## Evaporation II

Last week, we began exploring the piece Evaporation.  In particular, we looked at two aspects of the piece — randomness of both the colors and the sizes of the circles — and experimented with these features in Python.  Look at last week’s post for details!

Today, we’ll examine the third significant aspect of the piece — the color gradient.  The piece is a pure sky blue at the top, but becomes more random toward the bottom.  How do we accomplish this?

Essentially, we need to find a way to introduce “0” randomness to each color at the top, and more randomness as we move toward the bottom.  To understand the concept, though, we’ll be introducing 0 randomness at the bottom, and more as we move up.  You’ll see why in a moment.

Let’s first look at a linear gradient.  Imagine that we’re working with a $1\times1$ square — we can always scale later.  Here’s what it looks like:

The “linear” part means we’re looking at randomness as a function of $y^1=y.$  So when $y=0,$ we subtract $y=0$ randomness to each color.  But when $y=1/2,$ we subtract a random number between $0$ and $y=1/2$ from each of the RGB values.  Finally, at the very top, we’re subtracting a random number between $0$ and $1$ from each RGB value.  Recall that if an RGB value would ever fall below $0$ as a result of subtraction, we’d simply treat the value as $0.$

Why do we subtract as we go up?  Recall that black has RGB values $(0,0,0),$ so subtracting the randomly generated number pushes the sky blue toward black.  If we added instead, this would push the sky blue toward white.  In fact, you can push the sky blue toward any color you want, but that’s a little too involved for today’s post.

The piece Evaporation was actually produced with a quadratic gradient.  Let’s look at a picture first:

That the gradient is quadratic means that the randomness introduced is proportional to $y^2$ for each value of $y.$  In other words, at a level of $y=1/2$ on our square, we subtract a random number between $0$ and

$(1/2)^2=1/4.$

You can visually see this as follows.  Look at the gradient of color change from $0$ to $1/2$ for the quadratic gradient.  This is approximately the same color change you see in the linear gradient between $0$ and $1/4.$  Why does this happen?  Because when you square numbers less than $1,$ they get smaller.  So smaller numbers will introduce less randomness in a quadratic gradient than they will in a linear gradient.

We can go the other way, we well.  If we use a quadratic gradient (exponent of $2>1$), the color changes more gradually at the bottom.  But if we use an exponent less than $1$ (such as in taking a root, like a square root or cube root), we get the opposite effect:  color changes more rapidly at the bottom.  This is because taking a root of a number less than $1$ increases the number.  It’s easiest to see this with an example:

In this case, the exponent used is $0.4,$ so that for a particular $y$ value, a random number between $0$ and $y^{0.4}$ is subtracted from each RGB value.  Note how quickly the color changes from the sky blue at the bottom toward very dark colors at the top.

Of course this is just one way to vary colors.  But I find it a very interesting application of power and root functions usually learned in precalculus — using computer graphics, we can directly translate an algebraic, functional relationship geometrically into a visual gradient of color.  Another example of why it is a good idea to enlarge your mathematical toolbox — you just never know when something will come in handy!  If I didn’t really understand how power and root functions worked, I wouldn’t have been able to create this visual effect so easily.

Now it’s your turn!  You can visit the Evaporation worksheet to try creating images on your own.  If you’ve been trying out the Python worksheets all along, the code should start to look familiar.  But a few comments are in order.

First, we just looked at a $1\times 1$ square.  It’s actually easier to think in terms of integer variables “width” and “height” (after all, there is no reason our image needs to be square).  In this case, we use “j” as the height parameter, since it is more usual to use variables like “i” and “j” for integers.  So “j/height” would correspond to $y.$  This would produce a color gradient of light to dark form bottom to top.

To make the gradient go from top to bottom, we use “(height-j)/height” instead (see the Python code).  This makes values near the top of the image correspond to $0,$ and values near the bottom of the image correspond to $1.$  I’ll leave it to you to explore all the other details of the Python code.

Please feel free to comment with images you create using the Sage worksheet!

As mentioned in the previous post as well, each parameter you change — each number which appears anywhere in your code — affects the final image.  Some choices seem to give more appealing results than others.  This is where are meets technology.

As a final word — the work on creating fractals is still ongoing.  I’ve learned to make movies now using Processing:

You’ll notice how three copies of one fractal image morph into one of another.  You can find more examples on Twitter: @cre8math.  Once I feel I’ve had enough experience using Processing, I’ll post about how you can use it yourself.  The great thing about Processing is that you can use Python, so all your hard work learning Python will yield even further dividends!

## Evaporation I

This and the next post will walk you through how to create digital art similar to Evaporation.  I’ll also show you some Python code you can use yourself to experiment.

There are three significant features of Evaporation. First is the randomness of the colors. Second — if you look closely — the sizes of the circles are also different; these are randomly generated as well. The third feature is the gradient of the color — from a pure sky blue at the top, to a fairly randomly colored row of circles at the bottom. We’ll look at the first two features today.

Let’s look at color. In the figure above, the small teal square at the left has RGB values of 0, 0.5, and 0.7, respectively. The larger square at the left consists of 100 smaller squares. The color of each of these squares is generated by adding a random number between 0 and 0.1 to each of the RGB values 0, 0.5, and 0.7 — with a different random number added to each value. In the middle square, a random number between 0 and 0.2 is added, so this creates a wider range of color values. For the right square, the random numbers added are between 0 and 0.3.

But there is no reason that the ranges need to the same for each color. In the images below, the red values have a wider range of randomness then the green, which is “more random” than the blue.

You can see that different ranges of random numbers create different color “textures.” This is where I think computer meets art — as a programmer, when it comes to creating random numbers, you have to specify a range for the randomness of each variable. The choices you make determine how your image looks. How do you make “artistic” choices? There is no easy answer, of course.

Another way to use randomness is by varying the size of the graphic objects. In the left square below, texture is created by randomly changing the radii of the circles.

In the middle square, the circles are all the same size, but random shades of gray. The right square varies both the size of the circles and their color. The final result depends on “how much” randomness is used. You can try it for yourself by altering the Python code linked to below — change the randomness and see what happens!

I think of my laptop as an art laboratory. It is a place to experiment with different ideas — change this parameter, increase randomness, try out different color combinations. Can you imagine creating any of the images in this post by hand? The computer allows us to perform experiments unthinkable to Rembrandt and Van Gogh. We may lose the texture of brush strokes or the dimensionality of paint on canvas, but what we gain by having a programming language at our disposal makes up for this loss.  At least in my opinion….

Now let’s look at how we can use Python to create these images.  You can experiment with this color and texture worksheet.

There is not much more to say here since a lot is explained in the worksheet. But as you are creating, keep a few things in mind.

1. Use descriptive variable names. This helps a lot when you’re looking a lines of code. Using variables a, b, c, etc., doesn’t help much when you’re looking at a program you wrote a few months ago.

2. Comment liberally! Notes to yourself are easy to use in Python — just start a line (or part of a line) with the “\#” character. You’ll thank yourself later.

3.  Save versions often! I won’t bore you with stories of using Mathematica to do some intense computations for creating digital art — and then read the “Mathematica quit unexpectedly” message on my screen — before I saved my work. I’ve encountered this in Python, too — if you use the online version, you’re connecting to an external server, which hopefully will not encounter problems while you’re working….

Also, as you change parameters, you may want to keep track of results you like. If there are a lot of these, sometimes I write the parameters as comments — so I can reproduce them later. Very important: don’t forget to keep track of the random number seed you use! The feel and texture of an image can vary dramatically with the random number seed, so don’t ignore this vital parameter.

One final thought.  In creating this type of art, I keep in mind the tension between structure and randomness.  You can use the computer to create randomness — but if there’s too much randomness, the image doesn’t seem to hang together.  But if there’s too much structure, the image can lose its interesting texture, and appear flat and purely two-dimensional.  So the choice of parameters in creating randomness is fairly crucial in determining how the final image will look.  And as I’ve said before — this is where technology meets art.  It is fairly easy to create a computer-generated image — but not as easy to create computer-generated art.  The difference is not exactly easy to describe — and certainly opinions will differ.  It is the questions which arise in describing the difference which are intriguing and exciting.

Enough philosophizing. Time to begin the artistic process!  Feel free to comment by posting any images you create using the Python code.

## Creating Fractals III: Making Your Own

Last week, we laid down some of the mathematical foundation needed to generate fractal images.  In this third and final post about creating fractals, we’ll discuss in some detail Python code you can adapt to making your own designs.  Follow along in this Sage notebook.

In order to produce fractal images iteratively, we need a function which returns the highest power of 2 within a positive integer (as discussed last week).  It is not difficult to write a recursive routine to do this, as is seen in the notebook.  This is really all we need to get started.  The rest involves creating the graphics.  I usually use PostScript for my images, like the one below discovered by Matthieu Pluntz.  There isn’t time to go into that level of detail here, though.

As far as the graphics are concerned, it would be nice to have an easily described color palette.  You might look here to find a wide range of predefined colors, which are available once we execute the “import mathplotlib” command (see Line 20).  These names are used in the “colors” variable.  Since each motif has four segments, I’ll color each one differently (though you may choose a different color scheme if you want to).

The loop is fairly straightforward.  On each iteration, first find the correct angle to turn using the highestpowerof2 function.  Then the segment to add on to the end of the path is

$({\rm len}\cdot\cos(\theta), {\rm len}\cdot\sin(\theta)),$

which represents converting from polar to rectangular coordinates.  This is standard fare in a typical high school precalculus course.  Note the color of the segment is determined by i % 4, since 0 is the index of the first element of any list in Python.

All there is left to do is output to the screen.  We’re done!  You can try it yourself.  But note that the way I defined the function, you have to make the second argument negative (compare the image in the notebook with last week’s post).  Be patient:  some of these images may take a few moments to generate.  It would definitely help the speed issue if you downloaded Sage on your own computer.

To create the image shown above, you need to use angles of 90 and -210 (I took the liberty of rotating mine 15 degrees to make it look more symmetrical).  To create the image below, angles of 90 and -250 are used.  However, 26,624 steps are needed to create the entire image!  It is not practical to create an image this complex in the online Sage environment.

How do you know what angles to use?  This is still an open question — there is no complete answer that I am aware of.  After my first post on October 4, Matthieu Pluntz commented that he found a way to create an infinite variety of fractal images which “close up.”  I asked him how he discovered these, and he responded that he used a recursive algorithm.  It would take an entire post just to discuss the mathematics of this in detail — so for now, we’ll limit our discussion to how to use this algorithm.  I’ve encoded it in the function “checkangles.”

To use this function, see the examples in the Sage notebook.  Be careful to enter angles as negative when appropriate!  Also, you need to enter a maximum depth to search, since perhaps the angles do not result in an image which “closes up,” such as with 11 and -169.  But here’s the difficult part mathematically — just because our algorithm doesn’t find where 11 and -169 closes up does not mean that the fractal doesn’t close.  And further, just because our algorithm produced a positive result does not mean the algorithm must close.  Sure, we’ve found something that produces many results with interesting images — which suggests we’re on the right track.  But corroboration by a computer program is not a proof.

At the end of the notebook, I wrote a simple loop illustrating how you can look for many possibilities to try at once.  The general rule of thumb is that the more levels required in the algorithm to produce a pair of angles (which is output to the screen), the more segments needed to draw it.  I just looked for an example which only required 5 levels, and it was fairly easy to produce.

So where do we go from here?  Personally, I’ve found this investigation fascinating — and all beginning from a question by a student who is interested in learning more about fractals.  I’ve tried to give you an idea of how mathematics is done in the “real world” — there is a lot of exploration involved.  Proofs will come later, but it is helpful to look at lots of examples first to figure out what to prove.  When I find out something significant, I’ll post about it.

And I will admit a recent encounter with the bane of a programmer’s existence — the dreaded sign error.  Yes, I had a minus sign where I should have had a plus sign.  This resulted in my looking at lots of images which did not close up (instead of closing up, as originally intended).  Some wonderful images resulted, though, like the one below with angles of 11 and -169.  Note that since the figure does not close up (as far as I know), I needed to stop the iteration when I found a sufficiently pleasing result.

If I hadn’t made this mistake, I might have never looked at this pair of angles, and never created this image.  So in my mind, this wasn’t really a “mistake,” but rather a temporary diversion along an equally interesting path.

I’ve been posting images regularly to my Twitter feed, @cre8math.  I haven’t even touched on the aesthetic qualities of these images — but suffice it to say that it has been a wonderful challenge to create interesting textures and color effects for a particular pair of angles.  Frankly, I am still amazed that such a simple algorithm — changing the two angle parameters used to create the Koch snowflake — produces such a wide range of intriguing mathematical and artistic objects.  You can be sure that I’m not finished exploring this amazing fractal world quite yet….

## Creating Fractals II: Recursion vs. Iteration

There was such a positive response to last week’s post, I thought I’d write more about creating fractal images.  In the spirit of this blog, what follows is a mathematical “stream of consciousness” — that is, my thoughts as they occurred to me and I pursued them.  Or at least a close approximation — thoughts tend to jump very nonlinearly, and I do want the reader to be able to follow along….

Let’s begin at the beginning, with one of my first experiments.  Here, the counterclockwise turns are 80 degrees, and the clockwise turns are 140 degrees.

One observation I had made in watching PostScript generate such images was that there was “overlap”:  the recursive algorithm kept going even if the image was completely generated.  Now the number of segments drawn by the recursive algorithm is a power of 4, since each segment is replaced by 4 others in the recursive process.  So if the number of segments needed to complete a figure is not a power of 4, the image generation has to be stopped in the middle of a recursive call.

This reminded me of something I had investigated years ago — the Tower of Hanoi problem.  This is a well-known example of a problem which can be solved recursively, but there is also an iterative solution.  So I was confident there had to be an iterative way to generate these fractal images as well.

I needed to know — at any step along the iteration — whether to turn counterclockwise or clockwise.  If I could figure this out, the rest would be easy.  So I wrote a snippet of code which implemented the recursive routine, and output a 0 if there was a counterclockwise turn, and a 1 if there was a clockwise turn.  For 2 levels of recursion, this sequence is

0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0.

The ones occur in positions 2, 6, 8, 10, and 14.

I actually looked at 1024 steps in the iteration, and noticed that the ones occur in exactly those positions whose largest power of 2 is odd.  Each of 2, 6, 10, and 14 has one power of 2, and 8 has three.

You might be wondering, “How did you notice that?”  Well, the iterative solution of the Tower of Hanoi does involve looking at the powers of 2 within numbers, so past experience suggested looking along those lines.  This is a nice example of how learning neat math can enlarge your mathematical “toolbox.”  You never know when something might come in handy….

There was other interesting behavior as well — but it’s simpler if you just watch the video to see what’s happening.

First, you probably noticed that each of the 18 star arms takes 32 steps to create.  And that some of the star arms — eight of them — were traversed twice.  This means that 18 + 8 = 26 arms were drawn before the figure was complete, for a total of 832 steps.  Note that the recursive algorithm would need 1024 steps to make sure that all 832 steps were traversed — but that means an overlap of 192 steps.

Now let’s see why some of the arms were traversed twice.  The 32nd step of the first arm is produced after 31 turns, and so the 32nd turn dictates what happens here.  Now the highest power of 2 in 32 is 5, which is odd – so a clockwise turn of 140 degrees is made.  You can see by looking at the first 33 steps that this is exactly what happens.  The 32nd step takes you back to the center, and then a 140 degree clockwise turn is made.

Now after the next arm is drawn, the turn is determined by the 64th angle.  But 6 is the highest power of two here, and so an 80 degree counterclockwise turn is made — but this takes you over the same arm again!

Note that we can’t keep traversing the same arm over and over.  If we add 32 to a number whose highest power of 2 is 6:

$2^6m+2^5=2^5(2m+1),$

we get a number whose highest power of 2 is 5 again (since 2m + 1 must be odd).  Since this power is odd, a clockwise turn will be made.

So when do we repeat an arm?  This will happen when we have a counterclockwise turn of 80 degrees, which will happen when the highest power of 2 is even (since an odd power takes you clockwise) — when looking at every 32nd turn, that is.  So, we need to look at turns

32, 64, 96, 128, 160, 192, 224, etc.

But observe that this is just

32 x (1, 2, 3, 4, 5, 6, 7, etc.).

Since 32 is an odd power of two, the even powers of two must occur when there is an odd power of 2 in 1, 2, 3, 4, 5, 6, 7, etc.  In other words, in positions 2, 6, 8, 10, 14, etc.

To summarize this behavior, we can state the following simple rule:  arms move seven points counterclockwise around the circle, except in the case of the 2nd, 6th, 8th, 10th, 14th, etc., arms, which repeat before moving seven points around.  Might be worth taking a minute to watch the video again….

We can use this rule to recreate the order in which the star arms are traversed.  Start with 1.  The next arm is 1 + 7 = 8.  But 8 is the 2nd arm, so it is repeated — and so 8 is also the third arm.  The fourth arm is 8 + 7 = 15, and the fifth is seven positions past 15, which is 4.  Mathematically, we say 15 + 7 = 4 modulo 18, meaning we add 15 and 7, and then take the remainder upon dividing by 18.  This is know as modular arithmetic, and is one of the first things you learn when studying a branch of mathematics called number theory.

The sixth arm is 4 + 7 = 11, which is repeated as the seventh arm.  You can go on from here….

There are still some questions which remain.  Why 32 steps to complete an arm?  Why skip every seventh arm?  Why are the arms 20 degrees apart?  These questions remain to be investigated more thoroughly.  But I can’t stress what we’re doing strongly enough — using the computer to make observations which can be stated mathematically very precisely, and then looking at a well-defined algorithm to (hopefully!) prove that our observations are accurate.  More and more — with the advent of technology — mathematics is becoming an experimental science.

I’ll leave you with one more video, which shows PostScript creating a fractal image.  But laying a mathematical foundation was important — so next week, we can look at how you can make your own fractals in Python using an iterative procedure.  This way, you can explore this fascinating world all on your own….

There are 10 levels of recursion here, and so 1,048,576 segments to draw.  To see the final image, visit my Twitter feed for October 9.  Enjoy!

## Creating Fractals

Recently, I’ve been working with a psychology student interested in how our brains perceive fractal images in nature (trees, clouds, landscapes, etc.). I dug up some old PostScript programs which reproduced images from The Algorithmic Beauty of Plants, which describes L-systems and how they are used to model images of plants. (Don’t worry if you don’t have the book or aren’t familiar with L-systems — I’ll tell you everything you need to know.)

To make matters concrete, I changed a few parameters in my program to produce part of a Koch snowflake.

The classical way of creating a Koch snowflake is to begin with the four-segment path at the top, and then replace each of the four segments with a smaller copy of this path. Now replace each of the segments with an even smaller copy, and recurse until the copies are so small, no new detail is added.

Algorithmically, we might represent this as

F  +60  F  -120  F  +60  F,

where “F” represents moving forward, and the numbers represent how much we turn left or right (with the usual convention that positive angles move counter-clockwise). If you start off moving to the right from the red dot, you should be able to follow these instructions and see how the initial iteration is produced.

The recursion comes in as follows: now replace each occurrence of F with a copy of these instructions, yielding

F  +60  F  -120  F  +60  F  +60

F  +60  F  -120  F  +60  F  -120

F  +60  F  -120  F  +60  F  +60

F  +60  F  -120  F  +60  F

If you look carefully, you’ll see four copies of the initial algorithm separated by turning instructions. If F now represents moving forward by 1/3 of the original segment length, when you execute these instructions, you’ll get the second image from the top. Try it! Recursing again gives the third image, and one more level of recursion results in the last image.

Thomas thought this pretty interesting, and proceed to ask what would happen if we changed the angles. This wasn’t hard to do, naturally, since the program was already written. He suggested a steeper climb of about 80 degrees, so I changed the angles to +80 and -140.

Surprise! You’ll easily recognize the first two iterations above, but after five iterations, the image closes up on itself and creates an elegant star-shaped pattern.

I was so intrigued by stumbling upon this symmetry, I decided to explore further over the upcoming weekend. My next experiment was to try +80 and -150.

The results weren’t as symmetrical, but after six levels of recursion, an interesting figure with bilateral symmetry emerged. You can see how close the end point is to the starting point — curious. The figure is oriented so that the starting points (red dots) line up, and the first step is directly to the right.

Another question Thomas posed was what would happen if the lengths of the segments weren’t all the same. This was a natural next step, and so I created an image using angles of +72 and -152 (staying relatively close to what I’d tried before), and using 1 and 0.618 for side lengths, since the pentagonal motifs suggested the golden ratio. Seven iterations produced the following remarkable image.

I did rotate this for aesthetic reasons (-24.7 degrees, to be precise). There is just so much to look at — and all produced by changing a few parameters in a straightforward recursive routine.

My purpose in writing about these “fractal” images this week is to illustrate the creative process in doing mathematicsThis just happened a few days ago (as I am writing this), and so the process is quite fresh in my mind — a question by a student, some explorations, further experimentation, small steps taken one at a time until something truly wonderful emerges.  The purist will note that the star-shaped images are not truly fractals, but since they’re created with an algortihm designed to produce a fractal (the Koch snowflake), I’m taking a liberty here….

This is just a beginning! Why do some parameters result in symmetry? How can you tell? When there is bilateral symmetry, what is the “tilt” angle? Where did the -24.7 come from? Each new image raises new questions — and not always easy to answer.

Two weeks ago, this algorithm was collecting digital dust in a subdirectory on my hard drive. A simple question resurrected it — and resulted in a living, breathing mathematical exploration into an intensely intriguing fractal world. This is how mathematics happens.

## Hexominoes and Cube Nets

I have always been fascinated by polyominoes — geometrical shapes made by connecting unit squares edge to edge. (There’s a lot about polyominoes online, so take a few moments to familiarize yourself with them if they’re new to you.)

Today I’ll talk about hexominoes (using six unit squares), since I use them in the design of my current website.  There are a total of 35 hexominoes — but I didn’t want all of them on my home page, since that seemed too cluttered. But there are just 11 hexominoes which can be folded into a cube — I did want my choice to have some geometrical significance! These are called nets for a cube, and formed a reasonable subset of the hexominoes to work with. Note that the count of 11 nets means that rotating or turning over a net counts as the same one. (And if you want an additional puzzle — show that aside from rotating or reflecting, there are just 11 nets for a cube.)

Now how should I arrange them? I also wanted to use the hexominoes for a background for other pages, so I thought that if I made a 6 by 11 rectangle with them, that would be ideal — I could just tile the background with rectangles.

This is not possible, however — I wrote a computer program to check (more later). But if you imagine shifting a row of the 6 by 11 rectangle one or two squares, or perhaps a column — you would still occupy 66 square units, and the resulting figure would still tile the plane. This would still be true if you made multiple row/column shifts.

So I wrote a program which did exactly that — made random row and column shifts of a 6 by 11 rectangle, and then checked if the 11 hexominoes tiled that figure. After several hours of running, I found one — the one you see on my home page. If you look carefully, you can see the row and column shifts for yourself.

Is this the only possibility? I’m not sure, but it’s the only one I found — and I liked the arrangement enough to use it on my website. If you look at some of the other pages — like one of my course websites — you’ll see a smaller version of this image tiling the background. However, to repeat the pattern in the background, I needed to make a “rectangular” version of the image:

The colors are muted since I didn’t want the background to stand out too much.  And you’ll notice that some of the hexominoes leave one edge of the rectangle and “wrap around” the opposite edge.  But if you look closely, you can definitely find all 11 hexominoes in this 6 by 11 rectangle.

This wasn’t my first adventure with hexominoes — a few years ago, I created a flag of Thailand since I was doing some workshops there. Flags are generally rectangular in shape.

But you can’t create a rectangle with the 35 hexominoes! Let’s see why not. Imagine a rectangle on a checkerboard or chessboard. When you place a hexomino, it will cover some black squares and some white squares.

Now some hexominoes will always cover an odd number of black and white squares — let’s call those odd hexominoes. The others — even hexominoes — cover an even number of black and white squares. As it turns out, there are 24 odd hexominoes and 11 even hexominoes. This means that any placement of all the hexominoes on a checkerboard will cover an even number of white squares and an even number of black squares.

However, any rectangle of 210 = 6 x 35 hexominoes must cover 105 white squares and 105 black squares — both odd numbers of squares. But we just saw that’s not possible — an even number of each must be covered. So no rectangles. This is an example of a parity argument, by the way, and is a standard tool when proving results about covering figures with polyominoes.

To overcome this difficulty, I threw in 6 additional unit squares so I could make a 12 x 18 rectangle — and to my surprise, I found out that the flag of Thailand has dimensions 2:3 as well. You can read more about this by clicking on “the flag of thailand” on the page referenced above — and see that the tiling problem can be solved with a little wiggle room. But no computer here — I cut out a set of paper hexominoes and designed the flag of Thailand by hand….