Creativity in Mathematics

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….

Roman Numeral Puzzles

Today, I’ll talk about a set of puzzles I created just for this blog. The problems you’ve seen before — CrossNumber puzzles and Cryptarithms — I’ve been creating for many years. But in writing a blog about creativity in mathematics, I feel I should occasionally create something new….

I call these “Roman Numeral Puzzles” since they involve using Roman numerals in an interesting way. If you’re not familiar with how Roman numerals are written, just search online. (The internet knows everything.)

So here’s how the puzzles go. Fill in the following grid with the digits 0–9 and the letter X so that each row and column is either a number or a mathematically correct statment. The 1 may represent either the number 1 or an I in Roman numerals, and the X may represent either a multiplication symbol or an X in Roman numerals. For example, the second row may be XXX (the number 30 in Roman numerals), or 5X6 (since $5\times6=30$ ), or 3XX, where the first X represents multiplication, and the second X the number 10. What an I or X represents in a row may not the same as what it represents in the corresponding column. And as usual, no number can begin with a “0.” Happy solving!

Day018Grid1

Let’s work through the solution for this puzzle together. Then there will be two more for you to try on your own.

First, look at the second column. It can’t be 100, since there would be no way to write 12 in the third row with a 0 in the middle. So the second column is a multiplication, and the only way to write 100 using three symbols is XXX, which we interpret as $10\times10.$

In the third column, there is no way to write 40 as just a number, so it must be the result of a multiplication. So far, we have

Day018Grid2

Now think about how we can write 40. There are only four ways: 4XX, XX4, 5X8, and 8X5, where in the first two cases, one “X” is the multiplication symbol, and the other “X” represents the number 10. Since the first and third rows must be multiplications and 8 is not a factor of 20 or 12, that leaves 4XX or XX4. But 10 is a factor of 20 (and not 12), so we’ve got to use XX4. Once we have this, the rest is easy to fill in:

Day018Grid3

These puzzles aren’t difficult to make. You can begin with a grid, and simply fill it in with symbols and work out the values for the rows and columns. Try to think of using numbers which can be represented in different ways. For example, in the $4\times4$ puzzle below, I used 13 since it might either be written as a Roman numeral XIII, or a multiplication like 1X13. Having some entries end in 0 means a multiplication by 10, but that might be represented by 10 or X.

I can’t exactly remember what prompted me to bring in Roman numerals this way. You just let your mind wander — thinking about puzzles you are already familiar with, pushing the boundaries a bit — until your mind just “snaps” and you’ve got a concrete idea to try. Better minds than I have tried to pin down the creative process, so I won’t try to do that here. But I’m not really sure we’ll ever really undertand it….

Now the biggest challenge is solving your own puzzle and making sure it has a unique solution. Sure, you might say “find all solutions” — but as a puzzle maker, you really want just one solution. This is somehow more satisfying — if you create enough puzzles, I think you’ll see why. Good luck! And if you come up with any of your own puzzles, post them in a comment.

Here’s the $4\times4$ puzzle.

Day018Grid4

And for a real challenge, here is a $5\times5$ puzzle.

Day018Grid5

I hope you enjoy solving these puzzles. Let me know how it goes!

On Mathematical Creativity II

Continued from last week….

Content is subordinate to engagement. Again, a few paragraphs will not convince you to favor this position if you do not already — but given my own experience as an educator, I stand by it. I am clearly at my best when both my students and myself are thoroughly engaged in the work at hand…those occasional days when students say, “I can’t believe class is over already!” I wish I had more of them.

The waters muddy. Comparatively speaking, it is easy to teach content to pre-service teachers. But teaching them how to engage their students is a challenge.

Of course this is misleading — is there really a “how” when it comes to engagement? There may be many techniques and methods for drawing students in to learning mathematics. But engagement is about relationship. And here we confront a fundamental of the human condition — our profound inability to relate to one another.

Perhaps this is an exaggeration, though I might cite any number of large-scale wars as evidence. In the classroom, the student-teacher relationship is the scaffolding of the learning situation. But I am rather at a loss at what more to say.

Do my students laugh in class? What about the student who spent much of the last exam in tears? And what about the student at the table in the corner who never talks to anyone else? Why won’t that student come to visit my in my office? Why does this particular student always seem angry? depressed? tired? lonely?

We each handle such situations differently. Teaching is idiosyncratic. But how we relate to our students as human beings ultimately creates our classroom. Imagine, if you can, walking into your classroom and being able to instantly capture the individual responses of your students seeing you walk in. How would you feel?

I maintain that it is quite important that students like me as a teacher. I enjoy some moderate success here; I do not think that I am the most popular teacher in my department, but nor am I the least. Students are more likely to be engaged if they enjoy my being in front of the classroom. Of course this is common sense, but a point which I find is downplayed in discussions of curriculum.

Curriculum, pedagogy, content, engagement, relationship. Curriculum can be successfully standardized only to a degree — purposefully vague, but unavoidably so. Here in the US, more colleagues than not (at least among my acquaintances, both at my current and former institutions) are constrained by the curriculum they teach rather than inspired by it. Is it truly a mystery why our students are not engaged? Currently, a curriculum is seen as a sequential list of topics — complete with learning goals and outcomes — together with a nominally meaningful way to assess whether the outcomes have been met. As this list grows, students become superficially exposed to a breadth of topics, but are never given the opportunity to think deeply about any of them. Perhaps this is because it is difficult to measure depth of thought.

Measurement drives curriculum. I need hardly mention the situation in the United States and the infamous No Child Left Behind Act. Accountability drives assessment. Of course measurement and assessment need not be the same, but in practice, there is little difference. Simply put, the analysis of the results of standardized assessments is currently the means by which we decide whether our teachers and schools are doing their jobs.

Thus has assessment become political. Parents must be appeased, administrators validated, and legislators satisfied. Of course it is always the children who suffer. By any number of indicators, our educational system is becoming less and less effective. Reasons given for this decline are legion, but there is no need for finger-pointing here.

We imagine that the solution to this dilemma is the ideal curriculum, packaged so that teachers everywhere can deliver the necessary content, with the end result being a sufficiently pleasing number. It matters little what that arbitrary number represents, but that is still what is being sought — a sufficiently high number.

It is as though we were training would-be artists by selecting a certain number of classical works of art, turning them into paint-by-number exercises, and then counting the number of times students cross over the lines. At the very least, a prospective artist should be able to color within the lines! And so, charcoal in hand (due to limited resources, all work is done in shades of gray), artists of the future are ushered out into an unfriendly world.

At university, everything changes. Colored pencils! Perhaps the student of art is amazed for a brief moment. But only until it is time to learn how to teach younger children how to color within the lines. And, of course, create their own paint-by-number exercises for their own students. Now if I just make the lines a little thicker, then more of my students will be able to color within them….

Allow me this poor analogy. Suffice it to say that our educational system does not foster mathematical creativity. The teaching of creativity cannot be standardized, nor can creativity be easily measured (by those who feel so inclined). Thus it has no place in a “curriculum.”

What is required is that we cease to think of education as delivering a curriculum.

So how must we think about education?

I shall certainly disappoint the reader by having no ready answer to this question. Or perhaps not, for any pithy answer would necessarily be glib and certainly be suspicious.

But we might say at least this: Our classrooms should foster mathematical creativity. It is a sobering thought to realize that most individuals go through their entire lives without appreciating mathematics as a creative endeavor. I would go further to speculate that most of these think mathematics is nothing more than advanced arithmetic.

The reader will surely be able to supply any number of reasons for why this is the case. Unfortunately, the current legalistic approach to educational reform — an approach centered around standardization, assessment, equity, etc. — only worsens the problem. Such trends essentially serve one purpose: to insulate students from poor teachers. We can no longer guarantee that a student graduating with a teaching degree is competent. Our standards — especially in mathematics — are too low.

Thus the teacher is put on the defensive. Innovation is now suspect, and the impulse toward creativity is dampened. Teach the standard curriculum and have your students pass the standardized tests — or else suffer the very real consequences.

We must get students excited about learning mathematics. Force-feeding content to unmotivated students simply doesn’t work.

We must get teachers excited about teaching mathematics. The enthusiasm a teacher has for teaching mathematics is communicated to her students.

We must foster creativity in our classrooms. This is not an answer to a particular question, but rather a focusing point for conversation about pedagogy.

We need a paradigm shift in the way we think about curriculum. As technology develops, the ways students learn and students’ attitudes toward learning change much more rapidly than our teaching strategies do. Yet the current approach toward curriculum emphasizes standardization and homogeneity, when in fact more flexibility is needed. Technology develops more quickly than standards change, so that much of what we teach students to be able to do by hand can be accomplished with a few keystrokes. It may be the case that most students, after they graduate, will rarely perform a mathematical calculation by hand. We simply cannot ignore this sobering fact.

So there is much work to be done. A teacher whose primary focus is to be creative, spontaneous, and engaging in the classroom is a very different teacher than one whose primary focus is to prepare students for a standardized exam. We must radically change the way we train teachers, and we must make teaching a more attractive profession for our especially talented students. We must acknowledge that our current way of thinking about curriculum and pedagogy is not adequate in our technologically advancing world — and find alternate, workable perspectives.

I shall not end with a few hopeful platitudes — frankly, the situation is not really hopeful at all. Education might be about empowering students to create their own Starry Nights, or teaching them to color within the lines of paint-by-number imitations. Which shall it be?

On Mathematical Creativity I

What follows is an essay I wrote about five years ago. I’ll let it speak for itself — but it’s quite a bit longer than my usual posts, so I decided to separate it into two installments. Here is the first part….

(Note: References are to when the essay was written, so that “I currently teach…..” refers to what I was teaching when I wrote the essay.)

What is mathematical creativity?

Forgive me for not answering this question. Better minds have attempted to do so, but no consensus has been reached. I am not confident that a definitive answer will be forthcoming any time soon.

So why ask?

Now this is an interesting question! Perhaps even answerable.

There are those who seek to quantitatively measure creativity in some limited way — but I am not among them. Nor am I convinced that this is a worthwhile endeavor. Of course you will agree with me or not — and I am fairly certain I will not sway you with a few hastily written paragraphs.

We might instead attempt to qualitatively describe mathematical creativity. To what end? Perhaps we might arrange for a team of educators to individually write condensed paragraphs about creativity, but then what is to be done with all the diverse responses? Certainly many such paragraphs have been written already. Consensus is still lacking.

Should I withdraw the question?

Allow me a tentative rewrite. Perhaps, “How might we foster mathematical creativity?”

Much better! But why? We could find an answer potentially useful. Knowing what one teacher did successfully in his classroom could give a colleague an idea which she can adapt for use in her classroom.

Well, this seems to be a promising beginning! A fruitful exchange of ideas, followed by a suitable adaptation, then finally an enthusiastic implementation, and oops! What went wrong?

Learning is situational; teaching is idiosyncratic. From this there is no escape.

Many of us are familiar with the situation where we have two sections of the same class, and what seemed to work wonders in the earlier section is, somehow, not so wonderful in the later section. Perhaps one section was right before lunch, one after. Or a particularly energetic student in one section was sick that day. Maybe a desperate email from a parent just before the later section is lingering heavily on our mind. Rather more likely, however, is just the fact that there are different students in the sections.

Now add to this inescapable fact — that no two classes have the same students — the additional inescapable fact that you are not your colleague. You bring very different backgrounds to your classrooms. Moreover, in creating the lesson, your colleague likely thought through many potential difficulties, then arrived at something he could truly be excited about — and communicated this enthusiasm and confidence to his students in a way which you could not quite match in your classroom.

Nothing went wrong — unless you expected your experience to be the same as your colleague’s. Fortunately, often times it is sufficiently close, but more frequently than we would like, it is not.

This is simply the usual give-and-take we as teachers experience when we are ourselves creative in the classroom. That a new idea is implemented flawlessly is rare; often many revisions are necessary before we are satisfied with the result. An artist may make several sketches before deciding on a particular composition for a painting. A similar patience is required for artistry in teaching.

This suggests that there is no such thing as a successful curriculum. For success is not derived from the structure of a lesson, no matter how cleverly devised. It should be obvious that teachers must be sufficiently well prepared; but sadly this is often not the case. As I have found from interacting with colleagues from around the world, teachers — especially those working with younger children — have meager backgrounds in mathematics. There is an uneasy tension between insisting that teacher candidates have adequate mathematical experience and the real necessity of having them be certified to teach.

For those of us teaching older students, issues of training in both mathematics and pedagogy are significant. I currently teach at a secondary school for students especially talented in mathematics and science. Some of my colleagues (myself included) had previously taught mathematics at university, while others’ careers primarily involved teaching at the secondary level. It should not come as a surprise that such diverse backgrounds result in different views on mathematical creativity — and what is needed to foster it.

As an example, I currently teach a course entitled Advanced Problem Solving. My approach to fostering creativity? Among other things, I have students write an original problem each week on a topic of their choice.

Now given the nature of the students in this course and the course content, students write problems involving logic, geometry, number theory, probability, recurrence relations, generating functions, and geometrical inversion, among others. I give them relatively little guidance, so that they are free to explore and create. I am moderately successful with this approach.

Would I recommend this approach for a new teacher just out of college? With these topics, I would be hesitant except for the most mathematically proficient teacher.

Does that mean new teachers should forego teaching problem posing until they have more experience? Certainly not. I hope to suggest that my style of fostering creativity in the classroom is intimately related to my background and experience — different teachers will take different approaches. Perhaps more importantly, this particular approach plays to my strengths. And — dare I confess? — I get excited about it.

I suspect that every educator knows precisely what I mean. There are courses you teach, and there are courses you are excited to teach. Likely there is no need to wonder in which courses your students are more receptive.

To be continued….

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.

Day015FourToOneWeb

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:

Day015FourToOneGeometryWeb

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:

FourSquaresToOne

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):

Day015FourSquaresToOne2

Day015FourToOneGeometryWeb

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.

Day015FourToOneExcerptWeb

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….

Geometrical Dissections I

Closely related to the problem of tiling the plane with polygons is that of dissecting one geometrical object into pieces that can be rearranged to form another. The classic example is the following dissection of an equilateral triangle to a square, attributed to Henry Dudeney, 1907.

Day014TriToSquare

Note that the pieces are exactly the same in both polygons. It’s not hard to appreciate the beauty and elegance of this “geometrical artwork.” And it’s not hard to imagine a puzzle based on this dissection — give the puzzler these four pieces with instructions to make both a triangle and a square from the same pieces.

This led me, once upon a time, to construct an Honors Geometry course centered around geometrical dissections using Greg Frederickson’s wonderful Dissections Plane & Fancy.

But even “simple” dissections — involving triangles and squares — weren’t so easy to create. For example, the pieces divide the base of the equilateral triangle into lengths Day014formula and these are some of the easier calculations! I won’t say more about that here — but you can read all about this dissection and many others in Greg’s book.

So I was fascinated by geometrical dissections — but I needed a way to make the idea accessible to my students. I thought — what could you create just by experimenting with dissections on ordinary graph paper?

Well, I have been answering this question for over 15 years now. I’ll start with some introductory ideas in this post, but this is definitely not the last word on dissections!

Let’s begin with the following puzzle. By cutting the rectangle along the grid lines, how many pieces are needed so you can also make the square with a corner missing? Day014Puzzle1A This seems like an easy puzzle to solve, as shown below. Day014Puzzle1B So yes, only two pieces are necessary — but one had to be rotated. Here is the question: can this puzzle be solved with just two pieces, but with neither piece rotated?

It turns out this is possible — but a solution requires a bit more creativity. Here is one way to do it: Day014Puzzle1C This is a solution technique commonly used in Dissections Plane & Fancy. Why bother? In the world of geometric dissections (and it is a growing universe, surely, as any internet search will show), finding a minimum number of pieces is the primary objective. But of all solutions with this minimum number of pieces, “nicer” solutions require rotating the fewest number of pieces. And rotating none at all is — in an aesthetic sense — “best.” It is also preferable not to turn pieces over, although sometimes this cannot be avoided for minimal solutions.

Another criterion for solutions might be whether they can be hinged or not, as Greg discusses extensively in Hinged Dissections: Swinging & Twisting. We won’t have time to explore this topic today.

Of course there is no reason you have to start with a rectangle, and also no reason why you need to restrict yourself to just one shape. The puzzle below shows how you can find a four-piece, rotation-invariant dissection from two smaller octagons to one larger octagon. Day014Puzzle2 It is important to note that the octagons here are not regular. A quick glance through Dissections Plane & Fancy will reveal that dissections involving regular polygons are generally rather difficult (as the initial triangle-to-square example amply shows).

Further, “two-to-one” dissections lend themselves nicely to a square grid, as the diagonal of a unit square has length square root of 2. Take a moment to study the octagon dissection again — paying particular attention to the side lengths — to see how this plays out. In the world of regular polygons, however, two-to-one dissections are in general quite difficult. Visit Gavin Theobald’s web page of two-to-one dissections to see some fascinating examples.

It is not hard to create dissection puzzles of your own — a pencil, eraser (!!!), and graph paper are all you need. What is difficult, however, is proving that you’ve found the fewest number of pieces. And when you have, proving that your dissection is unique. Uniqueness is virtually impossible to prove, but sometimes you can get a handle on minimality. For example, if the octagon dissection could be done in three pieces, one of the smaller octagons would have to be uncut. It’s not hard to see that there is no way of cutting the other smaller octagon into just two pieces to create the larger octagon.

What I enjoy about creating dissection puzzles is that there is not a single strategy you can use to solve them. You really need to use your imagination. Sometimes you might even surprise yourself by stumbling upon a really neat puzzle, like the one below.

Here, an 8 x 8 square with four holes (shown in black) can be dissected into three pieces to create a 6 x 10 rectangle, although one piece needs to be rotated. There is a simple elegance about this dissection which I find appealing. Day014Puzzle3

Another grid which lends itself to creating dissections is a grid of equilateral triangles. We won’t go into details here, but you can get the idea with the following dissection of an irregular dodecagon to an equilateral triangle in just five pieces. (The minimal dissection with regular polygons requires eight pieces.) Day014Puzzle4 I’ll leave you with two puzzles to think about. Of course, you can just make up your own. If you come with anything interesting, feel free to comment!

For this puzzle, my best solution is four pieces, without needing to rotate any pieces or turn any pieces over. (Black squares are holes.) Day014Puzzle5

For the last puzzle, my best solution is five pieces — but I had to turn over and rotate two of the pieces. Day014Puzzle6

A parting suggestion: when looking up dissections on the web, be sure to use the search words “geometrical dissections…..”

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.

Evaporation II

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Evaporation I

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

ColorSlide Puzzles

Part of the fun of writing a math blog is designing new puzzles. I wanted to come up with a puzzle which incorporated color and logic, so here’s the result!

In the grid below, slide the tiles either horizontally or vertically onto the grid so that each row and column contains exactly one each of green, cyan, magenta, and yellow dots. When two dots share the same square, they are combined, so that if a green and a red dot occupy the same square, the dot is yellow, while if a blue and a green dot occupy the same square, a cyan dot is produced. In other words, overlapping the tiles is like adding their RGB values. (If you aren’t familiar with RGB values, you can experiment in the “Continue reading” section by searching for Day002 in my blog.) Horizontal tiles may only overlap vertical tiles (and not other horizontal tiles), and vertical tiles may only overlap horizontal tiles.

Dots of the same color may never overlap — although this condition follows from the puzzle setup. Note that green, cyan, and yellow all require a green dot, so with four rows/columns, at least 12 green dots are necessary. Since there are exactly 12 green dots, no two green dots may overlap. Similar arguments show that no two red dots or two blue dots may overlap, either.

In the first part of the post, we’ll solve this puzzle so you see what’s involved. We’ll look at the design later.

Let’s start by looking at Column 1. Each row column needs three green dots, two red dots, and two blue dots — this follows from the fact the the RGB values of the four colors in the completed puzzle are (0,1,0), (0,1,1), (1,0,1), and (1,1,0). Since the vertical tiles in Column 1 already include three green dots, we know how to place the tiles in Row 2, since we need to avoid a fourth green dot in the first column. This results in

Now see that however the tiles are placed in Column 1, the dot at (2,1) — Row 2 and Column 1 — must be yellow. So (2,4) can’t be yellow, and therefore we know how to place the tiles in Column 4.

Note how the red and blue dots overlap to create magenta. Since there are no blue dots in the completed puzzle, we must place the tile in Row 3 so that the blue dot is covered by a green one (making a cyan dot).

Since green dots cannot overlap, we know how to place the tiles in Rows 1 and 4. Note the red and green dots overlapping in (4,4) to create yellow.

Since green dots cannot overlap, we know how to place the tile in Column 2. See how the overlapping dots create yellow, cyan, and magenta dots.

Now in Column 1, there can only one be yellow dot. So we know how to place the tiles in Column 1, and it is easy to slide the remaining tiles from Column 3.

That wasn’t so bad! Just a little knowledge of color theory and a bit of logic is all you need.

These puzzles are fun to create. However, there are a few issues to be aware of. The first arises since you want to avoid too many single dots — I used just one in the first puzzle, and I think that’s enough. It’s actually rather easy to create a grid with one of each color in each row and column — there are called Latin squares. The hard part is then creating an appropriate set of tiles. The difficulty lies with the fact that some squares are covered by just one tile, others by two.

In the following small grid, the numbers represent how many dots are needed to create a color in a certain location.

Suppose we wanted a vertical tile to cover a 1 and a 2 in column 3. Since (2,3) is covered by two tiles and vertical tiles cannot overlap, this means we must use a horizontal tile here. But then the only way to cover location (1,2) is with a horizontal tile and a vertical tile of height 1 (since the 1’s are already covered). But this is exactly what we want to avoid! Once I realized was happening, I first designed a grid with just 1’s and 2’s and a corresponding tile set, then added the colors later. In other words, I broke a harder design problem into two easier ones.

The other issue which arose was the question of uniqueness. It’s desirable to have a unique solution to puzzles like this — when you design a puzzle, you clearly have one solution, but can you be sure that there aren’t any others? This is not an insurmountable task, but for the larger puzzle below, it was more involved than I thought it would be.

So I leave you with a 6 x 6 puzzle to solve. Slide the tiles so that there is a red, green, blue, yellow, magenta, and cyan dot in each row and column. As mentioned above, the solution is unique. Feel free to post your own puzzles if you create them!