The Brainsturbator Fractal Toolkit
If you haven't already, I suggest you start with Miqel's collection of Visual Math galleries and explanations. It remains the best introductory page I've ever found -- educational and beautiful, and it's hard to beat that combination. If you're already perfectly hip to the fractal concept, you will still dig the gallery of Wada Basin fractals, which are even cooler than they sound.
This collection is focused on that which is useful, but if you're looking for that which is amazing to look at, you should go check out Miqel immediately.
Although everything is just numbers -- including the text you see on the screen right how -- fractals are best understood visually (if not viscerally, but perhaps that requires drugs).
Arthur C. Clarke's Fractals: The Colour of Infinity Sci-fi author and megagenius Arthur C. Clarke made one of the original documentaries on Fractals, and amazingly, it's still one of the best. At nearly an hour, this is a serious primer for primates with some time on their hands, but well worth it.
Chaos, Fractals and Dynamics -- a three-part series hosted and narrated by Robert L. Devaney. A little dry but still excellent.
Fibonacci, Fractals, and Financial Markets -- this is truly fascinating, hopefully the title alone is enough to get your attention. Basically, the classic black and white intellectual horror film Pi was more than a clever plot -- it was truly onto something. For a more detailed explanation of chaos and it's role in market systems, check out J. Orlin Grabbe's very accessable introduction.
Mandelbrot Zoom -- exactly what it says: a continuous 2-minute zoom into the infinitely complex Mandelbrot set.
"Dragon Fractal" tutorial a short lesson about hands-on creation of fractals on paper.
And for someone who wants the Heavy Weird, watch this: The 2012 Matrix Singularity Seminar: Fractal Time, which explores Terence McKenna's classic "Timewave Zero" theories in relation to 2012 and the impending, uh....whatever. What? That's not enough? Fine, I got you: Sacred Crop Circles and Divine Holographic Fractals! 10 minutes of non-stop mindbending, complete with vague terminology and Theories of Everything.
If all that doesn't do it for you, Cymatics, Fractals and the Spirit Molecule is worth checking out. Is there a connection between vibration, resonance, fractal math and the DMT experience? Of course there is....and this documentary is one attempt at explaining and exploring that terrain.
BIPT Library's Mandelbrot Collection
Fractals and Chaos: An Illustrated Course -- (Amazon Link) From 1997, an excellent general introduction with a ton of depth for anyone who wants to get heavy. This is a damn good read and I'm probably going to buy a copy shortly, because I refer back to it often.
Nonlinear Dynamics and Chaos: Where Do We Go From Here? -- This one is wild: a 2003 collection of essays envisioning what the future of chaos and fractal studies will be -- some truly visionary work in here. Topics include: Spontaneous Pattern Formation in the Visual Cortex, Many-Body Quantum Mechanics, Time-Reversed Acoustics and Chaos, and Higher-Dimensional Chaos. Download this now.
Dynamical Systems and Fractals -- (Amazon Link) Truly dense. This is a translation of a German textbook, oriented towards doing your own computer experiments in Pascal. This might be a useless dinosaur, but hey...I have it...you can, too.
Fractal Geometry: Mathematical Foundations and Applications, 2nd Edition -- (Amazon Link) Being a 2004 edition, this is the most current PDF in the BIPT library -- but then again, the math hasn't changed much since the Mandelbrot years.
Of course, if you're really into this stuff, nothing in the Universe will beat The Book: "Chaos and Fractals," by Peitgen, Jurgens and Saupe. It's 864 pages and offers one of the most complete, and fascinating, explorations of this infinite terrain you will find anywhere. Although it's fairly expensive -- even used copies are $26 on Amazon -- bargains do exist. I recently got a copy for $4.50 -- I realize that's not helpful information, I'm just celebrating.
White Papers and Original Research
Of course, to get into the really meaty stuff, you can't rely on popular and mass market titles -- you've got to burrow into the technical and academic literature. Even if you only understand 10% of what you read, at least it's the latest, most accurate 10% you can find. Better yet, every time you subject yourself to a white paper, you become more intelligent and you will understand future white papers that much better.
Fractal and Chaotic Dynamics in Nervous Systems -- one of Chris King's best essays, and that's saying a lot. We've covered New Zealand genius Chris King before, so refer back to Skilluminati if the guy fascinates you. This essay also has a quote I feel obligated to share, because it's a mild headfuck:
"The raw numbers game of neurogenesis [growth of the brain from embryo to adult human] suggest attractor dynamics may form an essential bridge between central nervous system genotype and phenotype. The 5 x 10^4 genes governing central nervous system development, around 60% of human genes, cannot informationally specify the connections for 10^11 neurons and 10^15 synapses."
Fractal Structure of the Galaxy Universe -- Cosmic fractals? Hell yeah. Could not recommend this one enough, it's a serious mindbender.
The Fractal Organization of Nature by John A. Gowan. BIG THANK YOU goes out to Prof. Gowan for making this publicly available -- it's a very ambitious and informative essay, well worth printing out and really sinking your teeth into.
Dimensional Analysis Scaling and Fractals -- an excellent 2003 paper looking at innovations in accurate measurement and the fractal mathematics which has made those advances possible.
Temporal Fluctuations In Coherence of Brain Wave Cycles -- very interesting stuff -- analyzing the patterns that emerge when EKG and EEG information is re-sampled and analyzed using geometric and fractal math.
The Fractal Flame Algorithm -- an explanation of one of the coolest equations going -- very arresting imagery, worth the download just for the pictures.
Thoughts on the Fractal Nature of Legal Systems -- I'm still unclear if this is just theoretical wanking or truly deep. Take a peek and see what you think -- one thing is for sure, Franz Kafka would probably dig this paper.
Is the Evolutionary Tree a Fractal? -- wowza. Truly large scale fractal math -- in French and English.
Early fractal pioneers realized immediately that their new tools would have huge implications for the modeling and measurement of living systems, but it's been a lot of uphill mathematical work to prove that, in the decades since. This section is a collection of recent research and articles about the cutting edge of this newly-formed, vaguely-understood, and still un-named science:
Algorithmic Botany -- a vast, high-quality selection of papers about L-Systems, which are the mathematical models for plant growth. If you're ready to dig into the real fractals -- the kind we live around, breathe oxygen from and occasionally eat -- this is The Fountainhead. Also check out their excellent lesson "Visual Models of Morphogenesis."
Fractal Analysis of Blood Vessels -- it's one thing to be stoned and realize your body is a fractal, but seeing mathematical proof is much more valuable. (Well, unless it's really, really good weed.)
DNA "Junk" noise is fractal -- good old Chris King was kind enough to save a copy of this 1992 article.
The Fractal Modeling of Plants -- everyone has seen the "fractal fern" by now, but modeling other vegetatation is much more challenging and complex task. This supercool website takes a close, illustrated look at the cutting edge in computer flora simulation.
Using Fractals to Grow Human Organs -- in terms of real-world applications of all this hippie bullshit, it's hard to beat the fact that a fractal matrix yields the best possible results in growing artificial replacement human organs. From dazzling acidheads to saving human lives -- this is a pretty major milestone in fractal research.
Magical Numbers in Nature -- a short but superb interview with mathematician Ian Stewart about the fractal geometry of, oh...the whole Universe.
DNA Barcodes Suggest Fractal Nature of Genome -- I'm still catching up on what a "DNA barcode" even is, but this is fascinating reading with HEAVY implications for the future of genetics.
The Final Iteration
This is not a completed object. I'm sure there's still outstanding material I didn't find -- let me know if something is obviously missing here. I want this to be useful a year from now, too. Get in touch with me if you've got resources to suggest -- especially if you've got more books in PDF, we love those. That's like water here at the BIPT, seriously.
So in closing, here's the Fractal Hubs: other collections of resources, gathered by other humans just as obsessed as myself.
Fractal Dot Org -- just going off domain names, this would have to be #1. An alarmingly high number of dead links, but those that still work are uniformly excellent.
The Chaos Hypertextbook -- if you have the time and the patience, this is a truly badass resource. Less pretty but equally jam-packed is Yale's free online college course about Chaos and Fractals.
FractoGene -- this site is run by Dr. Andras J. Pellionisz, who is in the habit of commenting anywhere he gets mentioned and might show up here, now that I'm linking to his work. A curiously large number of people have written him off as a crank, but I've found his material to be fascinating. It's not his only site, he also runs a personal archive, and JunkDNA.com
Wolfram MathWorld on Fractals -- Wolfram Mathworld is a seriously fucking amazing website and I refer to it constantly. Their presentation is clean, their resources are vast, and their scope is nearly infinite.
And finally, there's a whole scientific journal devoted to Fractals that's well worth exploring once you have the basics under your belt.