Is Math Art? Part II

Earth Day 2 (Aurora Australis and Bright Star Achernar)

When I started writing this post (several weeks ago!), I was aiming to answer the question posed in the title. I even thought I knew what I was going to say.

But — fortunately or unfortunately — these ideas are pretty complex. To write this post, I had to rethink my stance on some things I’ve said earlier in the semester. And that rethinking left me with even more questions.

Circle-style-warning

Warning: this post has many questions and few conclusive answers.

In the very first post on this blog, I asked,”Is math fiction?” In that discussion, I defined math as “the science of patterns.”

I’m not sure about that anymore.

Over the course of this year, I’ve written about math as a way to express relationships between things, as an exacting search for truth, and as a source of both cute puzzles and far-reaching ideas. In the post before last, I described my own deeply personal experience of math.

To my mind, “the science of patterns” is too vague to capture mathematics.

There is a sweet spot in the world of patterns that gives rise to good math. Good math is “interesting.” That term, “interesting,” gets thrown around by mathematicians a lot. It refers to a feeling people have about certain ideas and not other ideas, a feeling used to decide which patterns are worth studying.

Some “patterns,” like those of an argyle quilt, are too simple and lacking in meaning to be of interest to mathematics. Other logical structures are too complicated and discordant to be interesting. Not all patterns are math. And not all math is patterns.

Here is another possibility: because math is held to strict logical standards, one might try to define math as the study of logic. But math is more than pure logic. Just as important are guesswork, intuition, and creativity. If one is inclined to compare math to art, then logic could be called the medium of mathematics. It’s the paint, the charcoal, the clay, the language. If your image is not made from paint, it’s not a painting; if your idea is not logically consistent, it isn’t math. But the choice of what math to study — of which patterns to draw out, and which to leave be — is often a matter of taste. I don’t know any mathematician who feels he is grinding out dead logical statements.

I know mathematicians who study because they are curious about ideas, because they find them elegant or cute or perplexing and want to know more. A “nice” mathematical idea tugs at a person’s mind in a pleasant way.

All the diagonals of the elongated  pentagonal pyramid

I hope this blog has shown that side of math to people who didn’t know about it. Math is more than a tool to “feed big numbers into computers for some reason or other” (as Lockhart puts it). It is not just something people do to be smart or cryptic. It’s something people do because it’s fun, because they’re curious, and because it’s beautiful.

Personally, I think that joy is an amazing thing to be exposed to when you’re studying in a field.

Now, back to the original question.

Does that make math art?

Art is tricky to define. Perhaps Hardy, Lockhart, and others are simply using the term loosely. Or perhaps they see math in a way that I’m not advanced enough to see yet.

But I would say no. I don’t think math is art.

Still, I’m not done thinking about it. I haven’t really defined math, and I haven’t really defined art. This is only the beginning of a thought.

What do you think? What is math? Why do people do it? Is math art? Is “is math art” even a meaningful question?

Let me know what you think in the comments. Or ask something different. There are a lot more interesting questions to be asked, and we don’t have answers yet for most of them.

Isn’t that fun?

Earth Day 4 (Moon and Earth)

Is Math Art? Part I

Baseball String Art

Is math art?

For many, math brings to mind pencils and paper, obscure symbols, and dusty chalkboards. Even if you like math, you probably don’t think of it as being in the same category as painting or poetry.

Why ask if math is art?

Two written pieces prompted me to ask this question.

The first is a reflection by a mathematician looking back on his choice of career. The other is a critique of modern mathematics education.

Sphere Limit 1 AA11

A Mathematician’s Apology

Mathematician G. H. Hardy published “A Mathematician’s Apology” in 1940. In it, Hardy writes, “A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.”

The idea of math as patterns is something I’ve written about before. Hardy then articulates a further idea: “The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way.”

This description touches on artistry in math. But the idea that math’s patterns “must be beautiful” may be puzzling. What does Hardy mean by that?

A more recent essay, by an elementary school math teacher, takes an even stronger stance.

Lockhart’s Lament

Fifteen years ago, research mathematician Paul Lockhart decided to devote himself to elementary school teaching. In 2002 he wrote “A Mathematician’s Lament” as a critique of the current state of K-12 math education. The essay is brilliantly written: funny, sharp, and thought-provoking. (While I don’t endorse every one of Lockhart’s views, I highly recommend the piece.) Hardy spoke of patterns and beauty; Lockhart was much more direct:

“Mathematics is an art. The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such.”

Lockhart imagines a world where all arts are squeezed into the educational format that is now used to teach mathematics. In that world, art is stripped of meaning, reduced to something children hate and fear — because of the way it is taught. Lockhart exposes what he sees as a tragic distortion of the art of math:

“Everyone knows that something is wrong. The politicians say, ‘we need higher standards.’ The schools say, ‘we need more money and equipment.’ Educators say one thing, and teachers say another. They are all wrong. The only people who understand what is going on are the ones most often blamed and least often heard: the students. They say, ‘math class is stupid and boring,’ and they are right.”

Mathematics, Lockhart says, has been distorted beyond recognition. “The fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics.”

Finally, “Mathematics is the purest of the arts, as well as the most misunderstood.”

Why?

Hardy and Lockhart both viewed mathematics as art. And, from what I can tell, they are not alone. Many of the mathematicians I have personally encountered have expressed similar thoughts.

Some mathematicians feel that they are artists. Why? What quality of mathematics pushes its practitioners to describe it with words like “beautiful,” “elegant,” and “poetic”?

I’ll try to answer that in the next post.

Villarceau Variations

On Math and Meditation

Grand place, sun after rain
I find it almost impossible to explain the experience of being involved in math.

To describe something, a person uses words. If I have seen a pineapple before, and you have seen a pineapple before, and we both know the word “pineapple,” we can use the word “pineapple” to communicate. But if you’ve never seen such a fruit, I have to be creative to get you to understand. I might describe its stringy chewiness, bright yellow color, and piquant aroma; I might draw parallels to your other experiences, perhaps compare it to other foods. That way, we could communicate, even though you don’t have a previous concept of “pineapple.”

So here I am, every week in this blog, trying to explain math ideas in creative ways — such that, even if you’ve never savored a good piece of it, you can appreciate math’s unique flavor. This particular post will try to explain, in the best words I can find, what the experience of math is like for me.

What it’s not

  • Anything like what you did in middle school.
  • Adding, subtracting, multiplying, or other arithmetic operations. Or, manipulating algebraic symbols, remembering formulas, or drawing accurate pictures. In my experience, these are tools — useful, but not the heart of the experience. They are there to facilitate the more interesting ideas that come later.
  • Getting the “right answer.” That framework is pretty much irrelevant.

What it is (sort of)

Thinking about math involves multiple steps, much the way that baking a cake involves measuring, mixing, and applying heat.

Math starts with a translation step. When reading mathematics, you have to peel the words and symbols off the page, and turn them into a concept in your head.

If I read, for example, about an “open sphere S in R^4,” I first translate that symbolic language to semi-English words:  “the interior of a four-dimensional sphere.” Then I mentally review what “interior,” “sphere,” and “four dimensions” mean. (I’m not going to explain what these are here — I’m just bringing them to show that I translate symbolic statements into ideas I can understand.)

This translation process gives me a kind of mental picture of the sphere and its properties. I then hold that picture in mind and move on to the next phrase. A slightly sped-up version of this happens when I am sitting in class: when my professor refers to a difficult concept, I’ll pull up associations and mental pictures to make sense of what he or she is saying.

All of this translation can be difficult sometimes. It’s not to figure out anything deep — it’s just for seeing what exactly is being referred to. But when I encounter this type of challenge, even though it happens before I encounter the actual math, I’ll keep working at it, because I know that the concepts will be interesting once I get ahold of them.

Once the concept is in my head, there is … well … there’s the math.

waterfall

Math requires you to take a concept and stretch it, turn it over, invert it, or break it into pieces. It involves taking a logical argument and deciding whether it really makes sense, creatively employing tools to solve a problem, and often inventing tools on your own. More than almost anything else I know, it really works the mind.

The world of the concrete, of grass, clouds, chairs, and pencils, falls away. In its place, I consider an abstract scene: the grand, sweeping architecture of calculus; the straight, clean spaces of linear algebra; the chopped-up landscape of discrete mathematics.

The experience is completely absorbing in a way that I find almost meditative. My mind has no space for other thoughts. Everything else must quiet. With intense focus and hard thinking, I can often see the elegance of a mathematical pattern: something beautiful, outside of the self, that isn’t like anything else I know.

I can’t explain what math looks like from the inside, if you’ve never seen it, any more than you can explain the flavor of a pineapple to someone who has never tasted an acidic fruit. But I can say that pineapples are tingly, bright, sweet and little sour; and I can say that understanding math abstractly is like coming upon a mountain or a waterfall after a long hike. And that if the moment of understanding is like coming upon a waterfall, then, the experience of thinking is like the exhilaration of maneuvering through a dense growth of forest.

For me, climbing that path is almost as thrilling as reaching the destination.

Grand Canyon

The Mutilated Chessboard: Part II

Shadows and chessboard

Last post, we looked at a tiling of a complete chessboard:

Chess Board

Dominoes tiling 8x8

Then we cut off the corners of the board:

Mutilated checkerboard 3

The puzzle was to figure out whether or not there is a tiling of this board, and to explain either how to do it or why it can’t be done.

Don’t read further unless you’re ready to hear the solution!

Domino Spiral

Okay, here it is.

There is no way to tile the mutilated chessboard. It’s impossible.

One explanation is this.

Each domino covers exactly two squares: a white square and a black square. Try it yourself: imagine placing a domino on the board. You’ll see that no matter where you put your imaginary domino, it has to cover exactly one white and one black.

Mutilated checkerboard 3Since each domino covers exactly one white square and one black square, any board that can be tiled must have the same number of black squares as white squares. Otherwise, the dominos can’t match up with the board.

The mutilated chessboard has 32 black squares and 30 white squares. So it cannot be tiled by dominos. No matter how hard you try, there will always be two black squares left over.

That’s the reason it can’t be done.

I hope this explanation was clear. This is kind of a surprising answer. I for one would never have thought to look at the colors of the squares on the board. That’s a pretty creative way to go about solving the puzzle.

Keep in mind that there is more than one solution to this problem. The board can’t be tiled: that’s a fact. But there are many ways to explain why. I like this reasoning. You might not. Whether or not you find this interesting is a matter of taste.

Hope you enjoyed thinking about this!

The Mutilated Chessboard: Part I

Lafayette Park Chess Tables
Here is a little puzzle, just for fun. I may post hints in the comments if people ask.

Take an 8×8 chessboard.

Chess Board

Imagine we have a set of dominoes and we want to cover this chessboard. Each domino covers up exactly two squares. The goal is to cover the chessboard completely, in a way that none of the dominoes overlap or stick off the board.

First question: is that possible?

Answer: It is. Here is one way to do it. Each of the colorful rectangles below represents a domino.

Dominoes tiling 8x8

A setup like the one in the picture above, where the board is completely covered with dominoes that don’t overlap and don’t go off the board, is called a tiling.

Now take an 8×8 board and cut off the top right corner and the bottom left corner.

Mutilated checkerboard 3

Here comes the puzzle.

Can this board, without the two boxes on the corners, be covered completely with dominoes? Is there a tiling of this board?

There are two possible types of solutions to this puzzle. The format of the solution depends on whether a tiling exists or not.

  1. There might be a tiling that works. In that case, a solution would be the tiling: a picture of how to cover the board.
  2. There might be no tiling that works. In other words, it might be impossible to cover this board with dominoes, no matter how many combinations you try. In that case, a solution would consist of a reason why no tiling can work. If there can’t be a tiling, then why not? Can you prove it?

This is called the “Mutilated Chessboard” problem. It doesn’t involve a lot of mathematical sophistication. It does require creative problem solving. My favorite solution to this puzzle is clever, but simple.

Feel free to post thoughts or questions in the comments! I’ll try to respond, and maybe even post hints.

And no full solutions, please. We’ll get to that in the next post.

Featured Links: Fibonacci Sculptures

This video is not an animation. These are solid sculptures that, when spun under a strobe light, seem to move. The result is a little dizzying, but beautiful. Video here.

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Each piece is designed with the golden ratio in mind, and each has a Fibonacci number of spirals.

The creator of these sculptures, artist John Edmark, says he uses mathematics throughout his work:

I employ precise mathematics in the design and fabrication of my work. I do this neither out of a desire to exhibit precision per se, nor to exalt the latest technology, but because the questions I’m trying to formulate and answer about spatial relationships can only be addressed with geometrically exacting constructions. Mathematical precision is an essential ally in my goal of achieving clarity.

Many of his pieces have a sort of peaceful symmetry.

Articulated Spiral III

More is available on his website.

Math has beautiful order and symmetry, but it can be hard to see it in words on a page. It is interesting to see the ways that mathematical patterns are expressed in this artist’s work.

Habits of a Math Major

I’ve noticed that a lot of math people share the same quirks. While not everyone has the exact same habits, there are a few things I’ve noticed in myself as I’ve become more involved in mathematics.

Reams

1. Paper. There is scrap paper everywhere. Desk, backpack, folder. Stuck in between books. I have entire notebooks devoted to scratch work.

Buried Alive

2. Lots of hand motions when talking. I can’t just say “unit vector” – I have to say “unit vector” while jabbing my pen in the air at ninety-degree angles.

3. Describing mathematical ideas with words like “beautiful,” “fun,” “elegant,” or “cute.” As in, “That’s a beautiful theorem,” or “Here’s a cute problem.” This can surprise people who aren’t familiar with the deeper study of math. But it is actually quite common among people who do math and enjoy it.

Unit-distance:  An artistic drawing of the Petersen Graph

4. I have found that apparently I have given up certain expressions, like “ninety-nine percent sure” to mean “pretty sure.” If I’m really only pretty sure, I end up using a more appropriate number, like eighty percent sure. A bit weird, I know.

5. At unexpected times, I will suddenly, absolutely need to write something down. It could be on an airplane, in the shower, in the middle of some other conversation – all of a sudden, an idea strikes and needs immediate attention. I was recently waiting for a train in Penn Station when I had a great idea about a homework problem I was working on. I only had one piece of paper: a form I had to submit to my college advisor. I was so desperate that I almost wrote on the back of the form! And I probably would have, if I had not remembered that I had a doodle app on my phone.

Blizzard of 2015: Empty Penn Station

6. I’ve seriously considered getting a chalkboard for the wall of my room.

from imgkid.com

It makes me laugh to realize how many math habits I seem to have picked up. Many are actually useful – they help me do better work, or they help me appreciate math better. Some are just silly.

Interestingly, of all the eccentric people I’ve met in my college years, not one has fit the stereotype of the lonely, antisocial mathematician. Some are among the most personable and outgoing people I know.