Digital Art VI: End of Week 14

The last three weeks have been very intense!  The main focus has been on laboratory work, both with Processing and Final Projects.

Week 12 began with another round of Presentations on papers from past Bridges conferences.  This proved to be successful again — with topics ranging from geometrical furniture to zippergons to maps of Thomas More’s Utopia.  We all learned something new!

The time not spent on Presentations that week was devoted to working on projects.  Students were finding their stride, and their ideas were really beginning to take shape.

Two students were interested in image processing, so Nick has been working with them.  The initial problem was that doing anything in Sage involving image processing was just too slow — you’re code is essentially sent to a remote server, executed, and the results sent back.  Since we’re using the free version, this means performance is unpredictable and far too slow for computation-heavy tasks.

So Nick helped with installing Python, finding image processing packages, etc.  This process always takes a lot longer than you imagine, but eventually all issues were resolved.

While Nick was working on image processing, I was helping others.  One student was really interested in using L-systems — one of my favorite geometrical topics recently!   But all my work with Thomas involved code written in Postscript and Mathematica.  Which meant I had to rewrite it in Python.

This proved to be quite a bit trickier than I thought.  The student was looking at The Algorithmic Beauty of Plants, and was interested in modeling increasingly complex L-systems.  First, the simplest type, like that used to generate the Koch curve.  Next came bracketed L-systems, and then bracketed L-systems with multiple rules.  I did finally get all these to work; I intend to clean up the code and share it in a future post.

Week 13 was devoted entirely to Processing and project work.  I started the week with introducing a few new ideas.  What really inspired the class were the mouseX and the mouseY variables.  When your mouse is in screen space, the mouseX and mouseY variables contain the location of your mouse.  So you can put an object where the mouse is, or have the x-coordinate of the mouse in some way determine the color or the size of the object.  The possibilities are virtually endless.

Here’s a sample movie made with this technique.

Since the code is just a few lines long, it won’t take long to explain.  Here is the complete program:

mouseanddot

The drawPoint function will draw a point centered at the position of the mouse.  But in addition to those coordinates, the function takes an addition argument:  float(mouseX)/500.  This argument is 0 at the left edge of the screen, and 1 at the right edge.  (Recall the need for “float,” since otherwise Python will perform integer division and give 0 for any number less than 500.)

So the stroke command uses the parameter p to determine how much red is in the color specification.  When p is 0, there is no red, so the dot is black.  And when p is 1, the dot is red.  I used “p**0.4” as an illustration that interpolation need not be linear — the exponent of p determines how quickly or slowly the dot gets brighter as you move the mouse to the right.  Of course the dot also gets larger as you move your mouse to the right, as is clear by the strokeWeight function call.

I showed this example as I introduced the movie project.  Their assignment was to make a movie — anything they wanted to try.  The complete prompt is given on Day 34 of the course website, but I’ll give the gist of it here.  The main goal was to have students use linear interpolation in at least four different ways in their movie.  Of course they could use nonlinear interpolation if they wanted to, but it wasn’t required.

There was no length requirement — it’s easy to make a movie longer by adding more frames.  Just be creative!  Interpolation is a very useful tool in making movies and animations, and has a mathematical basis as well.  So I wanted that to be the focus of the project.  I also had them write a brief narrative about their use of interpolation.

This is Andrew’s movie.  You can clearly see the use of interpolation here in different ways.  It was also nice to see the use of trigonometry to calculate the centers of the dots.

 

Ella was interested in L-systems, and so Nick spent some time working with her on Python code during his weekly office hours.  Here is what she created.

 

Lucas wanted to use some interaction with the mouse, and he also had the idea of the sun setting as you moved the mouse down the screen.  Watch how the fractal clouds move with the location of the mouse as well.

 

So you can see how varied the use of interpolation was!  The students really enjoyed having control over the possible special effects, and created a wide range of interesting features in their movies.

That takes us to Week 14.  On Monday, we had a guest speaker come in, Chamberlain Fong.  I met him in Finland at Bridges this summer, only to find out he lives the next neighborhood over in San Francisco!  He gave a very interesting talk about taking pictures with a spherical or hemispherical camera lens, and the issues involved in printing the pictures.

The problem is essentially the same as creating a map of the globe — making a sphere flat.  There will always be distortion, but you have some control over what type of distortion.  You can keep angles the same, or areas the same — or some combination of the two.  And as these cameras keep getting cheaper, there will be a growing interest in making your spherical photos look realistic.

And although we had class on Wednesday, some students had already left for Thanksgiving Break.  So Nick and I were available to help on an open lab day.  Most of the students actually showed up, and we had a productive day working on movies (which were due Wednesday) and projects.

The next (and final!) post on Mathematics and Digital Art will survey the students’ Final Projects, so stay tuned!

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.

imag3969

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.

glassleaves_0
Leaves of Glass

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.

imag4281

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.

graphdodeca

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.

graphprism

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.

imag4318

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 specifically asked about how students’ attitudes about mathematics changed.  I got some encouraging responses.

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.

d11fractal

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.

af2

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

jn3

Safina went a different direction, and created this.

sm4

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.

Beetle2
Fractal beetle created with an iterated function system.

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.

affinetransformation

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.

Two2

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).

revsierpinski4

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.

af_1

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.

mc_2

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.

la_1

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

mj_1

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.

Day002Albers6Web3

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.

Day011Evaporation2bWeb

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.sample

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.

AQ_1

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.

JN_1

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

SM_1

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

EK_1

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!