Tag Archives: fractals

New Release: Fractals – 2017 Adult Coloring Calendar


I know how it is. You get to the end of November and you realize you forgot Mandelbrot’s birthday (it was Sunday). Or your friends ask you to come over for Pi day and then laugh at you when you bring pie instead (which they eat without thanking you). Well don’t worry, I’m looking out for you.

Introducing Fractals – 2017 Adult Coloring Calendar.

Did you know that you can make up math holidays just by choosing the right numbers? I mean, tomorrow is Fibonacci day because it’s 11/23 (it’ll really be cool in 42 years when it’s 11/23/58). I even made up my own holiday on January 26th: Koch curve day (because the approximate fractal dimension of the Koch curve is 1.26. It’s also E. H. Moore’s birthday apparently).


Feeling stressed? Why not color in some fractal bubbles or a cozy quilt? And take some time to marvel at the clean numbers and lines of the calendar template I created using a python script. There were definitely at least five tiny pixel adjustments to make sure the numbers lined up in diagonals just right.

Perfect for the math geek who also enjoys trivia and pretty colors. But even if your name isn’t Brian Buckley this calendar is a great for someone looking for something just a bit different in their date tracking this year. Available now on Amazon from the good people at Green Frog Publishing.

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What I never expected

I’ve been off WordPress for a while and so it’s been a while since I checked my stats, and I was surprised to learn that I had something like a 200% increase in traffic last week, and for the best reason.

My “Fractals You Can Draw” posts have always been the most popular ones on the site. In general I think the writing life is weird like that. You never know what 20 minute or hour long effort is going to be the one that really lasts. I’ve spent hundreds of hours writing this blog, but that week I spent getting my wife to draw fractals, building a Sierpinski triangle out of marshmellows and toothpicks, and frantically trying to update by C++ skills has been one of the more lasting efforts of the last five years for me.

But the best thing is every year around the spring and fall I get new referrals from schools. WordPress does a pretty good job of letting you know where traffic is coming from, and every year I find some new class, ranging from grade school to college that references one of my fractal posts. That’s really the reason I’m doing any of this. What I’ve learned since I started blogging and especially in the last year working on the “Fractals You Can Draw” book is that I really want to teach people. I like writing fiction, but I love writing about math.

Honestly I’m as shocked as the rest of you.

Right now I’m working on Chapter 5 of the new book (or trying to, it’s been a crazy couple of weeks). I’m learning about new ways to use the Fibonacci sequence to draw fractals, and I can’t wait to share what I’ve learned. I’m so excited about this stuff I even snuck in a half an hour to write on Monday night while I was waiting for my mom to finish her grocery shopping.

If you’ve found this blog through one of your math courses I’d love to hear from you. To all the teachers who included links to my posts in their courses, thank you. And thank you for teaching people about fractals. It’s one of the best ways to build a love of math.

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Down from the mountain

I was annoyed with John Oliver in late August when he said he’d be taking a month off. Watching him on Monday after work has quickly become one of the ways I can handle all of the nonsense going on in the world. And here I’ve been gone for nearly two months. Hope you guys didn’t miss me too much.

Short version of what’s been going on is this:

  • I got a new job in mid-August. Closer to home, interesting work, nice environment. Overall a big improvement. Still with the same company, just changed divisions, so a nice mix of old and new. I am even dressing nicer on a day-to-day basis.
  • Time that I am not spending with my wife or my work has been spent largely on writing the new fractal book. I am excited to share this with you as soon as possible. I am even considering new platforms for fractal art (I might venture out of the safe pastures of Twitter/Facebook/Wordpress).
  • I am handling this election about as well as the rest of you.
Photo of author taken mid-September

Photo of author taken mid-September

It’s this last point I want to talk about just a bit. Over at Previously.tv they’ve been doing a marathon diary of The West Wing, watching the entire series prior to election day. If you want to feel depressed about the election, I couldn’t think of a better way to do it than to watch what a fantasy American democracy looks like. My instincts are to watch or read something that bears more of a resemblance to reality, or even something far worse, so I can comfort myself in the knowledge that at least things aren’t that bad.

And so I’ve found myself immersed once again in a contemporary of The West Wing: the cyberpunk, post-human, political drama, comic book series Transmetropolitan. “Transmet” follows outlaw journalist Spider Jerusalem as he takes on corruption wherever he sees it, and pursues truth in relentless and foul-mouthed fashion. Much of the series’ narrative takes place during the administrations of two Presidents: “The Beast” and “The Smiler.” Sound familiar?

Admittedly, Transmet is as much a fantasy as The West Wing. It assumes journalism and truth can bring down Presidents. This certainly has been true, but in the fractured cable TV and social media news landscape of today it’s hard to say. But like all great science-fiction the series still has a lot to tell us about the moment in which it was written (1997-2002) and the moment in which we now live.

So I’m running a mini-marathon of my own, from now until election day. Twice a week (hopefully), I’ll be posting my thoughts on a volume of Transmetropolitan, both its place in the overall arc of the story, and what it has to say about our present moment. Already in rereading the comic I’ve seen it touch on themes of police brutality, transhumanism, racial profiling, political scandal, right-wing extremism, rampant consumerism, historical preservation, child prostitution, poverty and more.

If you want to read along I’ll post Volume 1 “Back on the Street” on Thursday (I’m using the new printing numbering, so this covers issues 1-6). Fair warning, the content is rough both in terms of subject matter and especially in the early issues, vulgarity. Also, as a dog owner I want to state clearly I do not agree with Spider Jerusalem’s attitudes/actions toward dogs. But if you can get past the rough trappings, you’ll find a gripping narrative with plenty of twists and turns, and even some genuine human feeling.

In the meantime I will try to write something that isn’t about comic books or fractals, but I’m not promising anything.



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Dispatches from writing a new fractal book

I spent a lot of the weekend making figures. If a picture is worth 1000 words, then my book would be 500,000 words long. There are a lot of things to consider when making an image for a book:

  • Since this will be a B&W book, what’s the best gray-scale values to look two-tone without looking faded?
  • What resolution will work best for the print size, but still work for an eBook (so I don’t have to make two of every picture, hopefully)?
  • How does the image flow with the text? Does it make it hard to read, or does it illuminate a point the text is making?

This last is the primary goal of pictures. Pictures serve two purposes in my book: to show people pretty images, and to show people how to make pretty images. A lot of the heavier math texts I read can still be very informative if they have good illustrations that make it clear what the writer is trying to say. Still, you don’t want to lean on the pictures too heavily, the text still has to make sense.

My early drafting process has been to write the chapter without illustrations. I insert an [ILLUSTRATE] tag into my text where I think an image would/should go, and then I move on. So far this seems to be working okay. I’m remembering from writing the first fractal book that what I think is a clear explanation and image isn’t always what people actually understand, so I may have to try a few different images, and change the text accordingly.

It probably doesn’t make sense to talk this long about pictures without showing you one, so here is a Minkowski sausage:


Neat, huh?

One other weird quirk of the early going is the feeling that I’ve written this all before. In some ways, this fractal book is the culmination of 5-10 years of thinking about fractals, as well as an intense period of six months research. I have run through the progression of thoughts that make up the first chapters dozens of times in my head, and even though this one of the first times I’m committing them to paper in this particular order, it still feels like something I’ve been doing all along. Probably some of that comes from blogging about fractals, that’s really where this recent and previous interest started.

Last thought of the day is a nice little moment at the library. I was checking out yet another fractal reference, and got to talking with the librarian about the new book, and how I got interested in fractals back in the 6th grade EPP program at school (more than 18 years ago now). Turns out, her son had just recently been part of the same program, and had loved fractals and all of the ways you could use them to make math beautiful. It’s for these kinds of kids and curious adults that I’m writing this book, and it was a nice little encouragement to know that they’re still out there.

Have a good week. I’ll check in when I can.


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On Research: Finding Sources on a Budget


A good portion of the research budget for the new book was selling my Inuyasha manga (which to be fair I hadn’t read in years). This is still better than my old budget for Fractals: A Programmer’s Approach, which consisted largely of buying fractal books I could get for $4 (1 penny cost plus $3.99 shipping). For both books I’ve made good use of the library, and the wonders of this thing called the internet, and I thought I’d share a little of what I’ve learned along the way.

Dover Pictorial Archive: A lot of what I’m looking for in this book are designs from different cultures and some background on drawing techniques, religious significance, history, etc. There are a number of great books on Amazon all pretty affordable from the Dover Pictorial Archive. But if you’re even more of a cheapskate, for a lot of these books you can look on the copyright page (using Amazon’s Look Inside feature) and find the original public domain book the new printing is based on. The Internet Archive will often have the original book for free. Now considering the fact that one of the books I did this for was originally in French (Les Elements de l’Art Arabe), there are some drawbacks to this approach. But then again, if your main interest is public domain designs, this isn’t a bad way to go.

Papers Submitted to Conferences: A number of the papers I’m working from are from various years of the Bridges conference. Most of this stuff is available for free online on the conference websites and includes some fascinating material, ranging from presentations of new techniques, to more general overviews.

Papers on Professor Websites: If you find a particular name coming up again and again, you might want to find their university page to get a complete list of their papers. Some will be available directly on their site. Just be sure to keep some idea of when you accessed them and their URL for your works cited.

Google Books: Let’s face it, you really might only need 5 pages of a 1000 page tome that costs over $100. Google books isn’t a bad way to find the little tidbits you need from books without forking over for the whole thing. It can be a little hit or miss as to whether you’ll be able to access the parts you need, but often you can get enough.

ILL (Inter-Library Loans): I haven’t done this a lot since I tend to get more done with digital sources, but ILL’s can be a good way to get that expensive book you really need (the one that every paper seems to cite). Just be sure to return the book on time or you’ll be slapped with pretty hefty fines.

Your own collection: I have a whole bookcase dedicated to fractals, some of which I’ve barely touched. Some material that didn’t really work for Fractals: A Programmer’s Approach might be just what I need for this one. Keep a running archive of the things you gather even if you never use them (I keep all my papers in Calibre). And you can always raise a little money by selling books you no longer need (though do this wisely because you’ll lose more if you end up buying the book again).

Google the Works Cited: Most papers and books will have a good bibliography. While not all of these resources will be accessible, some will be, and there’s a good chance they’ll fit in with your subject since the author based their work on them.

My last two tips don’t really have anything to do with money, but I think they’re valuable nonetheless:

Do the Works Cited as you go: I never do this, and I always realize later that I should have. If you’re pulling from a lot of diverse sources, you absolutely need to keep track of them, or you’ll have to do detective work to find them again.

Cite more than one author: Some of the areas I’m covering really haven’t been widely studied, but there are still ways to corroborate the research of one-off writers. Again the works cited is key, and some wider Googling. Just be careful if everyone else has based their work on your one original source without any new information. If they’ve been out there for a while, it’s probably fine, but it’s better to have multiple sources.

Friday’s post won’t be so work related. I’ll review something fun you’ll like.

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On Research: Trimming the Tree


I’m heavy in the middle of the research phase for my new book. I’m trying to read at least one academic paper a night with the goal of having roughly 100 sources for the entire book, and at least 5 sources for each chapter. The trouble is, every time I read a new paper I find five more I want to read. When you’re in deep on a subject, you’re eager to drink up all of these connections and new thoughts as much as you can. There are good reasons for this, the more research you do the better the book you’ll eventually write. But I still have to actually write this book.


I wrote an outline for the new expanded edition of Fractals You Can Draw before I entered the research phase. It shaped my initial gathering of material and gave me a good sense of the structure of the book and the kinds of topics I wanted to cover. And now several months later I realize it needs revision. I’ve added some topics, dropped others, and overall have a slightly different feel for what the book will be. The structure is the same, but in my research I’ve found myself fascinated by the different cultural expressions of fractals, sort of an abstract versus practical study all with fractals you can draw by hand, and something that was just a chapter in my original outline is now a whole part of the book.

An outline is a great way to shape your vision initially, and it’s a structure you need to continue to revisit as you get more into the execution. It can remind you of areas you still want to cover but haven’t gotten around to, and it can be a way to shape the new thoughts you’ve been having and see if they fit into the book you still want to write. As the outline matures it can be a good measure for judging the effectiveness of reading new material. Is the new material relevant enough to fit into the existing framework, or will it require a restructuring, and is that restructuring valuable? Some things will fit nicely, others won’t fit neatly but might fit the overall shape of the book, and still others might be for another book entirely. An outline can help divide your sources into these piles.

Source Limitations

I have neither unlimited money nor unlimited time. Practically speaking, my research consists of what I can do on my lunch breaks, off times, and evenings at home. Because of this, I’m more biased in favor of electronic sources, though I do have about two-dozen physical books on my source list as well. I can’t afford to pay for access to expensive databases, meaning some papers are just stuck behind a pay-wall.

Other sources are very informative, but a little hard to cite. Lectures from University classes, unattributed papers from supplemental course work, or images of cool designs from Facebook groups. Often these things aren’t great to use directly, but can point you in the direction of more traditional sources.

It can sound like these limitations are frustrating, but there is a lot of knowledge that is freely accessible, certainly more than you’d be able to read anyway. It might not actually add that much value to your book to spend tons of money on research, if the information can be found for free elsewhere. I’d love to be a world traveler, to go to the regions in India I’m reading about, but for now I’m happy to work from the writing of people who have.


In addition to writing a book with a bunch of cool fractal designs I have three goals for the book as a whole:

  1. Use fractals to introduce other interesting math subjects. Provide a good overview of fractals and math that leads people to want to learn more.
  2. Show how math and fractals are interconnected across culture and technique (math and art as universal languages).
  3. Choose fractals that can be drawn freehand or with minimal tools (compass, ruler).

Some subjects, like how fractals are used in weather prediction or chip design, are interesting to study but not really on the point of this book. At best they might be a few introductory paragraphs or notes here and there. So it doesn’t make sense to read long papers on how Hilbert Curves are used to organize data-clusters in computers unless I really want to talk about those subjects (which I don’t for this book at least).

Having a good sense of vision can eliminate the temptation for tangents, or for a knowledge dump. You’ll never be able to cover everything, and dropping in a bunch of the random trivia you’ve learned doesn’t make for particularly compelling reading. It’s better to try to say one thing well, than a dozen things poorly. Most academic papers have abstracts or intros that tell you what they’re going to be. If from the intro it doesn’t sound like the paper will contribute much to vision, it’s best to save it for another time.

These are just some of things I’m learning while putting together this book. What else have you learned from long term research projects?

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Try as you go

I’m working on an expansion of Fractals You Can Draw transforming it from a pamphlet I wrote in the space of a week to something that could be used in a 6-9th grade math class. I want the book to serve not only as an introduction to fractals, but as a gateway to other interesting areas of math, and even world culture.

But at the core, the book still needs to be fun to draw.

I’m heavy in the research phase of this book and for the last few weeks I’ve been studying a traditional form of drawing native to Southern India (in the Tamil Nadu region) called Kolams.


Kolams have a lot to teach the casual math enthusiast or the serious math student about fractals, symmetry, context-free grammars, hexadecimal encoding and countless other subjects. They also can be kind of tricky to draw as you can see from my increasing lack of skill from top to bottom. All of these are theoretically able to be drawn free-hand as one long continuous line, but it takes practice.

I keep free-hand notes in part to test the difficulty level of what I’m expecting people to draw. Even in the original series I drew a couple of the images, and the little red-haired girl handled the other two, which gave us both a sense of how long it took to draw each image and some of the difficulties involved. I’d known how to draw all of the fractals in that series of posts for years, but it took actually trying to draw them by hand before I really knew how they worked.

What I’ve learned from drawing Kolams is that it takes a lighter, freer touch than is my natural inclination. And maybe gel-pens that smear easily aren’t the way to go either. You can make some pretty images very quickly, but you need to get a sense of the flow as you draw, or you can easily go off track (as I did multiple times on the bottom image).

More generally it is important for the writer to be able to take a step back and engage with whatever they’re writing as a their final target audience. Especially when you’re down the rabbit hole of research, it can be easy to lose a sense for how easy or difficult a particular subject is, and you need to take the occasional application step back. This is good not only for assessing the level of difficulty, but also in solidifying the theory behind what you’ve been studying. There were properties of how Kolams were drawn that didn’t gel in my head until I’d tried to draw a few.

You are your first beta reader. It’s still important to get outside perspectives, but trying things yourself helps you discern what should actually be included in the first place, and what should be left out. Engage with your work in different ways: read it aloud, read it out of order, try to actually follow your how-to directions without any outside info, color in your coloring book, etc. Whatever your genre, there’s more than one way to look at your book, and there’s value in gaining that new perspective.

You can read the original Fractals You Can Draw series here or check out my other book from Green Frog Publishing, Adult Coloring Book: Fractals (adultcoloringbookfractals.com) with cover art by the little red-haired girl.


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