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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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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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Squirrel Rant: Mathematical Notation


Since this post veers into very inside baseball territory, let’s start with how we got here.

Have you ever had that book that you had to take with you everywhere? It could be the latest exciting story you were reading, or a really handy reference guide, or something with deep sentimental value. Maybe this book has been a passenger in your car; you take it out to lunch even though you know you’re not going to read it while you eat because you might get sauce on the pages. You just want to pick this book up and hold it, flip through the pages, feel the weight of it as you toss it from hand to hand. Maybe you even… smell the pages. That sort of book.

My latest book of this type is “A Perspective in Theoretical Computer Science: Commemorative Volume for Gift Siromoney“:


I know, it’s cliché. Everyone loves this book and has a copy next to their nightstand.

No? Just me, then? Okay.

Why this book is important is that it is the end of a long trail of looking through references in papers, from a stray mention of Kolam patterns in a L-System book by Przemyslaw Prusinkiewicz to the work of Darrah Chavey, Paulus Gerdes, Marcia Ascher and countless others. This book has some of the best grounding in the computer science behind drawing curved line patterns using context-free and array grammars, and contains images from work in the field. It is perfect because it is obscure and exactly the right book at the right time, even though it was written almost 30 years ago.

Which brings me to mathematical notation.

As a fledging programmer one of the first things they teach you is to use meaningful variable names. A variable is an object that holds a value. So if I’m counting oranges, I might have a variable called numOranges. What I wouldn’t have is a variable called ‘o’. ‘o’ could be anything: orangutans, oscilloscopes, Timothy Olyphants, etc.

Math on the other hand tends to use one letter symbols all the time. You’d think this was okay, because the symbols only ever have one meaning. A ‘+’ symbol is a plus symbol. ‘π’ is pi the number.

Except when π is an iteration symbol, or a time stage symbol. And ‘x’ means multiplication, until you graduate to Algebra where a dot is now multiply and x is a variable. And ‘+’ could mean turn right if we’re talking L-systems.

Math papers in general make an assumption about mathematicians that isn’t always correct. It assumes they can write in a way that can be understood. They understand their field technically, but not in common language, or even readable technical language. Now I’m not picking on the Siromoney book too much. The text is very readable. Some of the paper’s problems come from problems in reproduction. Older books like this one had papers submitted physically and then photocopied into a full book. This was done a typewriter or an early word-processor (courier font is kind of a tip off that we’re looking at a typewriter), so some subscripts and superscripts are incorrect, and some symbols have to be hand-drawn after the paper is typed.

The bigger offenders are the ones who use a symbol without any explanation. I remember after staring at it for a few seconds that ∪ means union, but it might be nice to have some handy definitions at the back or in the text. Defining your terms is often a necessary part of making any proof, or explaining any new concept. It never hurts to make sure your audience is on the same page you are. Especially if you plan to have terms mean something other than their common meaning, like ∪ meaning recursive level, or some such.

For a while now, my goal with this new book has been to take complicated concepts and explain them in ways that make sense to everyone. These papers often do the opposite, take something simple and explain it in a complicated way. Fractals actually aren’t that hard to understand. More and more after reading these papers it feels like I’m translating from some arcane and obscure language, with symbols that change in meaning from one place to another.

It’s confusing, and it doesn’t have to be this way. We can be rigorous AND we can be clear. If you can explain a concept to a 6th grader, then you really understand it.


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Sharing someone else’s culture

I spent part of the weekend reading about creation myths and fables of the Chokwe, an ethnic group that lives roughly in Angola. This is part of my research for the new book, which is expanding to have an extensive “Ethno-mathematics” section (i.e. Math and Fractal Drawings from around the world), specifically the Chokwe tradition of Sona. Sona are drawings made in the sand while telling a story or riddle. It’s one of the ways in which Chokwe elders impart knowledge and fables, though from what I’m reading the Sona tradition is dying out. They bear a resemblance to the Kolam of the Tamil-Nadu community in India, and even to traditional Celtic knots.


Math from other cultures is becoming really intriguing to me, and it’s an area I don’t think is covered enough in public education. Colonial era westerners often made the assumption that these “primitive” peoples didn’t understand some of the higher concepts of technology and mathematics, but if my studies have taught me anything it’s that we westerners were a little behind the curve (so to speak). At the very least, learning about how other cultures look at math and art can help us to see connections between ideas from new perspectives.

But one of the things I am wondering about is how to tell these stories respectfully. Some fables and tales are very private, specific to a culture, and not something that is intended to be shared with outsiders. Now obviously, since I don’t have the resources to travel to Angola myself, I’m getting these stories from people who’ve already spread them around. The genie is out of the bottle, so to speak. But it’s still important to consider their meaning, rather than to just include them as a pretty picture.

A lot of Adult Coloring Books have mandalas, in fact mandalas seem to be the stand-in term for most circular patterns in coloring books. I don’t think there’s anything wrong with enjoying these patterns, or designing new ones, and coloring them as a loose form of meditation. But at the same time I think it is also important to be respectful and understanding of the tradition. We want to learn and educate ourselves about a type of drawing, not just appropriate it.

Sometimes meanings for things change. The Kolam tradition seems to have had religious significance in the past, but now it is more a form of artistic expression by women in the Tamil community. Celtic knot constructions have a triune grid which reflects the triune nature of God, but also look really good on leather bound notebooks.

I’m a guy who wants to spread art and cool designs for their own sake, while also trying to explore some of the deeper meaning these traditions have to the cultures that created them. And I want to do that in a way that honors those traditions, without sharing them merely because they are exotic or different. The best way, at least for me, is showing the connections between some of the more abstract concepts of fractals, and their origins before they really came into their own (the days of computers and Mandelbrot). I’ve been thinking about fractals as something that is a new concept in math, but their origins may be much older.

I’m still working this stuff out, but I hope my intentions if nothing else can shape the writing in a good direction.

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On Research: Dive Deep or Swim Around?


I’m not going to lie. I’m not sure how on point this picture is, but I had to post it for the little red-haired girl.

Just a brief thought today:

When writing a book on diverse subjects, should you read all your sources on one topic then move onto the next, or should you cherry-pick and keep bouncing from one topic to another?

I’ve done a little of both for the expanded edition of Fractals You Can Draw, though my inclination is to go for the cherry-picking or “swimming around” approach. I’ve got a few reasons for this:

  1. Keeps each subject fresher – Spending a week or several reading about one subject can be taxing. I inevitably find myself skimming over passages that I then have to re-read. Reading one topic at a time can give you a solid understanding of the subject matter and helps you better understand concepts as they are presented by different authors. But it can also make you feel like you’re reading the same thing over and over.
  2. Helps you to discover interconnected ideas – With a book that is going to be a survey of different topics, it is important to have a through-line that ties everything together. Reading about a new topic each time can help you to see what’s different about each, and what commonalities they share. And you get some of the same benefit as you would when reading different authors on the same subject; you solidify ideas by seeing them presented from different angles.
  3. Each section is better balanced for research and time – I have a source list that’s currently about 150 papers and 20 or so books. I will not have time to read them all, and I’d like each chapter to be well-balanced in terms of the number of authors and sources. I don’t want to rely too heavily on any one author’s perspective. Since I don’t have infinite time I need to make sure I’m actually covering all the areas I want to write about.
  4. Helps you eliminate topics that don’t fit the theme – If you have better sense of what the whole book is shaping up to be, then you can eliminate sources that don’t fit your book. And reading from diverse sources can change your idea of what the book will be. Something that was going to be a chapter in my first outline is now about 30-40% of the final book.
  5. Gives you new ideas for sources – When you see how ideas connect, you’ll discover new angles and areas to research you hadn’t thought of at the beginning. It’s best to find these topics as early as possible so you can get outlines approved by a publisher (if you have one) and so you can make a better writing and research plan.

So what do you do? Do you dive deep, or float around?

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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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