Take a point called z on the complex plane...

Relevant theme song for today's blog post.

So I've finished a Sunsarity one-shot that I hope to release soon once I clean it up and get cover art for it, but in the meantime, I'd like to share some art I've made in the past few years, because I think it's pretty cool. Here's one example, and I'll be sharing and talking more about them below the break. You can find more on my DeviantArt account at https://www.deviantart.com/haycartesiangeometry.

Fractal Flames are a type of Iterated Function Systems (IFS) which can be used to create fractals using nonlinear iterated functions in addition to affine ones, using log-density display and color-by-structure, as outlined in this paper from 2003. None of this really matters other than to say that it uses some special techniques to create some truly breathtaking art using a special class of mathematical functions. Essentially, fractal flames are just graphs of a type of mathematical object, generated using some process which I only have a vague understanding of.

Luckily, one doesn't need a very deep understanding of the algorithms involved to create art. A number of programs abstract away the math and provide a simpler interface for designing these works of art and rendering them. The one I use is Apophysis which also helpfully has options to randomly generate fractals. If you do some googling you can find simple tutorials that can help you create your first fractal, and also explain what some of the buttons and sliders do. It takes quite a bit of work to develop an intuitive understanding of how tweaking one parameter will affect the resulting art, but it's well worth the effort in my opinion, and the way Apophysis provides quick previews is really helpful for developing that sense.

Anyways, one major aspect of fractal flames is that instead of a black-and-white coloring scheme, they take in a user-defined color gradient to determine the colors of the resulting fractal. You can find a link to some MLP gradients to get you started here, though admittedly I'm not too happy with how some of the color palletes came out on those, so I wrote my own script to generate my own. If you're interested in the particular gradient I used for one of my pieces, you can go to the DA page for it and save the parameter file in the description as a .flame file, then open it with Apophysis to extract the gradient out (you should be able to find tutorials on gradients online, and if you get stuck feel free to PM me).

So without further ado, here are some highlight from my own gallery. Again, you can find all of them and more at https://www.deviantart.com/haycartesiangeometry.

The first one, which I showed above, and which I've reposted below, is based on the Rainbow Dash color gradient. I wanted to try to emulate the feeling of breaking the sound barrier, the moment before right before the Sonic Rainboom. It's one of the first fractals I generated, and still one of the most breathtaking fractals I've made. The bulk of the work of this fractal was done by the "fan2" and "radial blur" transforms, the former of which created that "shattering" effect, and the latter which helps smooth it out. (Source)

Linear transforms can be used to "tile" fractals, and create a nice "tapestry" effect. Apophysis also has a "diamond" transform which, when you combine it with Rarity's color scheme, gives you something like this: (Source)

You may recognize the above image as being one half of the cover art for my fic The Queen And The CEO. The other half was created using Twilight's color scheme and the "rectangle" transform, coupled with a bunch of linear transforms of varying sizes to create a circuit-like pattern. Those familiar with fractals may notice that it also seems Sierpinski-esque, and I did end up using a base set of transforms similar to what is used to generate the Sierpinski Triangle to kick-start this piece. (Source)

Beyond the Mane 6, the use of Julia Set-inspired transforms can create some "blossom-like" effects that one often associates with fractal art. Combine that with Zecora's color scheme, and you get something that looks like it would be at home in the Everfree Forest. (Source)

Here's another example of the tiling effect, coupled with some sinusoidal transforms to create a mind-bending labyrinth inspired by Starlight's vault. (Source)

(Edit: this one's too big to preview, so I've replaced it with a low-res version. Please follow the link for the full-res image.)

One piece of advice that I can give is that creating "thin" triangles in Apophysis will create "sharp" shapes that, with the right coloring, can look like jagged crystals that a certain King might summon. (Source)

To be perfectly honest I have no idea how I managed to make this one. Apparently some very obscure combination of uncommon transforms can create a treble clef-like shape, and when I saw this I knew I had to try Octavia's color scheme with it. (Source)

This one is a good example of how just a few simple transforms can create something that looks really complex. This is just three transforms with linear and julia-based variations, and colored with Flurry Heart's color scheme (I actually tried Celestia and Cadance's colors on it and they came out really nice too, but I think Flurry's is the best). (Source)

Finally, since I am incredibly predictable, I've been experimenting with combined gradients that have the color schemes of two different characters. You can probably guess which two characters I've been experimenting with. I haven't really found a fractal that screams "RariLight" so far, but here's a couple of my attempts that I think are really pretty nonetheless.

So there you have it. A look into what kind of stuff I get up to in my spare time when I'm not writing. I hope you've enjoyed this quick tour of my gallery and please check out my DA for more art. My current priority is getting a RariLight fractal that really speaks to me and putting that up on DA, so hopefully that'll get posted soon. For now, have a sneak peek at my next story. I had a lot of fun writing it, and I hope you all enjoy reading it when it comes out.

Osto Vinya nauva ustaina!

Comments

[Ice Star]

These get my cover-making senses itching.

[Undome Tinwe]

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Feel free to use anything in my gallery as cover art and modify it however you wish (with appropriate credit of course).

[Ice Star]

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Of course! Ty! They’re very aesthetic!

[Cynewulf]

This is an incredibly high tier post

[SigmasonicX]

It's fascinating to hear the methodology behind these.

[UrsaMajor]

Oooh, pretty mathematics! Love me some fractals. These are gorgeous.

[FanOfMostEverything]

Fractals! I adore fractals, and these are some truly gorgeous ones. Though I will say that the heart works a lot better for Cadence and Shining Armor. I'll definitely have to give this software a try, especially since there's a Derpy gradient in the pack you provided.

[Fillyfoolish]

...This is the nerdiest thing I've read all day.

.Thank you.

[Reese]

Neat and nice; thanks. :)

edit:
5322117
5322128
...Why did those get downvoted?

[Undome Tinwe]

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images-wixmp-ed30a86b8c4ca887773594c2.wixmp.com/f/c4781703-b228-4370-8114-3cef86545798/de26gug-b7881d0b-0e7b-4e18-909b-48d29d126c3a.png/v1/fill/w_1192,h_670,q_70,strp/loveshield3_by_haycartesiangeometry_de26gug-pre.jpg?token=eyJ0eXAiOiJKV1QiLCJhbGciOiJIUzI1NiJ9.eyJzdWIiOiJ1cm46YXBwOiIsImlzcyI6InVybjphcHA6Iiwib2JqIjpbW3siaGVpZ2h0IjoiPD03MjAiLCJwYXRoIjoiXC9mXC9jNDc4MTcwMy1iMjI4LTQzNzAtODExNC0zY2VmODY1NDU3OThcL2RlMjZndWctYjc4ODFkMGItMGU3Yi00ZTE4LTkwOWItNDhkMjlkMTI2YzNhLnBuZyIsIndpZHRoIjoiPD0xMjgwIn1dXSwiYXVkIjpbInVybjpzZXJ2aWNlOmltYWdlLm9wZXJhdGlvbnMiXX0.zrirt2MutAWTg_WiVWvp65U-BZypHk-q_n0PXcBq9CE

You might have a point about ShiningCadance working better with the heart. It pops quite a bit more with their color scheme. Thanks for the suggestion!