## Digital Art V: End of Week 11

It’s been an exciting three weeks in Mathematics and Digital Art!  We began Week 9 after our Fall break with a talk by Carlo Sequin.  Here’s Carlo with one of his sculptures at the University of California, Berkeley.

I met Carlo through the Bridges conferences; he currently sits on the Board.  Since he’s so close by, I thought it would be great to have him visit my class.

His talk centered around the computer graphics programs he wrote in order to design sculptures like the one you see in the picture above.  Carlo included several different parameters, allowing an incredible variety of images to be generated.

Then he focused on a few of his sculptures and described the design process from conception to final sculpture.  He even remarked how he got lucky once — he and his team forgot to measure the width of the door they needed to take the sculpture through, and it fit with just a few inches to spare….

Friday, Day 24, was our first Project day.  Students’ ideas were still somewhat vague — they did have to write a Proposal, but still didn’t have a clear direction.  We made some progress, though the next Project day helped quite a bit more.

On Monday, Day 25, we dived into working with Processing.  I kept in mind the comments from students’ response papers which I mentioned in my previous post — they really wanted to learn more about code.

Now coding is a precision endeavor, as there is no room for error as far as syntax is concerned.  So after working through a simple example, I presented them with the following movie.

Admittedly it’s not a blockbuster…but their lab work was to duplicate this movie as precisely as possible.  I gave them a basic template (see the May 2016 archives for the start of a six-part series on Processing which I used as a basis for their lab work), but they didn’t just have to tweak numbers — they also had to add new elements.

The main mathematical tool involved was linear interpolation, which we went over in some detail in class.  The next class, I had them work on recreating the following movie.

This proved a bit more challenging.  Some students at first thought there was a rotation involved — but it’s just linear interpolation again, on a somewhat larger scale.  Here are the prompts I gave them:

• the screen is 500 x 500 pixels;
• the dots are always 25 pixels from the edge;
• the colors are the standard red, green, blue, and orange;
• the smaller dots are 100 pixels wide, and the larger ones are 200 pixels wide.

The main challenge was working in screen space, since they needed to calculate the exact centers of the circles.  Of course students progressed at different rates, and they were finally getting the main point — you can use linear interpolation to morph any aspect of an image which depends on a numerical parameter.  As a mathematician, I was used to thinking along these lines all the time, but it’s a concept that takes a while to really sink in if it’s new.

Friday, Day 27, was our next project day.  We worked at fleshing out more details of the students’ various projects.  Two students wanted to explore image processing, so Nick continued working with them to download the appropriate packages and get Python installed on their computers.  Such projects are never as easy as they sound, but Nick did get everything to work.

While he was doing that, I circulated with the other studets, discussing their projects and answering a few questions about making movies, if they had them.  It was a productive day, and everyone left class with a much clearer idea of their project than they had the week before.

On Monday, we continued our work with Processing.  I went through an example from my earlier blog posts — making a movie which morphs a Sierpinski triangle — in some detail, explaining it line by line.  Again, the most challenging part was converting from user space to screen space.  A fractal which fit in a unit square had to be scaled and moved to fit nicely into screen space.

This took just about the entire class.  Although we had done work previously with iterated function systems, we did have to take some time to review certain aspects of the code we used before.  Then on Day 29, I had them duplicate the following movie.

In order to focus on the coding, I began the movie with a fractal they had on a previous quiz, so they knew which affine transformations to start with.  They had to figure out how to modify one of the transformations to produce the final image, and then use linear interpolation to create the movie.

This proved challenging, but everyone made good progress during the lab.  They are supposed to finish by Monday.

On Friday of Week 11, we had our second guest speaker — Shirley Yap from California State University, East Bay.  I met Shirley last February when Nick and I went to a regional meeting of the Mathematical Association of America.  She was in charge of organizing the Art Exhibition Nick and I had pieces in.

After showing a few interactive examples from her web page, Shirley focused her discussion on the following piece she created.

She talked a lot about the challenges of making a physical piece, rather than a work of digital art.  For example, she actually wanted to use glass, but is was not possible to etch in glass given the tools she had available.  So she had to settle for acrylic, which is very easy to smudge if you aren’t careful.  You see, the individual squares can pivot where they are screwed in, so the artwork is interactive.

There were also size requirements, since she had to be able to take in on the airplane with her to a conference.  Other issues arose — in some ways working digitally is a lot easier.  As Shirley remarked, once you drill a hole, you can’t undrill it…but it’s easy to change a parameter in digital work.

What was really nice was that Shirley talked about mathematical envelopes (one of my favorite topics; I’ve written about it before on my blog), and the curves she used to make her envelopes — Bernstein polynomials.  She took the time to go through a few simple examples, so that students got a sense of what these curves are like.  It was a nice example of yet another topic in mathematics students hadn’t seen before being used to create art.  Truly, mathematics is everywhere….

Stay tuned for the next update of Mathematics and Digital Art!

## Digital Art IV: End of Week 8

The last two weeks were focused on a study of polyhedra.  While not strictly a digital art topic, I thought it important for students to develop a basic three-dimensional vocabulary in the event they wanted to do further study in computer graphics.

We began with the Platonic solids, naturally, looking at enumerating them geometrically and algebraically.  The algebraic enumeration involved solving the usual

$\dfrac1p+\dfrac1q=\dfrac12+\dfrac1E.$

This proved challenging, especially when I gave some additional, similar Diophantine equations for homework.  We also took time to build a dodecahedron and an icosahedron.  This occupied us on Day 17, Day 18, and part of Day 19.

Since the first half of the semester was nearing its end, it was time to begin thinking about Final Projects.  So I took the rest of Day 19 to help students individually choose a topic, and assigned the Project Proposal over the weekend.

We had a brief discussion of graph theory on Day 20, which involved looking at the adjacency graphs of the vertices of the Platonic solids, such as the one for the dodecahedron.

I introduced much of the basic vocabulary, using the chapter I wrote in my polyhedra textbook as a guide.  Of course there is only so much progress to be made during a single class, but I did want to indicate how two apparently different areas in mathematics are related.

The homework involved untangling adjacency graphs, such as the one below.

This is just a triangular prism, although drawn a little unconventionally.  This assignment again proved more difficult than I thought, even with Euler’s formula to help in calculating the number of faces.  So we spent extra time on Day 21 going over the homework, followed by a very brief discussion of duality.  And as students were having difficulty narrowing their focus for their project proposals, I spend the rest of Day 21 talking individually with them as they started building a few rhombic dodecahedra.

Over half of Day 22 was taken up by a quiz on their homework; I didn’t want there to be much time pressure.  The last part of this class was spent creating an in-class sculpture with rhombic dodecahedra.  I chose this dual model for them to build as it is space-filling.

I was surprised at how much they really got into it!  I do hope we have time for a similar project at the end of the semester.  We were in a bit of a rush for time, but still managed to create something intriguing.

Last time I mentioned I assigned a short response paper getting feedback from the students about how the course was going so far, and I said I’d share some of their comments.  All (anonymous) quotes are from student papers.

I really like how hands-on the course is, and how there is a good balance between lecture and lab time.

This was a common opinion — and validates a major feature of the course design.  I am glad students appreciate it!  Another student made a similar remark about the lab time.

I enjoy the lab assignments that we get because I like the designs I create.  It allows us to put to practice what we have learned with each lesson.

I was used to thinking of math as just something I had to do, that would probably be useful later in life, but wouldn’t really pertain to whatever I wanted to do with art.  I’ve realized that I would actually really like to use this kind of math in my art in the future, because I never realized what kind of things I could make with this medium.

As someone who wasn’t the best at geometry in high school (I’m more of an algebra person), I think this class has given me a practical use for all the things I learned in high school that I found difficult to grasp or uninteresting.

I was pleased to read responses like this, since again, this reflects an overarching purpose of the course — see how mathematics can actually be used in practice.  One student even went so far as to say,

I would also like to mention something, though this might not be considered significant, I never thought I would have to use matrices ever again in my life.

A few students remarked on using mathematics in the creative process.

At first, I was a little unsure of the role that mathematics took in graphic design, but as soon as we started playing with Sage, I noticed that it affects almost every aspect brought forth in the image.

Making “rules” for you art was something very different for me…..combining the left brain and the right brain creates incredible pieces of work….

Some students made more specific comments.  One student liked the presentations the best.

Looking through the different papers in the Bridges archive and hearing everyone’s presentations really made me realize the extent that mathematics is related to so many other topics.

In addition, I asked for specific suggestions for improvement.  By far the most common remark was that students want to learn more about how the Python code in Sage works.  I was really encouraged when I read those comments!  We will start to learn Processing in a few weeks, and I’ll make sure we discuss the code in more detail.

One student really liked the discussion board I set up for a few of the assignments — it is not difficult to create discussion boards for future assignments, so we’ll try that again.  Another remarked that it would be good to learn about the printing process — and I certainly agree!  But as I remarked in previous posts, I thought the logistics of this challenging endeavor would be too difficult to implement.  It is certainly a future goal.

But overall, I was very glad to read how students were enjoying the course — and also pleased about their suggestions for improvement.  So I’ll work hard at making the second half of the course even better than the first!

## Digital Art III: End of Week 6

Recall that at the end of Week 4, we had just begun a lab on affine transformations and iterated function systems.  At the beginning of Week 5 (Day 11), students were supposed to finish the exercise they began on Friday, and try another.  This was the new prompt:  Create a fractal using two affine transformations. For the first, rotate by 60 degrees, then scale the x by 0.6 and the y by 0.5. For the second transformation, rotate 60 degrees clockwise, scale the x by 0.5 and the y by 0.6, and then move to the right 1.  I provided a link to the fractal which should be produced, shown below.

Again, this proved to be a challenge!  The reason is that the computer is somewhat unforgiving.  To produce the image above, every calculation has to be correct.  Here were some common issues:

1. There was difficulty interpreting the statements as geometric transformations.  In particular, getting the order of the matrix multiplication correct, and interpreting a clockwise rotation as a rotation by -60 degrees.
2. Performing the matrix multiplication correctly.  Some students were using online calculators, and some had trouble with converting degrees to radians.
3. Translating the results into Python code; the affine transformations needed to be converted to the form $(ax+cy+e, bx+dy+f).$

I also gave them an exercise to reproduce one of the fractals they needed to analyze for homework the previous week.

This occupied us for all of Week 5, since I also wanted to give them time to work on their upcoming assignment.

Week 6 was our first Presentation Week.  Since we’d only need two days for the presentations given the class size, I spent the first day of the week giving my Bridges talk on producing Koch-like fractal images.

The class was interested to learn more about this algorithm, and I had hoped to spend a week devoted to the topic.  But because of the Processing work we’ll be doing later — animation of fractals using iterated function systems — I didn’t want to rush through the preliminary work on affine transformations and IFS.  Any misunderstandings now would surely be problematic later.

The presentations were to be on papers from the Bridges archives.  All papers since the Bridges conferences began (which was in 1998) are archived online.  The assignment was simple:  find a paper in the archive that’s at least four pages long, and give a five-minute presentation on it to the class.  I did create a discussion board where they posted the title/author of their selections so there wouldn’t be any duplicate presentations.

These presentations went very well.  Most students’ presentations were close to ten minutes long, and the enthusiasm for their presentations was quite evident.  I must admit that I learned a lot, too — I had not read most of the papers they selected.

I had students do simple peer evaluations, and included the item “I would like to learn more about this topic after hearing the presentation.”  They selected a number from 1–5, with 5 meaning “most interested.”  The overall average score was about 3.9 — meaning the papers piqued their curiosity.  In addition, I wanted to get additional ideas about what to include in the few days I set aside for special topics at the end of the semester, or what new ideas to incorporate into next semester’s class.

The most popular paper (4.75 average) was On Generating Dot Paintings in the Style of Howard Arkley, which Madison said reminded her of some of the textures we created during a few of our lab sessions.  The next highest average was 4.25, so this was the clear favorite.  I hope we’ll have time to explore it further later on.

And that took us to the end of Week 6!  I had hoped to have some time in the lab to have students find interesting polyhedra nets they’d like to build, since the next two weeks will be devoted to polyhedra.  No, it’s not really a digital art topic — but it is an area of expertise.  I wanted students to have some exposure to three-dimensional geometry since if they continue to study computer graphics, they will certainly move from two into three dimensions.

Now it’s time to look at some student work!   I’ll focus on one fractal from the assignment on iterated functions systems.  Here is the prompt:  Create a morphed Sierpinski triangle, based on the code in the Sage worksheet. The idea is to have your fractal look like it was derived from a Sierpinski triangle, but just barely. Someone looking at it should wonder about it, and maybe after 30 seconds or so, say “Hey, that looks like a Sierpinski triangle!”

One student created the following image, and wrote:

I think this fractal looks like many sets of pine trees, and I like the way that it shears out. I made the whole thing the same color (gray) to make the fact that it was derived from the Sierpinski triangle less obvious.

Julia creating the following fractal image, and remarked that at one point, she went “too far” and had to backtrack a bit.

Safina went a different direction, and created this.

She wrote,

The stretch and rotation reminds me of a tree, more specifically a Christmas tree….I wanted to emulate a sort of Christmas feel because I was listening to Christmas music.

So motivation can come from anywhere!  I’ll post more of their images on my Twitter feed, @cre8math.

Again, you can see how my students are really embracing the course and being very creative with the assignments.  I’m looking forward to seeing more of their work.

I just assigned a brief response paper asking students how they felt about the course so far, and how it has changed their ideas about art, mathematics, and computer science.  I’ll report on their comments in my next summary in two weeks.  Stay tuned!

## Digital Art II: End of Week 4

It’s hard to believe it’s been four weeks already!  Recall that we ended Week 2 with work on an Evaporation piece.  We’ll see examples of student work after a brief recap of Weeks 3 and 4 (Days 6–10).

In the past two weeks, we focused on affine transformations and their use in creating fractals with iterated function systems.  This was not intended to be a deep discussion of linear algebra, but a practical one — how affine transformations reflect the geomtetry of fractal images.

Days 6 and 7 were an introduction to the geometry of affine transformations.  I like to emphasize the geometry first, so I worked to find a way to simply and unambiguously describe an affine transformation geometrically.

The blue square represents the unit square, and the red parallelogram the transformed square.  The filled-in/open black circles on the parallelogram represent how (0,0) and (1,0) are transformed, respectively.  Additionally, the pink fill in the parallelogram represents that fact that there is a flip (that is, the determinant of the linear part of the transformation is negative).  Strictly speaking, this coloring isn’t necessary, but I think it helps.

We then worked on writing the affine transformation in the form

$T\left(\begin{matrix}x\\y\end{matrix}\right)=\left[\begin{matrix}a&b\\c&d\end{matrix}\right]\left(\begin{matrix}x\\y\end{matrix}\right)+\left(\begin{matrix}e\\f\end{matrix}\right)$

We did this in the usual way, where the columns of the matrix represent where the unit basis vectors are transformed, and the vector added at the end is the translation from the origin.  These can all be read from the diagram, so that the picture above describes the affine transformation

$T\left(\begin{matrix}x\\y\end{matrix}\right)=\left[\begin{matrix}-2&-3\\2&0\end{matrix}\right]\left(\begin{matrix}x\\y\end{matrix}\right)+\left(\begin{matrix}-2\\1\end{matrix}\right).$

To help in visualizing this in general, I also wrote a Sage worksheet which produces diagrams like the above picture given the parameters af.  In addition, I showed how the vertices of the parallelogram may be found algebraically by using the transformation itself, so this meant we could look at affine transformations geometrically and algebraically, as well as with the Sage worksheet.   (Recall that all worksheets/assignments may be found on the corresponding day on the course website.  In addition, I have included the $LaTeX$ code for the assignments for those interested.)

I explained several different types of affine transformations — translations, reflections, scalings, and shears.  On Day 8, we saw how to write the affine transformations which describe the Sierpinski triangle, and students got to play with creating their own fractals using the accompanying Sage worksheet.

On Day 9, I took about half the class to go through three of my blog posts on iterated function systems.  I emphasized the first spiral fractal discussed on Day035, paying particular attention to the rotation involved and matrix multiplication.

Most students weren’t familiar with trigonometry (recall the course has no prerequisites), so I just told them the formula for rotation matrices.  We briefly discussed matrix multiplication as function composition.  I gave them homework which involved some practice with the algebra of matrix multiplication.

On Day 10, we began with going over some previous homework, and then looked at what transformations were needed to producing the fractal below (as practice for their upcoming homework).

We then engaged in the following laboratory exercise:  Create a fractal using two affine transformations.  For the first, rotate by 45 degrees, then scale the x by 0.6 and the y by 0.4, and finally move to the right 1.  For the second transformation, rotate 90 degrees clockwise, and then move up 1.

This was practice in going from a geometrical description to a fractal — they needed to perform the appropriate calculations, and then enter the data in the Sage worksheet and see if their fractal was correct (I gave them a link to the final image).

This turned out to be very challenging, as they were just getting familiar with rotations and matrix multiplication.  So we’ll need to finish during the next class.  I’ll also give them another similar lab exercise during the next class to make sure they’ve got it.

Last week’s digital art assignment was another success!  Again, I was very pleased with how creative my students were.  If you look back at the Sage worksheets, you’ll see that the class was working with color, texture, and color gradients (like my Evaporation piece).

One student created a texture with a lot of movement by keeping the circles separate and using a wide range of gray tones.

Maddie took advantage of the fact that the algorithm draws the circles in a particular order to create a scalloped texture.  This happens when the circles are particularly large, since the circles are drawn in a linear fashion, creating successive overlap.

Another student also used this overlapping feature with a color gradient.  He describes his piece as follows:

This piece is what I like to call “The Hedge” as it reminds me a lot of those tall square hedges that are in mansions. It’s as if the light is hitting the top of the hedge and dispersing down into the shadows and thickness of the leaves. I especially like the leafy effect the overlapping circles give.

Finally, Madison experimented with altering the dimensions of the grid to create a different feel.

She thought that making the image wider made the evaporation effect more pronounced.

These are just some examples of how students in my Mathematics and Digital Art course take ideas I give them and make them their own.  They are always asking how they can incorporate one effect or another into their work, and Nick and I are glad to oblige by helping out with a little code.

As I strive to keep my posts a consistent length, I won’t be able to share all the images I’d like to this week.  So I’ll be posting additional images on my Twitter feed, @cre8math.  Follow along if you’d like to see more!

## Digital Art I: End of Week 2

Mathematics and Digital Art is well underway!  As promised, I’ll be giving weekly or biweekly summaries of how things are going.  You’re also welcome to follow along on the course website.

Day 1 was the usual introductory class, with a discussion about the syllabus and what the course would be like.  (Remember I blogged a four-part series on the course beginning at Day045.)  I wanted to make it clear that there would be both mathematics and coding in the course.  Some students ended up dropping the course the first week as a result, leaving the class with nine students.

On Day 2, I discussed representing color on the computer — using RGB values (both integer and real), as well as using hexadecimal notation.  Since the course has no prerequisites, we really did need to start at the beginning.  Students did some exploring with various RGB values using a Sage worksheet.  (Links to all the worksheets and other websites referenced in class can be found on the course website.)

Josef Albers was the topic of Day 3.  I’ve talked a lot about his work before, so I won’t go into great detail here.  I let students try out a Sage worksheet — they had an assignment to create their own version which was due in about week, so we spent half the class starting the project.

Essentially, they needed to choose a color scheme and size for their piece.  But then they needed to use five different random number seeds, and decide which of the five resulting images looked best to them, and why.  So the assignment included a brief narrative as well.  The complete prompt is on my website.  (And I think I’ll just stop saying this — just know that anything I mention in these posts will be discussed in more detail on the course website.)

The next two classes focused on creating a piece based on Evaporation.

Again, I discussed this in some detail before (Day011 and Day012).  The basic idea is to use randomness to create texture, both with color and geometry.  Add to this the idea of a color gradient, and that’s enough to get started.

I took a bit of time on Day 5 to explain in some detail about how to create color gradients using a function like $y=x^p,$ where $p>0$ — it isn’t exactly obvious if you’ve never done it before.  Having Nick as my TA in the class really helped out — I was at the board, and he was at the computer drawing graphs of functions and varying the parameters in the routine to produce color gradients. We worked well together.

It turns out that having two of us in the computer lab is really great — debugging takes time.  Students are gradually getting accustomed to the Sage environment, and  learning to go back and redefine functions if they get an error message indicating that something is undefined….

I have really been enjoying the laboratory experience!  My students are wonderfully creative, taking the basic motifs in directions I hadn’t considered.  I’ll illustrate with a few examples from the first assignment.  In this post and future posts, all work is presented with the permisson of the student, including the use of his/her name if applicable.

I should also remark that my intent is not merely to post pieces I subjectively like.  True to the title of my blog, Creativity in Mathematics, I intend to illustrate creative, original ideas.  I may feel that some students have been more successful with their ideas than others — but I’ll share that with the students individually rather than discuss it here.

In order to talk about the students’ creativity, however, let me show you the image they started with.

Andrew experimented with minimalism, creating a series of 2 x 2 images.  This is the one he said he liked the best because of the symmetry and contrast.

Julia was interested in exploring different geometries and created a few drafts which deviated from the rectangle-within-rectangle motif.  Here is one of them.

Safina took a slightly different approach.  She wanted to create some variation by using trapezoids in the centers of the larger rectangles.

Ella really worked with the geometry, creating a lot of visual contrast — especially with the asymmetrical polygons.

So we’re off to a great beginning!  Right now, Nick and I are having to help a lot with editing code since it’s a new experience for most students.  But they are eager to learn, and as you can see, willing to explore new ideas, going well beyond where they started.  We can look forward to a lot more interesting work as the semester progresses!

P.S.  If you are ever interested in any of the students’ work and would like to discuss it further, send an email to vjmatsko@usfca.edu and I’ll forward it on to them!

## Writing a Math Blog

A friend and colleague recently suggested (thanks, Sanza!) writing a meta-post about creating a Math Blog, and I thought now might be a good time.  It’s been about four months since my first post, and I’ve found my stride (metaphorically, at least).

What motivated me to write a blog?  I realized I’d written a lot over the years in the form of puzzles and problems that I wanted to share.  But I had trouble thinking of a good venue — except perhaps writing a book — and it suddenly dawned on me that a blog might be the way to go.

As I mused upon the idea further, I began thinking about my artwork as well, and wanting to share a lot of those ideas, too.  I wanted the blog to be accessible — which doesn’t mean devoid of some interesting mathematics — as well as fairly novel.  There are good blogs which cull stuff from all over the internet, but I didn’t want to write one of those.  I thought that a creativity thread would be just what I wanted — to show mathematics as a creative endeavor.  And having taught for many years, I realized that this was one aspect of mathematics lost on most students at any level.  (As discussed in two recent posts.)

And so a blog was born.  Or conceived, I should say.  I settled on an audience of advanced middle school students to undergrads — but I was aiming at a fairly sophisticated student.  One who isn’t afraid of mathematics or programming, and is willing to dive into something new.

Now I’ve written a lot over the past several years, including a few books, so I thought that the writing would be fairly manageable.  But knowing that projects are often more involved than originally imagined, I decided to draft my first ten blog posts and get some feedback before I even started.  This was in August, before the semester began, and I thought it a prudent move so I didn’t get stuck in a content bind right away.

That work definitely paid off — and it made me think a lot about what I wanted to write before I launched the blog.  I wanted a nice blend of art, puzzles, teaching ideas, and geometry; drafting some initial posts helped me to organize those thoughts.

I decided early on to incorporate programming, for a few reasons.  I would have to say that the computer is perhaps the most important tools I use as a mathematician.  I think nothing of writing a Mathematica routine to test out a conjecture say, a million times, before I dive into looking for a proof.  And as an artist, well, the blog speaks amply to that point.

But I had just started learning Python in January of this year for a course I was teaching, and using it on the Sage platform.  I felt it was important that anything I did with programming should be accessible and open source, and Sage fit the bill perfectly — just click on the link!  Nothing to download or install.

But more importantly, I wanted to use programming to help illustrate the creative process — and encourage others to be similarly creative.  Making a puzzle, designing some artwork — not mysterious endeavors, but realizable projects made easier with the help of a few lines of code.

At that point, I had the basic setup in mind, and went for the first post!  You might have noticed (those of you following from the beginning), that the “Read More” sections have disappeared.  I originally thought to divide the content into two sections, so the reader might digest it in smaller chunks.

But with the help of the statistics gathered by WordPress, I noticed the following phenomenon.  When I began making movies and included one in the main body and one in the “Read More” section, the latter was hardly ever played.  So the chunking plan seemed only to succeed in having readers look at only half of my posts…..

So at that point, I decided to eliminate the “Read More” sections — and therefore also the idea of including a puzzle at the end of each essay for those who weren’t particularly interested in such things.  They’d have to endure….

I settled, then, upon writing one-section posts of about 1000 words.  This is long enough to say something interesting, but not too long to lose a dedicated reader.

I’ve received some good feedback so far, but the readership is still fairly small.  Now that I’m accumulating enough content, one of my next steps is to reach out to some colleagues and perhaps former students to help me publicize my blog.  More and more schools are teaching Python, and I think some of my posts on art and programming would make interesting projects for students taking an introductory programming course.

I’d also like to do some guest blogging — having other friends and colleagues describe their creative processes.  I haven’t decided exactly what form that will take yet, but that doesn’t need to be decided immediately.

One neat side effect is that I’ve got to meet some interesting people online through their comments, and not all are from the US.  I’m surprised by the geographic diversity of viewers — it’s fascinating how the internet transcends national boundaries.  I’m hoping to meet more people as the blog evolves.

Is it worth it?  So far, I’d say yes.  I’ve had many interesting conversations as a result of blog posts, and I enjoy putting my thoughts down on paper (metaphorically, that is).  Aside from the time invested (which is not insignificant), the only other cost involved was upgrading WordPress so there wouldn’t be any ads on my blog — I was quite surprised when I test posted and was informed that there might be ads!

For the would-be blogger, then, no good advice — a blog is a very personal endeavor, and sometimes you’ve just got to jump in and give it a go.  But this is my story — and I’m sticking to it!  Good luck if you’re willing to give it a try.

Word count is now 1,011, meaning it’s time to go.  You get pretty good after a while at putting your thoughts into 1000-word chunks….

## Geometrical Dissections II: Four to One

This week, I’d like to discuss a piece of artwork which began as a geometrical dissection — I call it Four to One.

I thought it would be interesting to discuss the process of creating such a piece from beginning to end.  The creative process is not really mystical, but because we so often only see the finished product, it may seem that way at times.

It all began about 15 years ago, when I was teaching the Honors Geometry course I mentioned in last week’s post.  In Greg’s book Dissections Plane & Fancy, he takes a few chapters to discuss dissections from squares to other squares — frequently two different squares to one, or three to one.  But there was very little about four-to-one dissections, so I thought I’d explore this avenue a bit more.

I can’t recall precisely how I arrived at the identity

$15^2+36^2+48^2+64^2=89^2.$

I might have written some for loops — but computers were not as fast back then….  Likely I used something like Lebesgue’s formula on p. 80 of Greg’s book, which gives a formula for creating three-to-one square dissections.  Then if one of those squares could be written as a sum of two others, I’d have a four-to-one dissection.  In particular, once I (might have!) found out that

$39^2+48^2+64^2=89^2,$

I could use the fact that $15^2+36^2=39^2$ (which is just a multiple of the Pythagorean triple $(5, 12, 13)$) to obtain the possibility of a four-to-one dissection described above.

Now this only suggested the puzzle, not the actual dissection itself.  And there certainly is a dissection — at the very least, we can cut up all the squares into $1\times1$ unit squares and reassemble!

This is hardly an elegant solution; but I did come up with the following one:

I liked it because each square was cut into just two or three pieces, which was particularly nice.  Moreover, only one piece needed to be rotated.  But even though the number of pieces is relatively small, there is still the possibility that a dissection may exist using fewer pieces.

Of course my original solution was sketched on a yellowing piece of graph paper — but what to do with it now?

My first attempt looked like this:

I was thinking of creating various pathways through the dissected squares so that when they were rearranged, the pathways would still line up.  I abandoned this approach, however.  I can’t remember exactly why, but the results didn’t appeal to me — and besides, the paths themselves actually had nothing to do with the dissection puzzle itself.

But then I had the thought — which was in fact a real challenge — can I communicate what’s happening with the dissection using only one square?  In other words, could I depict the geometrical dissection by just showing the largest square without giving the viewer the four smaller squares?  I think what might have moved me in this direction is that there was just no elegant way of putting all five squares together in a composition.  There were just too many corners.

So I though of overlapping the smaller squares onto the largest square, as shown below (note:  you’ll notice an error in the geometry, but as it was a draft I discarded, I didn’t bother to fix it):

Now if you look very carefully, you can find all the pieces of the dissected squares in the largest square.  There is some overlap, of course — but smaller circles were overlaid on larger ones so colors from both circles could be seen.  (I copied the original dissection again so it’s easier to compare.  I used different colors as the images were created at different times, so watch out! )

I liked the idea — I felt I was getting somewhere.  But I wasn’t happy with the colors.  Now creating mathematical art makes you hungry — I can clearly recall driving to lunch while I was in the middle of this project, and I can even remember the road.  It was Fall in Princeton, NJ, and the leaves had already turned color.  No more oranges and reds — but lots of greens and yellows, as well as browns from the tree trunks.  My color palette!

What intrigued me about the idea was the fact that I was working with a very abstract, almost purely mathematical problem — and here I was, thinking about using organic colors from nature, from my life experience.

Now I had already been working with the ideas from Evaporation, and realized if I was using an organic palette, I couldn’t have the circles be regular, precise — and the colors couldn’t be pure either, just like you might find hundreds of shades of yellow in a Fall forest.

So, as shown in this close-up of Four to One, the colors were varied by using random numbers just as was done for Evaporation, but there were no extremes — each piece of each square had to be clearly recognizable if the dissection was to be clearly seen.

The sizes of the circles varied as well, helping to contribute to a natural texture.  Here, you can clearly see how smaller circles were overlaid on larger circles for the two-color effect.  The smaller circles, however, had only about one-fourth the area of the larger ones, so it was clear which color was dominant.

And there it is!  The creative process is not magical, not mystical — in fact, much of the time it seems to consist of failed inspiration….  Consider yourself lucky if your first attempt turns out to be your last as well — but more often than not, creativity is an iterative process involving constant revision.

So my advice is to stick to it!  Don’t worry if the first attempt isn’t what you imagine.  Now I used Mathematica to create this image — and I’ve been programming in Mathematica for over twenty years.  So I’m pretty good at taking an idea and implementing it fairly quickly.  But if you’re relatively new to programming, you’ve got to be patient with your programming skills as well.  I can tell you though — it’s worth it.  Don’t let anyone else tell you any different….

## Writing Original Problems

How do students view mathematics?

Not surprisingly, many (if not most) students see mathematics as a set of known problems to be solved — changing a few numbers here and there, perhaps — but essentially, all of mathematics is known.

Mathematicians have rather the opposite view — we’re just scratching the surface.  There is so much more underneath.

As I mentioned in my first post, mathematics is creative.  What makes this difficult for students to appreciate is that the artistic medium is that of abstraction, and without a real understanding of abstraction in mathematics, the creative aspect is hard to see.

But there is a way to help students experience the creative side of doing mathematics — and that is by having them write their own Original Problems.  I began thinking about this while I was teaching a course in problem solving at a magnet STEM high school — and being an avid problem writer myself, I imagined that having students write problems would help them solve problems.  Whether this is true or not is difficult to determine.  Regardless, an assignment was born….

What really got me interested in this assignment was the student comments at the end of the semester.  One student wrote,

Anyone can write tedious, difficult problems that review core math subjects, but to write problems in a novel, challenging, and refreshing manner, one must be imaginative. I feel that this creative side of math is an often overlooked aspect of the field as many believe math to be an extremely black-and- white, rigid, and boring subject.

I was intrigued by the fact that even though students were not prompted to address creativity in writing their course evaluations, some spontaneously did so.  As a teacher, I was delighted — an unanticipated side effect of an assignment designed for another purpose was somehow more significant to me than the intended outcome.

Fueled by this success, I introduced the assignment in an Honors Calculus section I taught, and students responded positively again.  Then I incorporated writing Original Problems into a traditional calculus classroom, then precalculus, then algebra — and students kept getting it.  Posing problems was no longer an assignment just for advanced students.

What does the assignment look like?  I break it down into four sections.  First, Motivation.  Where did the problem come from?  For some students, they might start looking in their textbook at interesting problems.  For others, they take inspiration from their daily life.  One calculus student said he came up with his problem because he dropped his backpack down the stairs and had to retrieve it — and he immediately thought of this as a displacement/velocity problem.  Another student was doodling figure eights, and created a problem about ice skating on a figure-eight shaped rink.  It is remarkable what students can create, given the opportunity!

Second, the Problem Statement.  This is actually quite difficult for some students.  And we teachers know the challenge of writing a test whose problems can be interpreted in only one way.  Now that I’ve moved on from the STEM high school to teaching university again — and work with a different set of students — I now assign the Motivation and Problem Statement as a separate assignment.  That way, I can give written and verbal feedback to students and help them craft a well-stated, manageable problem.  This has been very helpful for the students, and the quality of their final submissions has improved.

Third is the Problem Solution.  This is fairly self-explanatory, but a few comments are in order.  I like to give students wide latitude in selecting a problem of interest to them — sometimes they want to challenge themselves with a difficult problem.  In this case, a partial solution is fine.  The point is to get them writing mathematics — and a partial solution to a difficult problem often involves more mathematics than a complete solution to a more routine problem.

Finally, there is the Reflection.  I only ask for a few sentences or a paragraph — enough to give me a sense of how students are responding to the assignment.  These can be very revealing, and you sometimes get students who really appreciate the assignment and understand its purpose.  All four sections are to be included in the final submission.

You might be interested in a recent Original Problem prompt.  This assignment is highly adaptable.  I’ve had colleagues who wanted to narrow the focus because the assignment seemed to broad.  Suggesting a specific application — such as the Pythagorean theorem — will give students a starting place.  In my mind, the assignment is about creativity, writing, and self-determination.  Let students choose a topic to create a scenario and write about it, and they start to get a handle on what creativity in mathematics is all about.  There is no one way to accomplish this.

I should say a few words about grading these assignments.  At their broadest, these assignments read like short essays.  But they’re all different, so you can’t really develop a rhythm in the grading process.  So Original Problems take more time to grade — this semester I’m just giving two assignments, so it’s more manageable.  I do think it’s important to give at least two assignment, so students have a chance to improve.  Generally, I’m more lenient when I grade the first assignment, since often this is the first time students will have encountered such an assignment.

To encourage creativity, I tell students that if they just do the assignment  — and get their mathematics correct — they won’t earn lower than a B.  I don’t want them worrying about grades (and we’re stuck with them for a while!), although some inevitably do.  I rarely give a C, unless it’s evident a student waited until the last minute, or a student worked below their potential.  I do believe that for an assignment like this, you should evaluate students relative to themselves, not their peers.  More able students should be pushed — and frankly, most of them appreciate it when you do push them.

Many students really do begin to understand the creative aspect of mathematics after doing these assignments.  They really do enjoy getting to choose their own problem — and though it is sometimes challenging to come up with a way for them to develop a particular idea, I rarely tell them to just choose another topic.  I try to find some avenue they can pursue.

So I encourage you to give Original Problems a try!  Let me know how it goes.  For additional reading, you can find an article about writing Original Problems in Publication 10 on my website.  There is also a discussion of several student problems in Chapter 6 of Mathematical Problem Posing.  It really is time to have all students experience creativity in mathematics.  This is one of the main purposes of writing this blog, after all.

## What Is Mathematics?

Mathematics is creative.

Unfortunately, this is lost upon many — if not most — students of mathematics, in large part because their teachers may not understand mathematical creativity, either.  One way to address this issue is to have students write and solve their own original mathematics problems.  This seems daunting at first, until students realize they are more creative than they were led to believe.  (I’ll discuss this more in a later post.)

The difficulty is that the creative dimension of mathematics is a bit elusive.  Give a child crayons and ask her to draw a picture, sure — but give a student some ideas and ask him to create a new one?  To appreciate mathematical creativity, you need some understanding of the abstract nature of mathematics itself.  To create mathematics, you need imagination much like you do in any of the arts — or other sciences, for that matter.

Over the years, I’ve created my fair share of mathematics.  How much of it is really new is hard to determine — how do you know if any of the billions of other people in the world already created something you did?  (Proof by internet search notwithstanding.)

This blog is about sharing some of my ideas, problems, and puzzles.  Some were created years ago, some are new — and I will consider myself lucky if some are entirely original.  I truly did have fun creating them, and I enjoy writing about them now.

I’m hoping to convey an enthusiasm for mathematics and its related fields — in other words, all human knowledge — and to share something of the creative process as well.  The creation of mathematics is not a mystical process, and needs no explanation to a mathematician.  But we can surely do more to make this enlivening process accessible to all in a time when it is certainly necessary.

As you follow, you’ll notice a heavy emphasis on programming.  Every student should learn to program — and in more than one language.  Perhaps this should be an axiom in the 21st century, but we’re not even close.  So many of the tools I use are virtual — the ability to write code to perform various tasks is essential to my creative process, as you’ll see.  In fact, many posts will have links to Python programs in the Sage platform (don’t worry if you don’t know what these are yet).  These tools are all open source, and available to anyone with internet access.

Finally, blog posts will usually have a “Continue reading…” section.  Some posts (like this one) will be essays on teaching, creativity, or a related topic.  Since not everyone may be so philosophically minded, the “Continue reading…” sections of these essays will be a puzzle or game.  Enjoy!