Mathematics and Digital Art: Update 2 (Fall 2017)

We’ve just completed Week 8 of the Fall semester, so it’s time for the next update on my Mathematics and Digital Art class!  As I had mentioned before, the major difference this semester was starting with Processing right from the beginning of the semester.

It turns out this is making a really big difference in the way the class is progressing.  The first two times I taught the course, I had students work in the Sage environment for the first half of the semester.  The second half of the semester was devoted to Processing and student projects.

Because students only started to learn Processing at the same time they were diving into their projects, they were not able to start off with a Processing-based project.  As it happened, a few students actually incorporated Processing into their final projects as the second half of the semester progressed, but this was the exception, not the rule.

But last week, we already started making movies in Processing!  Starting simply, of course, with the dot changing colors.

 

This was a bit easier to present this time around, since we already had a discussion of user space vs. screen space earlier in the semester.  So this time, I could really focus on linear interpolation — the key mathematical concept behind making animations.

Next week will be a Processing-intense week.  I’ll delay some topics — like geometric series — to a little later in the course so we can get more Processing in right now.  The reason?  I really think many students will involve Processing in their final projects in a significant way.  I want to make sure they have enough exposure to feel confident about going in that direction for their final projects.  I’ll let you know what happens in this regard in my next update of Mathematics and Digital Art.

Now for some examples of student work!  For the assignment on iterated function systems, students had three different images to submit.  The first was a Sierpinski triangle — I asked students to create an image simultaneously as close to and as far away from a Sierpinski triangle as possible.  The idea was that a viewer should recognize the image as being based on a Sierpinksi triangle, but perhaps only after staring at it for thirty seconds or so.

This is Sepid’s take on the assignment.  On many of her pieces, she experimented with different ways to crop the final image.  This has a significant effect on the image’s final appearance.

Day115Sepid1.png

 

This is Cissy’s submission for the Sierpinski triangle.  In this piece (and the others submitted for this assignment), Cissy remarked that she really enjoyed experimenting with color.  I commented that I thought color choices were among the most difficult decisions to make as far as elements of a work of digital art are concerned.

Day115Cissy1.png

The second piece was to involve only two affine transformations.  This is often a challenge for students, but there really is an enormous variety of images that may be created using just two transformations.  In addition, one of the transformations needed to involve a rotation by a non-trivial angle (that is, not a multiple of 45°), and students needed to submit a picture of their calculations as well.

One student was trying to create an image that looked like an animal footprint.  She remarked that she did consider a different color palette, but in the end, preferred to go with monochromaticity.
Day115L2.png

Interestingly, Terry also used a simple color palette.  She remarked that it was challenge to use just two transformations — and because of this minimalist requirement, decided to go with a minimalist color palette.  In addition, her resulting fractal reminded her of birds, so she set the fractal against a white moon and gray sky.

Day115Terry2

For the third submission, there were no constraints whatsoever — in fact, I encouraged students to be as creative as possible.  There was a very wide range of submissions.  One student was fairly minimalistic, using a highly contrasting color palette.

Day115A

Jack’s piece was also fairly minimalistic.  I should remark that we took part of a lab one day for students to do some online peer commenting; Jack (and others as well) remarked that he used the advice of another student to improve an earlier draft of his piece.  In particular, he adjusted the stroke weight to increase the intensity of the colors.

Day115Jack.png

Tera based her work on the Sierpinski triangle,  but also included reflections of each of the three smaller components of her version of the Sierpinski triangle.  She remarked that the final image reminded her of a snowflake, or perhaps a Christmas sweater.

Day115Tera.png

Alex’s inspiration came from The Great Wave off Kanagawa, by Katsushika Hokusai.

The_Great_Wave_of_Kanagawa.jpg
Courtesy the Wikipedia Commons.

 

First, he created the fractal image, experimenting with various color combinations.  When he was satisfied with his palette, he added the boat and the white circle to suggest a black moon.  A rather interesting approach!

Day115Alex.png

As you can see, I’ve got quite a creative class of students who are willing to experiment in many different ways.  It’s interesting for me, since there is no way to predict what they’ll create next!  I look forward to seeing what they create when they really dive deeply into Processing and begin making animations.

In the next update, I’ll report on how students involve Processing in their project proposals.  In addition, they will have submitted their fractal movie projects by then, so there will undoubtedly be many interesting examples of student work to exhibit.  Stay tuned!

Beguiling Games II: Nuh-Uh!

First, we’ll look at the game I left you with in the last installment of Beguiling Games.  These were the rules of yet another variation of Tic-Tac-Toe:  if during the game either play gets three-in-a-row, then X wins.  If at the end, no one has three in a row, then O wins.  Does X have a winning strategy?  Does O?  Why?

We’ll show that X has a winning strategy.  As with usual Tic-Tac-Toe, X starts at the center.  If O goes in any square which is not a corner, then X will win as a result of the winning strategy in Tic-Tac-Toe.  So all that’s left to do is look at the case when O goes in a corner.  Let’s suppose it’s the upper right corner.

Then X goes in the lower right corner.  O must block, or else X will get three-in-a-row and win (since X wins if either player gets three-in-a-row).  So now, the board looks like this:

Day114a

Take a moment to see if you can figure out X’s winning strategy.  Do you see it?  All X needs to do is place in the center of the bottom row.

Day114b

Look at the top row.  Someone eventually has to place an X or O in this location.  No matter which one, a three-in-a-row will be created, and X will win!  So in this version of Tic-Tac-Toe, X has a winning strategy.

Today’s game will involve a different dynamic — cards and logic.  But first, let me give a little bit of context.  I have often used the book Problem Solving in Recreational Mathematics by Averbach and Chien when I’ve taught an introduction to proofs course.  There are many interesting problems in this book, and the chapter on logic has several problems involving Truthtellers and Liars.

So you have to solve problems by figuring out who are the Truthtellers and who are the Liars.  I like to use these problems to get students writing arguments in complete sentences since not only are they fun to work out, but they don’t involve notation.  Using notation correctly is another issue entirely, and I prefer to deal with that later on in the course.

As before, there’s a story to go along with the game….

Lucas, Mordecai, Nancy, and Ophelia decided that enough was enough, and so ended their Saturday afternoon’s cramming for the National Spelling Bee.  Wondering what they might do to pass the time, Lucas suggested, “Let’s play Nuh-Uh!”  Everyone enthusiastically agreed.

And so a game of Nuh-Uh! — whose creators, incidentally claim that it is the only game on the market which “Tortures your Mind, Warps your Character, and Impoverishes your Soul — all at the Same Time!” — commenced.

So Lucas shuffled the deck of eight cards; four of the cards had the word Truthteller on them, and four had the word Liar.  Dealing from the left, Lucas first dealt a card to Mordecai, then one to Nancy, one to Ophelia, and then finally one to himself.  Cards are dealt face down so a player can only see his or her card.

Per the rules of Nuh-Uh!, Lucas, then Mordecai, Nancy, and finally Ophelia made the statement “I am a Truthteller.”  Of course, such a statement was consistent with each player’s card, regardless of what was written on it.  Thus ended the first round.

In subsequent rounds, each player passes his or her card to the player on the left, and makes a statement consistent with the new card.  Thus, if Lucas passed Mordecai a card with Liar written on it, Mordecai would have to make a statement which is false.  The statement a player makes is based on the knowledge of the cards he or she has seen, and any other information which may be deduced from the statements of the previous players.

The same player begins each round of making statements.  So Lucas began each round in this game.  Once the game is over, the deal passes to the left, so Mordecai would deal and begin each round with a statement.

After a player makes a statement, players may, beginning with the player on the left of the one who just spoke, “Declare” — that is, say what card each of the players held during the first round.  The declaring player then looks at the cards held by the players to decide if he or she has won.  If an error is made in declaring, the player drops out and play continues; otherwise the cards are turned over and the deal passes to the next player on the left.

So the players passed their cards to the left, and thus the second round of making statements began.  Lucas started off with “Ophelia told the truth in the first round.”  After a polite few seconds of pause to give someone a chance to declare, Mordecai said, “Lucas also told the truth in the first round.”

Another brief pause ensued before Nancy stated, “There is at least one liar at the table.”  Note:  there are four Truthteller and four Liar cards, so it is possible that all players were dealt a Truthteller card.

After Nancy made her statement, one of the players had enough information to declare and win the game.  Which player declared?

Yes, there are enough clues to solve this!  But a word of caution — the solution to the puzzle only involves answering the question, “Which player declared?”  There may not be enough information to give a more complete answer….

Good luck!  In the next installment of Beguiling Games, I’ll give the solution to this logic puzzle, and give a geometrical two-player game to analyze as well.  Happy solving!

Creating Animated GIFs in Processing

Last week at our Digital Art Club meeting, I mentioned that I had started making a few animated gifs using Processing.  Like this one.

GIF

(I’m not sure exactly why the circles look like they have such jagged edges — must have to do with the say WordPress uploads the gif.  But it was my first animated gif, so I thought I’d include it anyway.)

And, of course, my students wanted to learn how to make them.  A natural question.  So I thought I’d devote today’s post to showing you how to create a rather simple animated gif.

sample

Certainly not very exciting, but I wanted to use an example where I can include all the code and explain how it works.  For some truly amazing animated gifs, visit David Whyte’s Bees & Bombs page, or my friend Roger Antonsen’s Art page.

Here is the code that produces the moving circles.

gif3

I’ll assume you’ve done some work with Processing, so you understand the setup and draw functions, you know that background(0, 0, 0) sets the background color to black, etc.

The idea behind an animated gif which seems to be a continuous loop is to create a sequence of frames whose last frame is essentially the same as the first.  That way, when the gif keeps repeating, it will seem as though the image is continually moving.

One way to do this is with the “mod” function — in other words, using modular arithmetic.  Recall that taking 25 mod 4 means asking “What is the remainder after dividing 25 by 4?”  So if you take a sequence of numbers, such as

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 11, 12, 13, 14, …

and take that sequence mod 4, you end up with

1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, ….

Do you see it already?  Since my screen is 600 pixels wide, I take the x-coordinate of the centers of the circles mod 600 (that’s what the “% 600” means in Python).  This makes the image wrap around horizontally — once you hit 600, you’re actually back at 0 again.  In other words, once you go off the right edge of the screen, you re-enter the screen on the left.

That’s the easy part….  The geometry is a little trickier.  The line

x = (75 + 150 * i + 2 * frameCount) % 600

requires a little more explanation.

First, I wanted the circles to be 100 pixels in diameter.  This makes a total of 400 pixels for the width of the circles.  Now since I wanted the image to wrap around, I needed 50 pixels between each circle.  To begin with a centered image, that means I needed margins which are just 25 pixels.  Think about it — since the image is wrapping around, I have to add the 25-pixel margin on the right to the 25-pixel margin on the left to get 50 pixels between the right and left circles.

So the center of the left circles are 75 pixels in from the left edge — 25 pixels for the margin plus 50 pixels for the radius.  Since the circles are 100 pixels in diameter and there are 50 pixels between them, there are 150 pixels between the centers of the circles. That’s the “150 * i.”  Recall that in for loops, the counters begin at 0, so the first circle has a center just 75 pixels in from the left.

Now here’s where the timing comes in.  I chose 300 frames so that by showing approximately 30 frames per second (many standard frame-per-second rates are near 30 fps) the gif would cycle in about 10 seconds.  But cycling means moving 600 pixels in the x direction — so the “2 * frameCount” will actually give me a movement of 600 pixels to the right after 300 frames.  You’ve got to carefully calculate so your gif moves at just the speed you want it to.

To make displaying the colors easier, I put the R, G, and B values in lists.  Of course there are many other ways to do this — using a series of if/else statements, etc.

One last comment:  according to my online research, .png files are better for making animated gifs, while .tif files (as I’ve also used in many previous posts) are better for making movies.  But .png files take longer to save, which is why your gif will look like it’s moving slowly when you use saveFrame, but will actually move faster once you make your gif.

So now we have our frames!  What’s next?  A few of my students mentioned using Giphy to make animated gifs, but I use GIMP.  It is open source, and can be downloaded for free here.  I’m a big fan of open source software, and I like that I can create gifs locally on my machine.

Once you’ve got GIMP open, select “Open as Layers…” from the File menu.  Then go to the folder with all your frames, select them all (using Ctrl-A or Cmd-A or whatever does the trick on your computer), and then click “Open.”  It may take a few minutes to open all the images, depending on how many you have.

Now all that’s left to do is export as an animated gif!  In the File menu, select “Export As…”, and make sure your filename ends in “.gif”.  Then click “Export.”  A dialog box should open up — be sure that the “As animation” and “Loop forever” boxes are checked so your animated gif actually cycles.  The only choice to make now is the delay between frames.  I chose 30 milliseconds, so my gif cycled in about 10 seconds.  Then click “Export.”  Exporting will also take a few seconds as well — again, depending on how many frames you have.

Unfortunately, I don’t think there’s a once-size-fits-all answer here.  The delay you choose depends on how big your gif actually is — the width of your screen in Processing — since that will determine how many frames you need to avoid the gif looking too “jerky.”  The smaller the time interval between frames, the more frames you’ll need, the more space those frames will take up, and the longer you’ll need to upload your images in Gimp and export them to an animated gif.  Trade-offs.

So that’s all there is to it!  Not too complicated, though it did take a bit longer for me the first time.  But now you can benefit from my experience.  I’d love to see any animated gifs you make!

Mathematics and Digital Art: Update 1 (Fall 2017)

About a month has passed since beginning my third semester of Mathematics and Digital Art!  As with last semester, I plan on giving updates about once a month to discuss changes in the course and to showcase student work.

The main difference this semester (as I discussed a few weeks ago) was starting with Processing right from the beginning.  From my perspective, the course has run more smoothly than ever — and some of my students are already really getting into the coding aspect of creating digital art.

I do believe that beginning this way will pay off when we get to making movies.  Since we’ll already know the basics and understand the difference between user space and screen space, I can focus more on the interactive abilities of Processing — such as having features of the displayed image change by moving the mouse or pressing different keys on the keyboard.

The first two assignments were essentially the same as last semester.  We began with discussing color and the work of Josef Albers, emphasizing the fact that there is no such thing as “pure color” — colors are only perceived in relation to other colors.

Again, I was surprised by the diversity of the images students created.  Like last year, a few students experimented with a minimalist approach.  Here is what Alex generated using just a 2-by-2 grid of squares.

Day112Alex

I should point out that outlining the geometrical objects (using the strokeWeight function) is not “pure” Albers — you aren’t really seeing one color on top of another due to the black outlines.  But I did have students submit three pieces, insisting that one of the pieces was created only by changing the parameters in the original Albers routine, as shown in the following submission.

Day112Linh2

Here is Courtney’s submission on this theme, again created only by changing parameters to the drawing routine.

Day112Courtney

 

Most students — I think in part due to the fact that we started discussing code even earlier than previous semesters — really pushed the geometry far beyond the simple idea of rectangles within rectangles.

While toying with various geometrical motifs, Tera found something that reminded her of a rose.  This influenced her color palette:  reds and pinks for the flowers, with a green background, meant to suggest that the flowers were in a garden.

Day112Tera

Cissy explored the geometry as well.  Note how keeping the stroke weight at zero — so that the geometrical objects have no outline — creates a more subtle effect, especially since the randomness from the dominant color is not too pronounced.

Day112Cissy

The second art assignment, as in the previous semesters, was to explore creating textures using randomness in both color and shape.  As with the first assignment, I wanted students to submit one piece which involved only changing the parameters to a given function.  In this case, the function created a grid of gray circles, with both the intensity of the gray and the size of the circles having some degree of randomness.  I think it is important that students do some work within given constraints — it really challenges their creativity.  Here is Terry’s piece along these lines.

Day112Terry

 

The second piece was based on a function which created a grid of squares of the same size, but random colors.  Here, there were no constraints — students could modify the geometry in any way they wanted to.  Several were quite creative.  For example, Sepid approached this task by choosing both shape and color to create an image reminiscent of a stained glass window.

Day112Sepid

The third piece involved a color gradient (see my previous posts on Evaporation).  If you look back at these posts, you’ll recall that a color gradient can be created by increasing the randomness of the colors as you move from the top of the image to the bottom using a power function:  f(y)=y corresponds to a linear gradient, f(y)=y^2 corresponds to a quadratic gradient, etc.  Different effects can be created by varying the exponent.

As I was discussing this in class, one student asked what would happen if you used a negative exponent.  I had never thought about this before!  Jack used this idea in his piece, which he said reminds him of looking at a fire.

Day112Jack

It turns out that using a negative exponent creates a gradient beginning with black on the top.  Why is this?  As the image proceeds lower down the screen, the algorithm subtracts values from the RGB parameters proportional to y^n, where y=0 corresponds to the top of the image, and y=1 corresponds to the bottom of the image.

So if the exponent n is positive, there is very little randomness subtracted.  But if the exponent is negative, a lot of randomness is subtracted, since now the numbers near 0 are on the denominator.  Because the RGB values only go up to 255, subtracting a large degree of randomness leaves nothing left — in other words, black.  Now some of the numbers will end up being negative  near the top– but putting all negative numbers in a color specification in Processing does in fact give you black.

Another student also worked with yellows and reds to imitate fire in another way.  Instead of making small circles, he made larger circles with quite a bit of overlap, creating a rather different effect.

Day112Ali.png

And Rosalie found in interesting say to create stripes with the algorithm.  I had not seen this effect before.

Day112Rosalie

So that’s it for the first update of the Fall 2017 installment of Mathematics and Digital Art.  As you can see, my students are already being quite creative.  I look forward to seeing their work develop as the semester progresses!

 

Bay Area Mathematical Artists, I

Yesterday was the first meeting of the Bay Area Mathematical Artists at the University of San Francisco!

It all began one balmy Friday evening in Waterloo, Ontario, Canada at the Bridges 2017 conference….the Mendlers and I hosted a pot luck dinner at our AirBnB, and we realized how many of us were from the Bay area.  In fact, we remarked upon the fact that nine of us were actually on the same flight from San Francisco to Toronto for the conference!

Bridges participants do really form a community.  There is a spirit of sharing and mutual appreciation for each others’ work.  We really do cherish those few days each year when we can all come together.  The only drawback is that Bridges comes around just once a year.

So throughout the evening, between chowing down on grilled fare and sipping a glass of beer or wine, the idea of informally gathering now and then kept cropping up.

But as we all know, ideas do not automatically become reality.  They have a tendency to wither if not watered and fertilized….so I decided to take up gardening.

I had the advantage of being associated with a University, so I could arrange a meeting space.  Location was also somewhat convenient — some of us were to the northeast in Oakland and Berkeley, and others were to the southwest in Santa Clara and Scotts Valley.  It might be nice to move around occasionally so not everyone has to drive as far all the time.  But since the meetings are on Saturdays, at least traffic is not so much of a bother.

And then come the emails!  Yes, lots of them….  The main decision to be made was deciding on a format.  I thought informal was best — I sent out a call for speakers, and put them on the docket on a first-come, first-served basis.  I wanted to take away the stress of competing for time; if there were more speakers than we had time for, we’d just start where we left off the last time.

The other reason for this is that I wanted to encourage students from my Mathematics and Digital Art class, as well as members of the newly formed Digital Art Club, to participate as well.  I think it is important to let mathematical artists of all levels have a place to share ideas and get feedback on their work.

So for our inaugural meeting, we had three speakers:  Chamberlain Fong, Karl Schaffer,  and Dan Bach.

Chamberlain’s talk was entitled The Conformal Hyperbolic Square and Its Ilk.  He discussed different ways to transform circular hyperbolic tilings (particularly those of Escher) to square images.  Chamberlain did give a version of this talk at Bridges in 2016, but included more recent results as well.  For more information, you can contact him at chamberlain@yahoo.com.

001title

 

Karl Schaffer’s talk was entitled Dance’s Center of Attention Mass.  Inspired by Joseph Thie’s Rhythm and Dance Mathematics and Kasia Williams’ idea of “Center of Attention Mass,” Karl is interested in graphically showing where the center of attention is by weighting the position of each dancer on stage.  He went so far as to contact Thie — now in his 80’s — and they are actively collaborating together.

Apoll. Circles.png 

Karl is also giving the lecture/demonstration Calculated Movements at the Montalvo Art Center next March.  There is more information here.  You can reach Karl at karl_schaffer@yahoo.com.

Finally, Dan Bach’s talk was entitled 3D Math Art and iBooks Author.  Dan is keen on creating highly interactive math books which engage students of all ages.  He gave a practical talk demonstrating the software he uses, including examples of converting graphics to various different formats since it is not always a simple task to take a 3D image created by one software package and import it into another.  You can reach Dan at dan@dansmath.com.

DanBachSlide1

 

After the talks — which included ample room for Q&A — we had a brief discussion on the future of the group.  I wanted to make it clear that while I am willing to keep things going in the current format, it is really up to the group to decide how to run our meetings.  We opted to keep things going the same way for next month — but suggestions for the future included workshops, or perhaps themed sessions, like a series of talks on polyhedra.  Participants were encouraged to think of other ways to use our time together as a topic of discussion for the next meeting.  Keeping it informal means lessening the pressure of submitting talks/papers for conferences, etc.

Then dinner!  Most of us were available for a meal afterwards.  There were two nice options nearby — a cafe with sandwiches and salads, and an Indian restaurant with a buffet.  I went with the group who preferred Indian food — and truly, a good time was had by all!  We left for dinner at about 5:30, and I finally had to break things up shortly before 8:00, since some of us had a ways to drive home.  We could clearly have kept talking for quite a while….

So our first meeting of the (tentatively named) Bay Area Mathematical Artists was a success!  There were a total of 15 of us present, including three students from USF — a very respectable number for a first time event.  We plan to meet approximately monthly, modulo the University schedule of classes and holidays.

I’ll post summaries each month of our meetings, including a brief synopsis of the talks, workshop(s), or whatever other form the meetings take.  Feel free to contact the speakers for more information about the talks they gave this weekend, and don’t hesitate to spread the word to others who might be interested!

 

Using Processing for the First Time

While I have discussed how to code in Processing in several previous posts, I realized I have not written about getting Processing working on your own computer.  Naturally I tell students how to do this in my Mathematics and Digital Art course.  But now that I have started a Digital Art Club at the University of San Francisco, it’s worth having the instructions readily accessible.

The file I will discuss may be used to create an image based on the work of Josef Albers, as shown below.

0001

See Day002 of my blog,  Josef Albers and Interaction of Color, for more about how color is used in creating this piece.

As you would expect, the first step is to download Processing.  You can do that here.  It may take a few moments, so be patient.

The default language used in Processing is Java.  I won’t go into details of why I’m not a fan of Java — so I use Python mode.  When you open Processing, you’ll see a blank document like this:

Day110Screen1

Note the “Java” in the upper right corner.  Click on that button, and you should see a menu with the option “Add Mode…”  Select this option, and then you should see a variety of choices — select the one for Python and click “Install.”  This will definitely take a few minutes to download, so again, be patient.

Now you’re ready to go!  Next, find some Processing code written in Python (from my website, or any other example you want to play around with).  For convenience, here is the one I’ll be talking about today:  Day03JosefAlbers.pyde.  Note that it is an Open Office document; WordPress doesn’t let you upload a .pyde file.  So just open this document, copy, and paste into the blank sketch.  Be aware that indentation is important in Python, since it separates blocks of code.  When I copied and pasted the code from the Open Office document, it worked just fine.  But in case something goes awry, I always use four spaces for successive indents.

Now run the sketch (the button with the triangle pointing to the right).  You will be asked to create a new folder; just say yes.  When Processing runs, it often creates additional files (as we’ll see in a moment), and so keeping them all in one folder is helpful.  You should also see the image I showed above; that is the default image created by this Processing program.

Incidentally, the button with the square in the middle stops running your sketch.  Sometimes Processing runs into a glitch or crashes, so stopping and restarting your sketch is sometimes necessary.  (I still haven’t figured out why it crashes at random times.)

Next, go to the folder that you just created.  You should see a directory called “frames.”  Inside, you should see some copies of the image.

Day110Screen2

Inside the “draw” function, there is a function call to “saveFrame,” which saves copies of the frames you make.  You can call the folder whatever you want; this is convenient, since you might want to make two different images with the same program.  Just change the folder name you save the images to.

A word about the syntax.  The “####” means the frames will be numbered with four digits, as in 0001.png, 0002.png, etc.  If you need more than 10,000 frames (likely you won’t when first starting), just add more hashtags.  The “.png” is the type of file.  You can use “.tif” as well.  I use “.tif” for making movies, and “.png” for making animated gifs.  There are other file types as well; see the documentation on saveFrame for more details.

Now let’s take a look at making your own image using this program.

Day110Screen3

If you notice, there are lines labelled “CHANGE 1” to “CHANGE 6” in the setup and draw functions.  These are the only lines you need to change in order to design you own piece. You may tweak the other code later if you like.  But especially for beginning programmers, I like to make the first examples very user-friendly.

So let me talk you through changing these lines.  I won’t bother talking about the other code right now — that would take some time!  But you don’t need to know what’s under the engine in order to create some interesting artwork….

CHANGE 1:  The hashtags here, by the way, indicate a comment in your code:  when your program runs, anything after a hashtag is ignored.  This makes it easy to give hints and provide instructions in a program (like telling you what lines to change).  I created a window 800 x 600 pixels; you can make it any size you want by changing those numbers. The “P2D” just means you’re working with a two-dimensional geometry.  You can work in 3D in Processing, but we won’t discuss that today.

CHANGE 2:  The “sqSide” variable indicates how big the square are, in units of pixels.  The default unit in Processing is always pixels, so if you want to use another geometry (like a Cartesian coordinate system), you have to convert from one coordinate system to another.  I do all this for you in the code; all you need to do is say how large each square is.  And if you didn’t go back and read the Josef Albers piece, by “square,” I mean a unit like this:

Day002Square

CHANGE 3, CHANGE 4:  The variables “sqRows” and “sqCols” indicate, as you would expect, how many rows and columns are in the final image.  Since I have 15 rows and the squares are 30 pixels on a side, the height of the image is 450 pixels.  Since my window is 600 pixels in height, this means there are margins of 75 pixels on the top and bottom.  If your image is too tall (or too wide) for the screen, it will be cropped to fit the screen.  Processing will not automatically resize — once you set the size of the screen, it’s fixed.

CHANGE 5:  The “background” function call sets the color of the background, using the usual RGB values from 0-255.

CHANGE 6:  The first three numbers are the RGB values of the central rectangles in a square unit.  The next three numbers indicate how the background colors of the surrounding rectangles change.  (I won’t go into that here, since I explain it in detail in the post on Josef Albers mentioned above.  The only difference is that in that post, I use RGB values from 0-1, but in the Processing code here, I use values from 0-255.  The underlying concept is the same.)

The last (and seventh) number is the random number seed.  Why is this important?  If you don’t say what the random number seed is (the code does involve choosing random numbers), every time you run the code you will get a different image — the computer just keeps generating more (and different) random numbers in a sequence.  So if you find an image you really like and don’t know what the seed is, you’ll likely never be able to reproduce it again!  And if you don’t quite like the texture created by using one random seed, you can try another.  It doesn’t matter so much if you have many rows and columns, but if you try a more minimalist approach with fewer and larger squares, the random number seed makes a big difference.

OK, now you’re on your own….  This should be enough to get you started, and will hopefully inspire you to learn a lot more about Python and Processing!

 

 

Beguiling Games I: Nic-Nac-No

It has been some time since I’ve posted any puzzles or games.  In going through some boxes of folders in my office, I came across some fun puzzles I created for a class whose focus was proofs and written solutions to problems.  I’d like to share one this week.

For the assignments, I sometimes wrote stories around the puzzles.  So here is one such story.  The date on the assignment, if you’re interested in such things, is January 16, 2003.  (I assume that you are familiar with the game Tic-Tac-Toe, as well as the fact that if both players play intelligently, the game ends in a draw.)  I called the game “Nic-Nac-No.”

Betty and Clyde, after their favorite breakfast of blueberry pancakes one sunny Saturday morning, began a Tic-Tac-Toe tournament.  They were reasonably bright children — taking turns going first, the initial 73 games ended in a draw.

“Just once, Clyde, couldn’t you try putting your O first on a side instead of in a corner?” prodded Betty.  “That way, it wouldn’t be the same boring game every time.”

“Well, it’s my turn to go first this time,” said Clyde, putting an X in the center.  “OK, now you show me how you want me to play so I can do it that way next time.”

“Oh, shut up, Clyde,” sighed Betty, putting her O in the upper left corner.  And so game #74 ended in a draw.

“Hey, I’ve got an idea!” exclaimed Clyde.  “Let’s make up different rules.  How about this:  the first one who gets three-in-a-row loses.  Whaddya think, Betty?”

“That’s so random, Clyde,” said Betty, secretly excited by the suggestion.

“No, it’s not.  And besides,” reasoned Clyde, “it’s got to be better than playing another game of Tic-Tac-Toe since you won’t ever try anything different.

“OK, potato brain.  Let’s try.  You go first.”

“Great!” exclaimed Clyde, until he realized Betty was trying to outmaneuver him.  He just realized that it you’re trying to avoid three-in-a-row, the fewer squares you own, the better.

Assuming Betty and Clyde play optimally, will the game be a win for Betty, a win for Clyde, or a draw?

I should remark that the idea of a misere game — where you turn the winning condition into a losing condition — is not original with me.  But most students have not considered this type of game, so misere versions of games often make for engaging problems.

Before I discuss the solution, you might want to try it out for yourself!  There are likely many strategies possible to produce the desired result; I’ll just show you the ones I thought were the most straightforward.

In my solution, Betty and Clyde use different strategies, but the end result is the same:  the game must end in a draw.  Let’s look at what strategies they might use.

It turns out that if Clyde starts in the center, he can use a strategy where he does not lose.  It’s fairly simple:  always play opposite Betty.  Thus, when Betty plays a corner/side, Clyde takes the opposite corner/side.

Why can’t Clyde lose?  First, it should be clear that Clyde can never make a three-in-a-row that passes through the center.  Since he always plays opposite Betty, any line of three passing through the center must contain two X’s and one O (recall Clyde started with an X in the center), and so is not a three-in-a-row.

What about a three-in-a-row along a side?  Since Clyde plays opposite Betty, if he ever placed an X to make three-in-a-row along a side, that would mean Betty already had three O’s in a row on the opposite side, and would have already lost!  So it’s impossible for Clyde to lose this way.

Since any three-in-a-row must pass through the center or be along a side, this means that Clyde — if he plays intelligently — can never lose Nic-Nac-No.

Now let’s look a non-losing strategy for Betty.  There is no guarantee she will be able to take the center square on her first move, so we’ve got to consider something different.  And we can’t just rely on playing opposite Clyde, since there is no opposite move if the takes the center first.  Moreover, it may be the case that Clyde uses some other strategy than the one I mentioned, so we can’t even assume that he does take the center on his first move!

To see a strategy for Betty, consider the following diagram:

Day109NicNanNo

Betty’s strategy is simple:  place an O in one of the squares marked A, one marked B, one marked C, and one marked D.

Note that this is always possible.  Even if Clyde does not play in the center on his first move, he can only occupy one square labelled A, B, C, or D.  Then Betty places her O on the other square with the same letter.  If Clyde does begin in the center, then Betty has her choice of first move.

Since it is always possible — and since Betty only has four moves — these comprise all of Betty’s moves.  But note that since Betty never has an O on two of the same letter, she can never get three-in-a-row on a side.  Further, since Betty’s strategy never involves a move in the center, she can never get three-in-a-row in a line going through the center square.  This means that Betty can never lose!

So if the two players play their best games, then Nic-Nac-No ends up in a draw.  And while these strategies do indeed work, I would welcome someone to find simpler strategies.

I’ll leave you with another version of Tic-Tac-Toe to think about.  Here are the rules:  if during the game either play gets three-in-a-row, then X wins.  If at the end, no one has three in a row, then O wins.  Does X have a winning strategy?  Does O?  Note that in this game, there cannot be a draw!  I’ll give you the answer in my next installment of Beguiling Games….