Dear reader
 
Sierpinski
 
With the if-else feature of the formula parser it's easy to write a escape time formula for the Sierpinski triangle. I suppose someone have done that before. Nevertheless here is mine:
 
formula#5
Jhsierpinski {;Jos Hendriks,2002
z=Pixel:
IF (real(z)<.5 && imag(z)>.5)
z=2*z-(0,1)
ELSEIF(real(z)>.5)
z=2*z-(1,0)
ELSE
z=2*z
ENDIF
|z|<p1
}

 
 
I know pictures made with this algorithm. For instance there are some in "Fractals everywhere" (Barnsley). These pictures show the layers due to how long it takes for the orbits to escape. The Sierpinski triangle itself isn't really visible. I was curious if a better picture was possible. It turned out that 3 parameters were important: number of iterations, bail-out value and the colormap. Below is a parfile with an example witch gives a quite good impression of the triangle.
 
With the possibilities of fractint there are with the formula another kind of pictures possible. I'm always eager to find "texture" like pictures. Pictures that has a kind of repetition, but doesn't reveal more details when zooming in. Below are two pars with examples.
 
I wrote another formula, based upon those creating the Sierpinski triangle. You could call it a lambdaSierpinski formula:
 
formula#6
Jhsierpinskim {
z=Pixel:
IF (real(z)<.5 && imag(z)>.5)
z=(2*z-(0,1))*Pixel
ELSEIF(real(z)>.5)
z=(2*z-(1,0))*Pixel
ELSE
z=2*z*Pixel
ENDIF
|z|<p1
}
The linearity of the formula gives parts of the plane, edged by straight lines. Some unlinearity( spirals etc) slips in cause of lambda. Below 2 parfiles as examples.
 
For simplicity I put the frm's and pars below together. So it's easier to transport them together. All the pars take just a few seconds to render.
p.s: When I rerendered the images from some of the pars below, the parser gives:can't understand the parameter color=etc. Nevertheless fractint shows the image, but it may use an other colormap then I intended. Because I created the pars in the usual way, I can't understand what went wrong. Anybody knows?
frm:Jhsierpinski {;Jos Hendriks,2002
z=Pixel:
IF (real(z)<.5 && imag(z)>.5)
z=2*z-(0,1)
ELSEIF(real(z)>.5)
z=2*z-(1,0)
ELSE
z=2*z
ENDIF
|z|<p1
}

 
frm:Jhsierpinskim {
z=Pixel:
IF (real(z)<.5 && imag(z)>.5)
z=(2*z-(0,1))*Pixel
ELSEIF(real(z)>.5)
z=(2*z-(1,0))*Pixel
ELSE
z=2*z*Pixel
ENDIF
|z|<p1
}
jhsiertriangle {;Jos Hendriks,2002
reset=2000 type=formula formulafile=sier.frm
formulaname=Jhsierpinski passes=1
center-mag=0.475632/0.48248/1.850399 params=10000/0 float=y
maxiter=100 colors=000<8>000000000<3>000zzz<88>zzz<4>zzzccc<143>ccc
}
lambdasier1 { ; Jos Hendriks, 2002
reset=2000 type=formula formulafile=alle.frm
formulaname=Jhsierpinskim passes=1
center-mag=+0.54606658322903670/+0.12909227378964890/946.2815
params=10/0 float=y
colors=000z02<15>j02<2>g02f02f02f02<55>O02O02N02<3>M020z2<15>0j2<2>0g20f\
20f20f2<55>0O20O20N2<3>0M200t<15>00d<2>00a00`00`00`<61>00G
}
lambdasier2 { ; Jos Hendriks, 2002
reset=2000 type=formula formulafile=alle.frm
formulaname=Jhsierpinskim passes=1
center-mag=+0.54887713642052600/+0.10850051419031680/445.3089
params=10/0 float=y
colors=00000e0e00eee00e0eeL0eeeLLLLLzLzLLzzzLLzLzzzLzzz000555<3>HHHKKKOO\
O<3>ccchhhmmmssszzz00z<3>z0z<3>z00<3>zz0<3>0z0<3>0zz<2>0GzVVz<3>zVz<3>zV\
V<3>zzV<3>VzV<3>Vzz<2>Vbzhhz<3>zhz<3>zhh<3>zzh<3>hzh<3>hzz<2>hlz00S<3>S0\
S<3>S00<3>SS0<3>0S0<3>0SS<2>07SEES<3>SES<3>SEE<3>SSE<3>ESE<3>ESS<2>EHSKK\
S<2>QKSSKSSKQSKOSKMSKK<2>SQKSSKQSKOSKMSKKSK<2>KSQKSSKQSKOSKMS00G<3>G0G<3\
>G00<3>GG0<3>0G0<3>0GG<2>04G88G<2>E8GG8GG8EG8CG8AG88<2>GE8GG8EG8CG8AG88G\
8<2>8GE8GG8EG8CG8AGBBG<2>FBGGBGGBFGBDGBCGBB<2>GFBGGBFGBDGBCGBBGB<2>BGFBG\
GBFGBDGBCG000<6>000
}
sier1 { ; Jos Hendriks, 2002
reset=2000 type=formula formulafile=alle.frm
formulaname=Jhsierpinski passes=1
center-mag=0.488878/0.841371/0.7771474 params=100/0 float=y
outside=imag
colors=000zzz<5>ttz<3>ppzpoypnxpmwplv<3>mhrlgqkfo<3>gbgfaed_caX_<2>PKPKG\
HGBNC9TD6ZC6`D6bD6dF5fF5hG5jH3lH3nH3pJ3rJ3tL1vJ3xH6zG9wFCvDFsCHrAKo9Nm7P\
l6Si5Vh3Ye2`d1ba0e`0fZ6ZSDRLKKGSC9Z53e000zOLR7l00k00i00h00f20e50d70dC0<3\
>ZN0YP0YS0XX1<3>Rf3Pi3Pm5<2>Lv6Ky7Jz7Kz5Jz7Hz9HyCGvDGsGFpHDmJDkLChNCePAb\
R9`S9YV7XX7UZ6R`5Oa5Ld3Je3Gh<2>17m05o02r00s00z00t20p52l93hD6dG7`KAXOCSRF\
OVGKzm0mY6YHGL0NJ0OH2PG6RFARDFSCJU9NV7RV<2>3bZ9VL5XJ2eZ0zm0to1mo6foAaoFV\
oKOpOJpUCpY5pZ1sa0pd0oe0mh0lk0il0ho0fr0es0bv0ay0`z0fz0Z<2>m0Di06e00b00d0\
0e10f32h75iC9kFClJFmLHoPLpUOrXRs`UtdYvf`wkbymezrizvlzyozzrzztzzrwwpttopp\
lmmkkiifefdbeaZdzlK<2>KOlezp6Gv9JsCKrDNpGOmJPlKSkNUiPVfRYeUZdX`bYb``dZbe\
YdhXfiU<2>moPppNrsLttKzzKHzKGzLAzH<5>Zz`bzdfzfkziozm
}
sier2 { ; Jos Hendriks, 2002
reset=2000 type=formula formulafile=alle.frm
formulaname=Jhsierpinski passes=1
center-mag=0.529848/1.20584/3.72083 params=100/0 float=y
outside=atan
colors=000zzz<5>ttz<3>ppzpoypnxpmwplv<3>mhrlgqkfo<3>gbgfaed_caX_<2>PKPKG\
HGBNC9TD6ZC6`D6bD6dF5fF5hG5jH3lH3nH3pJ3rJ3tL1vJ3xH6zG9wFCvDFsCHrAKo9Nm7P\
l6Si5Vh3Ye2`d1ba0e`0fZ6ZSDRLKKGSC9Z53e000zOLR7l00k00i00h00f20e50d70dC0<3\
>ZN0YP0YS0XX1<3>Rf3Pi3Pm5<2>Lv6Ky7Jz7Kz5Jz7Hz9HyCGvDGsGFpHDmJDkLChNCePAb\
R9`S9YV7XX7UZ6R`5Oa5Ld3Je3Gh<2>17m05o02r00s00z00t20p52l93hD6dG7`KAXOCSRF\
OVGKzm0mY6YHGL0NJ0OH2PG6RFARDFSCJU9NV7RV<2>3bZ9VL5XJ2eZ0zm0to1mo6foAaoFV\
oKOpOJpUCpY5pZ1sa0pd0oe0mh0lk0il0ho0fr0es0bv0ay0`z0fz0Z<2>m0Di06e00b00d0\
0e10f32h75iC9kFClJFmLHoPLpUOrXRs`UtdYvf`wkbymezrizvlzyozzrzztzzrwwpttopp\
lmmkkiifefdbeaZdzlK<2>KOlezp6Gv9JsCKrDNpGOmJPlKSkNUiPVfRYeUZdX`bYb``dZbe\
YdhXfiU<2>moPppNrsLttKzzKHzKGzLAzH<5>Zz`bzdfzfkziozm
}
sier3 { ; Jos Hendriks, 2002
reset=2000 type=formula formulafile=alle.frm
formulaname=Jhsierpinski passes=1
center-mag=0.527529/-0.153234/7.771474 params=100/0 float=y
outside=atan
colors=000<3>00n00z0C0<3>0Cn0Cz0P0<3>0Pn0Pz0a0<3>0an0az0n0<3>0nn0nz0z0<3\
>0zn0zzC00<3>C0nC0zCC0<3>CCnCCzCP0<3>CPnCPzCa0<3>CanCazCn0<3>CnnCnzCz0<3\
>CznCzzP00<3>P0nP0zPC0<3>PCnPCzPP0<3>PPnPPzPa0<3>PanPazPn0<3>PnnPnzPz0<3\
>PznPzza00<3>a0na0zaC0<3>aCnaCzaP0<3>aPnaPzaa0<3>aanaazan0<3>annanzaz0<3\
>aznazzn00<3>n0nn0znC0<3>nCnnCznP0<3>nPnnPzna0<3>nannaznn0<3>nnnnnznz0<3\
>nznnzzz00<3>z0nz0zzC0<3>zCnzCzzP0<3>zPnzPzza0<3>zanzazzn0<3>znnznzzz0<3\
>zznzzz000<12>fffiiimmm<3>zzz000<18>000
}