Artwork Endeavor Five Year Anniversary

Drawing has been an on-and-off hobby of mine since, well, I can remember. When we were little, my brother and I used to spread huge pieces of moving paper on the living room coffee table and create mural-like drawings.

Thinking back, I’m sure some of them were inspired by Mark Kistler’s Secret City drawings. We were sometimes shown these videos in school.

 

[youtube=http://www.youtube.com/watch?v=4tK70tHKhME&w=400]

 

 

I always liked to doodle every so often, but for the most part, I considered my brother to be the family artist.

Five years ago my wife and I moved away from Chapel Hill, NC. She had just finished her degree, and since I was still attending NCSU, we moved closer to Raleigh.

With all of our stuff in boxes, all I had to entertain myself were a couple old boxes of colored pencils (the date stamped on the box goes back to the Clinton administration).

I was working on this project called History of the Wiener Dog, which would become An Orthogonal Universe. I had just become stuck on what would become A Foundation in Wisdom, when I had an epiphany. Why don’t I try drawing out the scenes I’m writing?

With that, I immediately got started procrastinating. Below is the very first drawing I completed in summer 2008.

 

Helicopters

R: You look real goofy driving that old thing.
L: Yeah? Well, that’s the most hideous fedora!

 

The above drawing has nothing to do with the plot of An Orthogonal Universe, but it did have one significance. I would soon start making a real effort to improve as an artist.

I took my old Clinton-era colored pencils and colored Ry, the Squirrel.

 

Ry the Squirrel

Poorly drawn Ry the Squirrel!

 

There was a novelty in seeing some of the characters I spent the past year writing about “come to life” in a visual form. Maybe that’s why I kept going, even after I broke through the writer’s block.

 

Start of Artwork Endeavor

 

I consider the image below to be the first drawing that came after a point I said to myself, “hey, I think I’d like to give improving a shot.”

This is a scene from An Ember in the Wind, the sequel to A Foundation in Wisdom.

 

Poorly drawn Mara

Mara and the Puppets! Mara has always had a Wolfpack Red tunic. The colors of Locana, the city she lives in, are Carolina Blue.

 

Eh, that wasn’t very good. The “puppets picture” is an important one, though. Every so often I redo it. The problem with learning any skill, especially if you’re self-teaching, is that progress seems to come so slowly. Improvement is measured in months and years, which is problematic if you get discouraged early on.

Below is the last colored pencil drawing I ever did. It’s John Bartlebee and Sheridan, the protagonists of the An Orthogonal Universe series.

 

LeSabre '88

The stars of An Orthogonal Universe: John Bartlebee and Sheridan! John drives an ’88 Buick LeSabre. John operates his school of the classical sort out of his classic car.

 

Sometime later I realized I could scan my line-art drawings and color them on the computer. I sat down with my copy of The GIMP, and griatch-art’s tutorials, and gave it a go.

Hey, it’s the puppets picture again!

 

Mara 2

Slightly better Mara and the Puppets. This is the very last drawing I did with a mouse.

 

2009 – 2010

 

My wife got me a Wacom Bamboo pen for Christmas 2009. Once I got the hang of it, it made a huge difference.

But, starting in 2009, my free time dwindled considerably. In Fall 2008 I finished the last of the core coursework in my degree program, and began working with my research advisor.

Qualifier exams are notorious in just about any Ph.D. program. Despite their infamy, the workload only goes up once they’re over and done with. At least at NC State, though, the pressure goes down.

The qualifiers make up the last “gateway”, after which, few people leave the program. Still, I didn’t have much time for anything other than quick doodles. Here’s an illustration of Marcus, the protagonist of A Foundation in Wisdom.

 

Marcus

An old illustration of what is now chapter 12 from A Foundation in Wisdom. Marcus is contemplating the highway.

 

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2011 – today

 

I graduated in May 2010. After my first semester as a full-time professor, I began to have more free time.

Everything except my degree program was put on hold until I graduated. Afterward, I came up with the idea of an illuminated hypertext novel. I started creating a lot of full-color illustrations for A Foundation in Wisdom, one for each chapter. This version had 30 chapters.

Of the set, the image below is one of my favorites.

 

Eru

Chapter 17 of A Foundation in Wisdom. Marcus climbs the Mount of Mislor to meet with Eru, the wise man.
Random trivia: “Mislor” was my mom’s AOL username back in the mid 90’s. I think my brother came up with it.

 

If you’ve read the book, some of these scenes may look familiar. The one below won’t, unless you read the 2011 version.

 

Peoria

Peoria from A Foundation in Wisdom. This was an illustration of a scene in which Marcus and Peoria went inside a leaf. The scene was deleted from the final manuscript.

 

Vasigari

Vasigari, the Priestess from A Foundation in Wisdom. Poor Marcus – do legs bend like that? I still like that forest.

 

Like I said earlier, learning a skill can be frustratingly slow, unless you’re the patient type.

I like to think I’m of the “patient type,” at least, most of the time I am. But, sometimes, it’s easy to look at the work of people who have been at it much longer than you – and forget just that – that they’ve been at it longer.

That’s why I sometimes redo the “puppets picture.” It was the first illustration. This is the last version, and already it’s over a year old.

 

Mara 3

Mara and the Puppets – the latest version! Recognize the machinery?
Fun fact: The desk and lamp is a reference to the attic in Alone in the Dark.

 

It was completed in March 2012 (although it’s dated 1 April). At this point, I was still planning on releasing An Ember in the Wind as an “illuminated hypertext novel.” So I needed to redo this illustration for the new site, anyway.

This was also the last illustration I created with the intent of releasing An Ember in the Wind as an illuminated hypertext novel.

For various reasons, I pulled the project. A couple months later, I re-declared A Foundation in Wisdom as a novel, and began working with editor Kisa Whipkey to polish it.

While I was waiting for the manuscript to come back, I took requests from random people on deviantART.

 

 

superheroes

Someone on DeviantART asked me to draw her and her friend as superheroes. That’s a futuristic San Francisco in the background.

 

And, just because I felt bad for John Bartlebee and Sheridan, I redid their scene.

 

LeSabre 2

John Bartlebee and Sheridan make their return after 4.333 years. That’s a much nicer looking LeSabre!

 

Now we get to this year! Although I’m not doing illustrations for an “illuminated hypertext novel” anymore, I still enjoy drawing. Up above I mentioned griatch-art’s tutorials. Below is a scene depicting one of his characters.

 

Biltmore

griatch-art@deviantART asked people to draw his characters. That’s Ebb the Dragon, about to make a snack of the Biltmore estate.

 

We’re coming to the end! After nearly five years, I figured it was about time to try drawing myself. I tried using a mirror as a reference, but I kept getting myself backward.

Thinking about what happens next in a story can be hard work. Despite all those monitors, I still primarily use 15-cent Walmart notebooks.

 

self

Incomplete self portrait. Shown: me, hard at work revising A Foundation in Wisdom, while in actuality, putting the real task off.

 

And now, the latest illustration – completed five years after I drew the very first doodle in my notebook.

Even though I’m not working on an “illuminated hypertext novel,” I haven’t lost sight of why I started drawing these scenes in the first place. Drawing the scene out is a great way to break through writer’s block. And since I’m working on An Ember in the Wind, well, here’s Mara again.

 

Forest

Five years later, I finally drew a forest I’m happy with.
Mara still has her Wolfpack Red tunic. Go State!

 

After five years of self-teaching, I’m finally pleased with how I’m doing. As for the illustrations, I may have found a use for them.

I don’t think my experiments with the “hybrid novel”, or “illuminated hypertext” are done. But utility aside, sometimes it’s just fun to draw.

 

The World of Fractal Art for People Who Hate Math

Fractal Monster

Figure 1. Common metaphor for a final exam in mathematics.

 

Or, Fractal Art Using High School Math.

This week’s post is especially for those of you who never saw math as much more than a means to calculate when trains will meet each other. Many of us who didn’t appreciate mathematics in grade school tend to as we get older. Even so, it’s easy to overlook that there’s more to mathematics than its use as a dry, practical tool.

Fractal art is a relatively recent fusion between art and mathematics. It’s a world I often find myself sharing with my introductory-level classes (there are some amazing visuals out there) – but it’s not easy.

Look up the Wikipedia entry on how to generate art using the Mandelbrot set. The math gets to a high level very quickly. This is too bad, because fractal art has a large following. Communities like deviantART have lots of people who have learned how to generate images using software like Apophysis. It turns out, though, some to many enthusiasts, or fans of fractal art, may not have seen the math behind the software. Any artist should understand how their tools work. I’d expect an oil painter to understand how the dyes in their paints work. “Fractal artists” shouldn’t be any different.

Fractal art is often cited as an example of the practical use of complex numbers. There aren’t many. Look up “complex numbers in the real world,” and you’ll often see two examples: quantum mechanics, and electrical engineering – two fields that aren’t exactly elementary. And then you’ll see fractal art tagging along.

Complex numbers themselves can be unwieldy. Understanding how fractal art is generated will, in many cases, involve understanding convergence and divergence of sequences… and suddenly we’re getting to levels of calculus beyond what the average college student takes, let alone high school.

But it’s really not that hard. If you can multiply and add numbers, and make lists, you can make fractal art “by hand.” Or, at least, you can understand what the computer is doing.

There’s not a single procedure involved in generating fractal art, so we’re going to look at one example. Let’s make pretty, pretty pictures using the Mandelbrot set.

Okay, let’s get this party started! ~

The piece of paper we want to draw on has two dimensions, and complex numbers form a two-dimensional number system. If you were looking for a reason why fractal art relies on complex numbers, this would be it.

The dimensions in mathematics can be generalized beyond “length” and “height”, but since we want to keep things simple, we’ll stick to that.

First, in case you’ve never seen one before, here’s a complex number:

$latex 3 + 2i &bg=e6eaea&s=1$

$latex i &bg=e6eaea$ is the imaginary unit, defined so that $latex i^2 = -1 &bg=e6eaea$. This means we can take $latex i = \sqrt{-1} &bg=e6eaea$.

Hence, the number $latex 3 + 2i &bg=e6eaea$ has two parts: A real part (in this case, $latex 3 &bg=e6eaea$), and a multiple of $latex i &bg=e6eaea$ (in this case, $latex 2 &bg=e6eaea$), called the imaginary part. The two parts are added together to form a complex number.

The number line that you may remember from grade school associates numbers with a point set some distance from zero. Since complex numbers have two parts, the real part can represent horizontal distance (lengthwise), and the imaginary part can represent vertical distance (height-wise), from the center on the complex number plane.

complex number plane

3 + 2i

Time to make some art! ~

This is the image we’re going to make:

Mandelbrot setBut it currently looks like this:

Empty rectangle

Since we want to color all the pixels in the rectangle, and pixels are located by horizontal and vertical distance from zero, what we really have is a set of complex numbers.

The horizontal position of the pixel is the real part, and the vertical position of the pixel is the imaginary part.

For each pixel, we need to determine what color to light it. This is the hardest part to explain, because it would normally involve calculus. But there is another way to explain it.

Imagine drawing a circle on a concrete floor and dropping a golf ball in its center. It would bounce around a lot, and most likely bounce out of the circle. Every now and then it may bounce a lot, but still stay inside the circle. We’re going to create a “golf ball.” We’re interested in the number of times it bounces before it leaves the circle.

Actual math! ~

Even though this is titled, “…for People Who Hate Math,” we do need at least some math. Since the goal is to stay within the realm of high school mathematics, we can use an old friend: FOIL. Skip this section if you don’t need the refresher.

You only need to know how to do two things: add complex numbers, and multiply them.

Adding them is easy: you simply add their real and imaginary components.

Example: $latex (3 + 2i) + (2 + 4i) &bg=e6eaea$

The two real parts are 3 and 2. Add them together to get 5.

The two imaginary parts are 2 and 4. Add them together to get 6.

So the result is $latex 5 + 6i &bg=e6eaea$

Multiplying is not too bad, either. FOIL is an acronym for “First, Outer, Inner, Last,” which tells you which pairs of numbers to multiply.

Example: $latex (3 + 2i) \times (2 + 4i) &bg=e6eaea$

The two parts that come “first” in both of the complex numbers are 3 and 2. Multiply them to get 6.

The two parts that are on the “outer” are 3 and 4i. Multiply them to get 12i.

The two parts that are in the “inner” are 2i and 2. Multiply them to get 4i.

The two parts that are “last” are 2i and 4i. Remember, we’re multiplying both the numbers and the i’s. So the product is $latex 8i^2 &bg=e6eaea$. Since $latex i^2 = -1 &bg=e6eaea$, then $latex 8i^2 = -1 \times 8 = -8 &bg=e6eaea$.

So now we have the four products we need to add: 6, 12i, 4i, and -8.

Remember from above that we add complex numbers by adding their real parts and imaginary parts. So the 6 and -8 are added together to get -2. The 12i and 4i are added together to get 16i.

The result: $latex -2 + 16i &bg=e6eaea$.

Okay, great… so what? ~

Let’s go back to the bouncing golf ball. Our goal is to calculate where it lands on each bounce. We’ll use the number of bounces to determine what color to light the pixel that is currently marked by the red dot below.

Red dot

Out damn spot, out I say!

That red dot is located .45 units right of the center, and .8 units up. This means the red dot is represented by the complex number $latex .45 + .8i &bg=e6eaea$.

We’ll use the expression $latex z^2 + .45 + .8i &bg=e6eaea$ to calculate where the ball lands. Notice that $latex .45 + .8i &bg=e6eaea$ is the location of the pixel we’re lighting.

First, drop the ball at the dead center of a circle of radius 2. The ball’s position, z, is currently $latex 0 + 0i &bg=e6eaea$, or 0 right, 0 up:

Red dot at center

Invisible to people named “Costanza”.

We’ll substitute $latex 0 + 0i &bg=e6eaea$ for z in our expression. So the ball will land at:

$latex (0 + 0i)^2 + .45 + .8i &bg=e6eaea&s=1$

Since $latex 0^2 = 0 &bg=e6eaea$, this time we can ignore multiplying (FOIL).

$latex 0 + 0i + .45 + .8i &bg=e6eaea&s=1$

$latex = .45 + .8i &bg=e6eaea&s=1$

Boing!

Boing!

Looks like it’s still in the circle. Let’s do another bounce. The ball is currently located at $latex .45 + .8i &bg=e6eaea$. Substitute that number for z in the same expression as we used for the first bounce:

$latex (.45 + .8i)^2 + .45 + .8i &bg=e6eaea&s=1$

$latex = (.45 + .8i) \times (.45 + .8i) + .45 + .8i &bg=e6eaea&s=1$

This time, the computation won’t be so pretty. We can rely on FOIL to do the multiplication:

Product of the “firsts”: $latex .45 \times .45 = .2025 &bg=e6eaea&s=1$

Product of the “Outer”: $latex .45 \times .8i = .36i &bg=e6eaea&s=1$

Product of the “Inner”: $latex .45 \times .8i = .36i &bg=e6eaea&s=1$

Product of the “Last”: $latex .8i \times .8i = .64i^2 = -.64 &bg=e6eaea&s=1$

Put all of this back into the problem above:

$latex .2025 + .36i +.36i – .64 + .45 + .8i &bg=e6eaea&s=1$

$latex = .0125 + 1.52i &bg=e6eaea&s=1$

Red dot 3

Wheee!

We’re getting close to the end it seems. One more bounce might do it. I’ll spare you all the details of the arithmetic this time. The current position of the ball is $latex .0125 + 1.52i &bg=e6eaea$. Substitute that into the same expression we’ve been using for all our bounces:

$latex (.0125 + 1.52i)^2 + .45 + .8i &bg=e6eaea&s=1$

$latex = -1.86 + .84i &bg=e6eaea&s=1$

Red dot 4

Freedom!

Finally! It took 3 bounces. So 3 is the magic number.

What we do with the number 3 is use it to determine a color for the pixel located at $latex .45 + .8i &bg=e6eaea$.

If you’ve used software which lets you select a gradient to determine color, this is what the gradient is used for. The number maps to a location on the gradient, and the pixel is given that color.

We’ll use a simple grayscale. In the grand scheme of things, 3 is pretty small. Low numbers correspond to dark spots, so we have our pixel:

Colored dot.

All that work just to figure out the color of this one teeny, tiny spot. Good job, team!

To fill in the rest of the pixels, we’d repeat the ball bouncing process for each pixel location. The tricky thing to remember is that the location of the pixel changes the expression that is used – that is, the procedure we use to calculate where the ball lands on each bounce. So to determine what color to light the pixel at $latex .1 + .1i &bg=e6eaea$, we would use the expression $latex z^2 + .1 + .1i &bg=e6eaea$ instead of $latex z^2 + .45 + .8i &bg=e6eaea$.

In some circumstances, the ball may never bounce out of the circle. If we count enough bounces, eventually we just assume it never leaves. If you have fractal software, check your list of settings for “max iterations.” This is what that value is referring to – the threshold at which we say “the ball will never leave the circle.” In our color scheme, this would correspond to a white dot.

And so, if we repeat the procedure for every single pixel (this is why computers are handy), we get:

Mandelbrot Set

Ta-da!

What next? ~

Well, this is a pretty simple image – but it already highlights a few way you can customize it. Notably, changing the max number of iterations, or the gradient, will have an immediate effect. You can also change the “bounce expression.” Instead of using $latex z^2 &bg=e6eaea$, why not $latex z^3 &bg=e6eaea$? What about a circle of radius 3 instead of 2?

This is also only one type of fractal. Another example could use the Julia set.

Here are some toys if you want to try your hand at generating some fractal art:

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Be sure to check out their galleries!

A note on fractal art: ~

Fractal art, like digital art in general, may be looked down upon by some traditionalists. Regardless of whether or not it’s “art,” if it’s even worth debating, it’s hard to argue the results can’t be impressive. Here are some examples by people with a lot more patience than I have with this medium.

Fractal Galleries

deviantART

Fractal World Gallery

WU-Gallery

Have a comment or a piece of artwork you’d like to share? Please comment below!

Riddle for June 2013

I’d like to start a new feature on this blog: The Orthogonal Universe Riddle of the Month. A lot of these riddles may be mathematics or logic based, but you shouldn’t need anything more than basic arithmetic or geometry to solve them.

To make this more interesting, I’ve provided an answer submission form. If you think you’ve solved the puzzle, submit your answer along with your name and website. At the end of the month, I’ll announce the correct answer along with a “winners list”.

 

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June 2013.

A man bumps into his mathematician friend on the street that he hasn’t seen in quite some time. The man asks the mathematician how old his children are. The mathematician, who always replies in riddles said, “I now have three children. The sum of their ages is equal to the number of windows on the building in front of you and the product of their ages equals 36.” The friend then says, “I need one more piece of information.” The mathematician then replies. “My youngest child has blue eyes.” What are the ages of the mathematician’s three children?