Tag Archives: Math

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.

KolamAttempts

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:

KitchenShelves

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:

ShelvingKitchenL1

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:

ShelvingKitchenL2

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:

ShelvingKitchenL3

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):

ShelvingKitchenL4

And so on:

ShelvingKitchenL5

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

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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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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?

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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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