Tag Archives: Math

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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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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Making an Adult Coloring Book: Kitchen Shelves

A number of the images in my fractal adult coloring book came from looking at objects around me. This is a set of shelves in my parent’s kitchen:


Understanding and creating new fractals means breaking shapes down to their most essential features. For L-Systems, which make up the majority of the images in my book, we call these essential features the axiom, or base image. The shelves are made up of five sections, a larger center section, and four small squares connected on each side. You might think the most essential feature is the whole shelf, but actually the most basic shape is a single square:


For simplicity, I made the shelves equal on all sides, rather than using rectangles of different widths, but the basic principle is the same. The above square is the axiom of our L-System. With each iteration (stage), we add four squares to each side of the square(s) from the previous stage. If we add a smaller square to each side of this base image, we get something resembling the kitchen shelves:


Not a bad model, but not very interesting to color yet. So let’s add four new smaller squares to each side of the four squares we added:


This is getting better. Already we can see how adding smaller squares creates interesting overlapping sections. At this stage we added 16 new squares, so let’s add 64 new squares to the next level (four on each side of the 16 squares we added):


And so on:


The level of intricacy used for a final image has to balance the expected medium (colored pencils and gel pens) with the ability to create many different types of patterns. The above stage is the one used in the book, but when using a computer to color, we can go to even higher levels of detail:

021_Little Boxes (1)

The above image was colored by my wife, who consulted with me on the best images to select for the book, and suggested their level of difficulty.

Creating new fractals is about seeing the potential for art all around you, even in the most basic and mundane parts of our lives. Simple patterns can be expanded into something intricate and beautiful. And deciding how to color these new patterns adds an even greater level of artistic expression. A simple object can be transformed into a universe of variations. That’s what I enjoy most about creating the coloring book, seeing how others take a pattern and make it their own.

If you enjoyed this post and would like to learn more about fractals, check out my Adult Coloring Book: Fractals available on Amazon.


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Show Your Work


I’ve been reading a lot of math papers lately.

I’m a computer scientist by trade, and theoretically should be able to speak this language. In fact, we take so much math as computer science engineers that we can’t double major in applied math since that’s already built into our coursework (something that would have shaved at least half a year off my college life if I’d known it sooner).

Some of those later courses were tough, but even math students at the most basic level have encountered knowing the answer, but not knowing how to get to the answer. On a particular test problem where I ran into this situation, I wrote “Poof! And the magic occurs!” between the problem and my answer. Suffice it to say, that was insufficient explanation.

Math writing is inherently logical. You define your terms, make your propositions, prove your theorems, then move on to the next property of whatever you’re studying. The problem occurs when you forget to define your terms, or leave out a step, or assume everyone in the universe has the same base knowledge as you.

I spent at least an hour last night trying to figure out how to change Fibonacci words into generalized Fibonacci snowflakes. I was missing one crucial piece of information that I finally had to track down in one of the cited papers, that all the addition in these equations was mod 4. The moment of finally watching something work the way you expect it to can feel a lot like magic, but a lot of trouble might have been saved on my part if the author had bothered to work out the interim steps in the paper. There were many other places where they had done this, but this one lacking piece of information was right in the middle from one really cool graphic to another.

So how do we apply this more generally?

Constructing a story, particularly a mystery, is a lot like proving a math theory. You discover evidence, make some conclusions, and prove your theory. Sure, a good mystery has some misdirection. You don’t want the reader to arrive at your conclusion too quickly. But you want your solution, your ending, to be the satisfying and logical progression of what has come before. Put another way, you want your ending scene to be “earned” by what you’ve written before it.

The problem as writers is that we always know this universe of our story better than our readers, to the point that sometimes we don’t know if we’ve said all we need to make it clear to others. You may know a character’s motivation, but if you leave no sign of it in the book, then the reader doesn’t know why they should care. Bad mysteries often introduce a surprise villain at the end of the book, rather than in the first 20%, cheating the reader of the opportunity to engage with finding the solution.

I don’t think this means you need to beat your readers over the head with facts already in evidence. But if something is important to things you’re going to write later, be sure you’ve actually said it the once. Beta readers and editors are especially helpful in finding these sorts of flaws, as is having an outline where you work out all of these connections ahead of time.

And incidentally, and separately from the main point of all this, writing in a technical language is not always better than the vernacular. I understand that academic papers serve different functions and are targeted at different audiences than more general work. But math doesn’t have to be obscure. Part of the reason I’m slogging through all this work is to write something I can share with everyone. This is something to keep in mind when you’re tempted to insert a lot of techno-babble or overly sophisticated words into your stories. Sometimes telling a story clearly, plainly and succinctly is the best way to go.

Just make sure to show your work.


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