1 Introduction Attention: this is an INTERMEDIATE to ADVANCED tutorial, and it requires some previous knowled e o! how Apophysis # Chaoti$a and %aos # wei hts work. I will not o deep into e%plaining the inter!a$e and so on, so you may need to che$k some be inner tutorials first& The oal o! this tutorial is to o through a !ew o! the (hal!)plane( trans!orms ) trans!ormations that map the whole plane to a hal!)plane, such as !lux * and a !ew strate ies o! how to 'uild stru$tured I+, !ra$tals with them. Most o! the trans!orms # plugins mentioned in this tutorial $an 'e !ound in the Aposha$k Plugin -a$k: https:##phoeni%ke"'la$k.deviantart&$om/art#The)Aposha$k-Plugin--a$k-.//01234 Below, you can see a few e%amples o! what you can a$hieve using container trans!orms: 2 One transform Half-planes The most $ommon hal!)plane trans!orm is !lu%& Below, "ou $an see what happens when we appl" !lux on a $he$k board pattern ($he$k board, le!t, and flux, ri ht7: 8'serve how !lux takes the whole plane, and maps it to the ri ht hal! o! the plane, leaving the le!t hal! empt"& There are several other trans!orms that produ$e similar e!!e$t, among them DeltaA and e,$ale 6with s$ale equal to 0&97& Note, that while they are somewhat di!!erent, hal! o! the plane always remains empt"& Another trans!orms that produce a hal! plane are $rop (there are several crop trans!ormations availa'le, but I strongly re$ommend using smart$rop7 and 'ent3 (with bent3% = 0 or bent3% = -;, for e%ample7: Another e%ample o! one)trans!orm hal! plane is -lusRe$ip '" dark-'eam 6$an 'e downloaded !rom his deviantart pa e7& +inally the last one is the polar + linear framework by Fardareismai 6https:##www&deviantart&$om/!ardareismai#art#-olinear)Examples)149144;;97: 3 Basic filling Now that we know how to et a hal! plane, we ot to learn to !ill the other hal!& The 'asi$ idea is prett" simple: the same that fills the ri ht hal! fills also the le!t hal!& 3.1 Flipped Linear The easiest wa" to !ill a hal! plane is '" using a mirrored linear, whi$h means "ou =ust mirror whatever "ou have on one hal! to the other hal!& Follow the steps below: ;& ,tart with a blank flame 3& 8n first trans!orm, repla$e linear with !lu% >& Add another trans!orm, and !lip it hori?ontall" (or rotate it 1/0 de rees7 The result should look like this: I! you had any troubles, you can download parameters here: https:##paste'in.$om/BRJd$A@q Tweakin tips: • Moving or rotating the linear will break the pattern • Bou can get some really cool results by moving, s$aling and rotating the flux trans!orm • Do not for et that flux has a varia'le, whi$h you can edit change 3.2 Flipped Linear with Gap Bou can "up rade( this pattern by creating a central split and filling it with something& Basi$ step by step: ;& Move the post trans!orm for flux 0&; to ri ht& This will create an empty verti$al spa$e in your patterm. 3& Add a new trans!orm, repla$e linear with $"linder (use cylinder = 0&;7 >& To same trans!orm, add preC'lur = 10 2& ,wit$h to the trans!orm ta' and, for the post trans!orm, change Y from 1 to 10 9& +inall", redu$e wei hts to something between 0&0; - 0&09 Below, flux + linear framework a!ter step 1 (le!t pi$ture7 and the parameters for steps 4 and 5 (ri ht7: Bour fra$tal, a!ter step 5, should look like this: Advan$ed tweakin : i! "ou want to make something even more $ompli$ated, "ou $an !ill the ap with any verti$al stripe "ou $an $ome up with& Trans!orms such as ellipti$ and 'ipolar, or ma"'e a verti$ally tiled square& Below, flux + flipped linear with the gap filled with ellipti$: 3.3 Flipped half plane The !lipped linear !ills the empt" hal! with the same pattern we have on the other side, without adding any additional distortion to it& Instead, we $an use another hal! plane trans!orm 6either 3 o! the same, or 3 di!!erent ones7& Dere goes a step by step: ;& ,tart with a blank flame 3& Add a new trans!orm, repla$e linear with !lu% >& Add another trans!orm, repla$e linear with !lu%& 2& Sele$t the post trans!orm, and !lip it hori?ontall" (or rotate it 1/0 de rees7 Does not look much di!!erent from previous, ri htE Tr" moving around, rotating and s$aling one o! the !lux trans!orms to see how this set up ena'les "ou to $reate asymmetri$ patterns, like the one below: Bou can also repla$e flux with other hal! plane trans!orms& For e%ample, below you can see deltaa, es$ale and 'ent2 com'ined with flux: 4 Circle to Halfplane Another way to create a hal! plane is to take a unit dis$ and use curl to map it to hal!plane& Unit dis$s can 'e produ$ed by trans!orms su$h as hemisphere, s$ry or bub'le (and many others7 or frameworks su$h as dis$ and hypertile& A classi$al e%ample o! this approa$h is the Dou'le Bu''le te$hnique des$ri'ed here: https:##www&deviantart&$om/piethein21#art#Double)Bub'le)Tut)9>9/./1;> I! you look closel", you will see that the tutorial features two hal! planes, both made with hemisphere and $url& Below, you can see the whole plane (le!t7 mapped first to a hemisphere (middle7 and then, with curl, to a hal! plane (ri ht7& It is important to o'serve that there is no unique way to map a dis$ to hal!plane using curl& In this tutorial, I will present two di!!erent ways, leaving it to the user to e%plore the other possi'ilities& Below, two di!!erent setups for the s$ry + curl combination: 4.1 Curl Setup with curl_c1 = -0.5 In this e%ample, we will use a very simple setup, just to illustrate how the method works& ;& ,tart with a blank flame 3& 8n first trans!orm, repla$e linear with s$r"& >& Rotate it b" 90 de rees CCW, and s$ale it down b" 200% on$e 2& Add a linked trans!orm to it, and repla$e linear with $url, set curlC$; to -0&9 9& Move this linked trans!orm ; unit ri ht 4& Dupli$ate this se$ond trans!orm 1& +lip the post trans!orm hori?ontall" In this setup, ea$h curl trans!orm corresponds to a hal! plane& Here, I fill both halves with the same thing, but "ou can use something di!!erent, for e%ample flux or a hemisphere with linked curl (it may require some %aos7& This is how this fra$tal looks with only the trans!orm 2 visi'le: And this is the whole thing: I! you had any troubles, you can download parameters here: https:##paste'in.$om/>IMMn'hr 4.2 Curl Setup with curl_c1 = -1.0 This is very similar to the previous e%ample, the only di!!erence is how we set up curl: ;& ,tart with a blank flame 3& 8n first trans!orm, repla$e linear with s$r"& >& Rotate it b" 90 de rees CCW and s$ale up by 200H 2& Add a linked trans!orm to it, and repla$e linear with $url, set curlC$; to -; 9& Move the post trans!orm o! this linked trans!orm 0&9 unit ri ht 4& Dupli$ate this se$ond trans!orm 1& +lip the post trans!orm hori?ontall", then move it 1 unit le!t (so it is positioned at -0&9 Bour final result should look e%a$tly like the previous one: I! you had any troubles, you can download parameters here: https:##paste'in.$om/J0NEh=$D So, why all the hassle i! it looks all the same? The answer is: this is a$tually use!ul when you are using some weird unit dis$ patterns, as it gives you more fle%i'ility in how to distri'ute it over the hal! plane& 4.3 Filling Flux with Hypertile Now, we will use the curl tri$k to fill flux with hypertile& Ge will start with something like this: To make a basi$ hypertile filled with blurry cir$les, you may: • Do it from blank all by yoursel! • +ollow this tutorial: https:##www&deviantart&$om/tatas?#=ournal#Dypertile)5asi$s)9>9.2>;20 • Jra' those params: https:##paste'in.$om/AqD4R0KR And then: ;& ,tart with setting the opa$it" o! all the trans!orms to 0 3& Now, add a new trans!orm, and set its $olor speed to 1 >& Jo to the xaos ta' and swit$h to “toM xaos& This trans!orm should only send points to trans!orms with blur (trans!orm 2 i! you use my params7 2& ,wit$h to “!rom” xaos& This trans!orm re$eive points !rom both hypertile and the cir$le 9& 8n$e you set everything up, play with wei ht o! this trans!orm to get a better look.