## Envelopes II: Making Spirals

While I don’t intend to make a habit of midweek posts (though thanks for the idea, Dane!), last week’s entry was so popular that I thought I’d write a short post on how the spiral designs are created.  Here’s one of last week’s images to refresh your memory.

These are not difficult to create,  but because of all the overlap of lines, it’s not immediately obvious how they are generated.  The image below shows the first forty line segments drawn, and highlights the endpoints of these segments.

The first line drawn is horizontal, and the endpoints are 180 degrees apart.   More lines are generated by having the orange endpoint (starting at the right) move 4 degrees counterclockwise, while the green endpoints (starting at the left) move just 3 degrees.  As a result, the endpoints of the second segment are just 179 degrees apart on the circle.  The orange endpoints start gaining on the green endpoints, closing the gap by 1 degree with each successive segment.

So the final spiral includes exactly 180 segments, which may be expressed in a form which may be easily generalized (and which is used in the Python code).  Note that the “180” on the left side of the equation represents the number of segments drawn, and the “180” on the right indicates the angular distance that the endpoints of the initial segment are separated on the circle.

$180 = \dfrac{180}{4-3}.$

All 180 segments are shown below, where the endpoints are retained to illustrate the process (although note that many endpoints are written over as the algorithm progresses).

The Python code is fairly short (here is the Sage link — look at some of the earlier posts about using Sage if you haven’t used it before). The only mathematics you need to understand is the standard conversion from polar to rectangular coordinates — it’s much easier to describe the endpoints first in polar coordinates since the radius of the circle is fixed and only the angles the endpoints make with the center of the circle change.

Finally, I include an image created with contrasting colors on the background as well.  Create your own version, and post as a comment!

## Envelopes I

Most of us have probably created a mathematical envelope, although we likely didn’t call it that.  Below is an example which may easily be sketched on a piece of graph paper.  You can see the “move one up/down, move one left/right” method of determining the ends of the line segments.  I created this envelope of lines using Python — we’ll look at the code later so you can make some of your own.

I first began sketching envelopes when I was an undergraduate.  Of course they were aesthetically quite interesting — but as a mathematician, you cannot help but ask exactly what curve you’re approximating by drawing an envelope of lines.

Although it seems that you might be drawing tangents to a circular arc, this is not the case.  By continuing the pattern of moving one up/down and left/right, the envelope in the first quadrant develops into the following figure.

You can fairly easily see a parabola forming — and this parabola is in fact the curve whose tangents we drew on our graph paper.

How can we prove this?  We need a theorem about tangents to parabolas, which we state as it applies to the case at hand, as shown below.

Here is how the theorem goes.  Suppose P and Q are points on a parabola whose tangents intersect at R.  If X is a point on the parabola between P and Q, and if the tangent at X intersects PR at M and QR at N, then

[PM] / [MR] = [MX] / [XN] = [RN] / [NQ],

where we use “[AB]” to mean the length of the segment AB.  But in our case, [PR] = [QR], so this means that

[PM] = [RN]  and  [MR] = [NQ],

which validates our “one down, one right” method of creating endpoints of tangent segments.  This is because by moving down one and right one, we are preserving the relationship [PM] = [RN], and hence also [MR] = [NQ].

For the curious reader, this helpful theorem about tangents to parabolas is very nicely proved by Steven Taschuk in his Notes on tangents to parabolas on p. 5 (where the names of points in my diagram above correspond to the names of points in his proof of the theorem).

So they’re parabolas!  Not what you’d expect at first glance.

Of course back when I was beginning college, there was no easy way to create desktop graphics, and so I used the tried and true pen-and-paper method.  I took a photo of one of my favorites from that time period.

Drawing these figure takes patience and concentration.  I distinctly recall the first draft of this piece.  If you look at the left square, you’ll notice that the blue lines look like they’re underneath the red lines.  This isn’t hard to do — just draw the red lines first, and then when drawing the blue lines, just stop at the red and continue after.  But I lost focus for just a second, and drew a blue line right over the red lines — and then had to start over.  There was no erasing.

For those of you who want to try creating images by hand, a word about technique.  When using an ink pen, the ink tends to blot — and if you drag the pen along a ruler, you might get end up with a smear of ink.  So keep a napkin or paper towel handy, and wipe the tip of your pen on the paper towel after drawing each line.  It’ll save you a lot of grief.

Despite the relatively simple method of drawing envelopes of lines, there is ample room for creativity.  Here is a computer-generated version of a drawing dated December 24, 1986.  It’s another of my favorites, and I love the effect of bright color against black.

There’s no reason to stop at purely geometrical designs, either.  Here is an abstract half-face of a tiger, created with a simulated graph paper background.

And then there’s polar graph paper!  When you wanted a graph, well, you just drew it — there was no graphing calculator/computer to do it for you.  So you had graph paper handy.  Below is a design I created just for this post, based on some older sketches.

Here’s another image based on a sketch made on polar graph paper.

It’s been really interesting to experiment with the computer, since designs which might have taken hours to draw can be rendered in a few seconds (after you’ve done the programming, of course).

Envelopes are not restricted to being created by lines, however.  The figure below creates a cardioid from tangent circles.

A base circle (in red) is given, and a point on the circle is selected (the white point at the left).  Now for every other point on the circle (like the red point), create a circle by using that point and the white point as ends of a diameter of the circle (drawn in black).  These circles are internally tangent to a cardioid.

I don’t have room for the proof of this construction here, but there is a very nice book called Envelopes by Boltyanskii (just google it) which illustrates many additional examples of envelopes as well as techniques to determine what the resulting curve is.  Some of the techniques do involve calculus, so be forewarned!  This book was my real introduction to a study of mathematical envelopes.

Now it’s time for you to create your own envelopes!  You can click on the Sage link to follow along and alter the code as you see fit.

The routines aren’t really too complex.  Note the use of vectors in Python to make working with the mathematics fairly simple.  This is so we can compute the endpoints of the segments of the envelope easily.  The mathematics involved is illustrated in the following figure.

It looks more complicated than it is.  Since $P_i$ is between $P$ and $R,$ we may write $P_i$ as a weighted average of $P$ and $R.$  We must do so in such a way that $i=0$ gives the point $P=P_0,$ and $P_{n-1}$ is just above $R.$

Similarly, $Q_i$ is a weighted average of $R$ and $Q,$ and must be such that $Q_0$ is just to the right of $R,$ and $Q_{n-1}=Q.$  Take a moment to study the figure and see that it all works out.  Note that the variables in the code mimic those in the figure exactly, so at least there is visual proof that the labels are correct!

So I’ll leave you to go ahead and experiment.  Feel free to comment with images you create using the Python code.  And stay tuned for next week, when I’ll talk about creating random envelopes.  There’s some really interesting stuff going on there….

## 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!

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

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:

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.

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

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.

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.

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.

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.

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

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

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

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

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

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

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

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

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

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

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

My first attempt looked like this:

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

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

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

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

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

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

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

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

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

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

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