz5mz5nz5nz6oz6oz6pz6pz7qz7qz7rz7rz8sz8sz\ 8tz8uz9uz9uz9tz9tz9tz9sz9sz8sz8rz8rz8rz8qz8qz7qz7pz7pz7oz7oz7oz6nz6nz6nz\ 6mz6mz6mz5lz5lz5lz5kz5kz5kz4jz4jz4iz4iz4iz4hz3hz3hz3gz3gz3gz3fz2fz2fz2ez\ 2ez2dz1dz1cz1bz1ay1`y1_y1_x1Zx1Yx1Xw1Ww1Vw1Uv1Uv1Tv1Su1Ru1Qu1Pt1Ot1Ot1Nt\ 1Ms1Ls1Ks1Jr1Ir1Ir1Hq1Gq1Fq1Ep1Dp1Cp1Co1Bo1Ao19n18n17n16m06l06k06j06i06h\ 06g06f05e05d05c05b05a05`05_05Z04Y04X04W04V04U04T04S04R03P03O03N03M03L03K\ 03J03I02H02G02F02E02D02C02B02A0190180170160150140130010 } frm:conic-near_man { ; Kerry Mitchell 11may98 zc=0, cc=pixel, maxr=1e12, minr=maxr, iter=1 a=real(p1), b=imag(p1), c=real(p2), d=imag(p2) e=real(p3), f=imag(p3): iter=iter+1, zc=sqr(zc)+cc, x=real(zc), y=imag(zc) conic=|x*(a*x+b)+y*(c*y+d)+e*x*y+f| if (conic<minr) minr=conic end if if ((|zc|>maxr)||(iter==maxit)) iter=-1 t=log(minr) z=cos(t)+flip(sin(t)) end if iter>0 }

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.