Tag Archives: Research

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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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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How long does it take you to buy a book?


This will come as a surprise to exactly no one but I have a big fractal book wish list. I’ve actually become kind of obsessive about it since I’ve been focusing more on a specialized area of fractals, while still considering options for broader fractal surveys.

Back when I was writing A Programmer’s Approach, my method for selecting books was simple. Search “fractal” in Amazon. Buy any book that looked vaguely helpful and that cost $0.01 (+ $3.99 shipping). Of course even then there were special books that I would pay a little extra for, but overall I was looking for a broad survey of authors and perspectives.

Considering that I have a full bookshelf now of fractal books, and that the bookshelf has started to bleed over onto my desk, I do not need more general books.

But, and again this might surprise you, specialized books are expensive. A lot of the better fractal books fall into one of two categories: college textbook or obscure lecture notes from a math conference. In college spending $120 on a textbook was a necessary evil. In later life, especially one that expects it to take a while to make $120 from a fractal book, that price is a little steep.

I’ve started to camp on books, throwing their Amazon listing into a wishlist called the “buying queue” and I’ve noticed something weird. Usually, even an expensive book, will have two sellers who have the lowest price. These two prices will leapfrog each other down by a few pennies several times a day. It can sometimes take weeks of waiting, but you can knock a couple of bucks off the book’s price if you wait long enough.

However, if you wait too long and somebody snatches one cheap copy up, the other cheap copy shoots up in price to match the second lowest price, and they fight it out again. I’ve observed this behavior on comic books, DVD’s, regular books, etc. I’m pretty sure it must be a setting in the Amazon Marketplace, coupled with an algorithm. Either that, or all marketplace sellers are exhibiting the same behavior.

With the buying queue, a good five minute segment of my day is looking at a book, gleaning as much information as I can from the preview or the reviews, and deciding if this is the day I will buy it, or if it’s the day I decide to take it off my list entirely, or bump it down to a secondary wish list I check less often. I’ve had books I’ve debated over for months, doing the online equivalent of picking it up, flipping through the pages, and putting it back down again.

With reference materials in particular I want as little overlap as possible, while still getting something that builds on other material I have. I prefer electronic books just because I will read them more often, but still acknowledge that there’s nothing like flipping through a real book. I have limited shelf space, but I’m always willing to clear away the chaff for something great. And, probably most difficult, all of these books aren’t popular, so there’s virtually no reviews or sales rank to give me a sense of whether it is actually good. Occasionally I can find an academic review if I do some digging, but that only sometimes helps.

Do you think it’s too late to start a Kickstarter campaign so I can buy more books. I’d do it for my Star Trek comics as well, but I have a hard enough time convincing others that reading comic books is “research.”

Ah well. Maybe I’ll go to an actual bookshop this weekend and stare at those books for a while. Happy Friday all.

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Research Mode

Image Source: Tumblr

Image Source: Tumblr

I just got my first interlibrary loan yesterday (ILL for all you bibliotheque nerds). It is an interesting mix of adult responsibilities and genuine excitement. The book is due on April 30th (no renewals), and I will be fined $2.50 a day if it is not returned. There’s an envelope it must be returned inside, and a sleeve that is stuck to the outside cover. The cheapest I could have bought this book was $20, with most copies ranging more in the $50-$60 range. As I continue on the journey from a general interest in fractals to a more specialized exploration, there are only going to be more such books and loans (though I still have to fight off the hoarders mentality that I’d have if I had a University Library’s budget).

It struck me the other day how different the way I conduct research now was from when I was in highschool and college. The internet was a strong resource in both times, but where I’d be printing off papers in college and compiling them in a notebook, now I am just throwing things on my tablet. I found a 2000 page math encyclopedia on the Internet Archive the other day, and I can carry it around in my bag without any back strain.

Yet I still find myself working with paper when it comes to taking notes and working things out. Part of this is simply mobility, it’s easier to take notes on paper at idle moments than it is to use a computer. And part of it is that I believe as many do, that taking notes on paper is a better way to retain information and to organize thoughts. Plus it’s a way to make use of the dozens of notebooks that have piled up in my house that have yet to be filled with brilliant short stories.

I’m a little more specialized with these notebooks than college. I got into a genuine discussion with Brian over whether Moleskine is pronounced “Moleskin” or “Moleskeen” (I prefer the later even though it is likely wrong). And I have all different sizes, larger stay at home notebooks for rough work, smaller reporter pads for technical notes, and mid-size for more general information. My “go bag” has a tablet, an eReader and six notebooks!

And even when I find myself frustrated with pay-walls for articles, or expensive books, I am amazed at how much knowledge is just out there for free. Even with the potential for steep fines, getting a book from an inter-library loan was cheaper and almost as quick (if not faster) than buying the book myself. I do admit to some impatience with having to wait for physical materials, both waiting for them to arrive, and waiting for time to read them at home. It’s why I’m a fan of writing affordable eBook reference materials. But sometimes there’s nothing like a good primary source from an author whose name you need a pronunciation guide for.

How do you do research? Are you still a pencil and physical book sort of person, or is Google the way to all knowledge?


One of the results of previous research projects was Fractals: A Programmer’s Approach. If you’re looking for a gateway to understanding fractals, particularly how to make them, it’s not a bad place to start, and it’s free on Kindle Unlimited.


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Information Glut

One of the details that struck me in watching some of the early X-Files episodes was how Mulder and Scully looked at the case-files. In the first episode Scully is reading a newspaper clipping that has been taped to a piece of paper. I remember preparing reports and research in highschool and early college. I tended to use very “dead tree” methods, photocopying articles out of books, printing out stuff from online, and shoving all of this material into large black binders.

How much things have changed in the intervening years.

Now my process involves a combination of Google searches and bookmarking web-pages, and downloading scholarly articles, cataloging them in Calibre, and sending them to my Kindle to review. As I’m preparing material for another book, I’m amazed at all the stuff I downloaded during the production of Fractals: A Programmer’s Approach that got thrown on a flash drive and never looked at again.

In addition to making me wonder what the plural of thesis is, looking at all of these papers (many of which are frankly far above my head even with the pretty pictures) I’m struck by how I’ve only scratched the surface of this subject. Part of me thinks at some point I should study another area of significance, maybe global warming, or even other areas of math. But the truth is this one field is so rich, and touches so many parts of life, science, and engineering, that I’m probably going to spend the rest of my life working on and writing about fractals.

One of the nicknames for a PHD is “Piled Higher and Deeper” meaning you know an incredible amount about a very narrow range of things. I’m not going after a doctorate, at best I might be trying to be the next Martin Gardner, but I still find myself amazed at just how much access I have to knowledge that would have seemed unthinkable 10-15 years ago. I’ve downloaded course slides from university classes in the Netherlands, dissertations from Germany, and papers from dozens of conferences.

I’m still old fashioned in some ways. I may load all this stuff onto a Kindle, but I keep a notebook handy to take notes. And I still refer to my old printouts, if for nothing else but to find the books and articles the material came from. And I write books as signposts along the way as a way of encapsulating what I’ve learned, for fear that the knowledge is somehow fleeting. I look back at some programs I wrote in highschool, or even a few years ago for programmer’s approach and wonder, how the hell did I do that (Green ink is very important not only for other programmers but for yourself)?

How do you compile your research? Is everything on the computer, or are you still a very physically oriented sort of person?

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