GLeaM and STEAM: The Intersection of Math and Art

One of my favorite memories of math was on a random weekend morning in elementary school. That week, my teachers had brought up the area and perimeter of rectangles, and I was struck by a question: When does the area of a rectangle equal its perimeter?

I had noticed that both a 3 by 6 rectangle (or a 6 by 3 rectangle) and a 4 by 4 rectangle met the criteria, but I was plagued by the idea that I might not have them all. Maybe there was some obscure rectangle with dimensions much, much further down the number line that I had yet to discover. How could I be so sure? (I was considering only integer dimensions, by the way, but I didn't really know that term at the time.)

Pulling out a piece of paper and drawing it out, I realized that it occurred only when the "grid" within the rectangle has only four squares which do not directly touch the perimeter.

A rectangular drawing from the Excel archives on my computer. Notice how there are exactly four squares in the center that do not touch the perimeter.

I noticed that the perimeter is exactly four more than the number of squares that touch the perimeter (because there are four corners, which each contribute one extra unit segment), so for the area to be equal to the perimeter, it would also have to contain four more units than the number of squares that touch the perimeter, which exhibits itself as four squares in the center. Since there are only two ways to rearrange these four squares into a rectangle: 1 by 4 and 2 by 2, we know that these are the only two rectangles with integer dimensions whose area equals their perimeter. (There is, in fact, a way to prove this algebraically as well, though that is tangential to the point, so check out the Appendix at the bottom of this article if you'd like to see it.)

For me, this insight brought up two questions. First, why had my teachers not taught me this? I didn't fully understand the structure of our math curriculum well enough to critique it like I do now, but this did seem like the perfect open-ended question for my teachers to pose next. Second, why don't we immediately gravitate towards visuals? Problems like these were often understood as 3*6 and 2(3+6) rather than as visual objects. Isn't math just art in disguise? Why do we shy away from drawings and hide our paintbrushes if they are so revealing?

Of course, I do not tell this story in any way to praise younger me (who certainly did not phrase this idea in anywhere near as concise and clear terms) but because it is one of my favorite examples to pose those two questions. I've heavily addressed the first question in two earlier pieces: "Math: The Poetry of Ideas" and "Stop Teaching Our Students to Fear Math," so let's look at the second question and see if we can grant ourselves a valuable perspective on the intrinsic connections between math and art—and show just how many horizons open up when you begin to view math as an artform in and of itself.

What if at school you had to take an "art class" in which you were only taught how to paint a fence? What if you were never shown the paintings of Leonardo da Vinci and Picasso? Would that make you appreciate art? Would you want to learn more about it? I doubt it. You would probably say something like this: 'Learning art at school was a waste of my time. If I ever need to have my fence painted, I'll just hire people to do this for me.' Of course, this sounds ridiculous, but this is how math is taught, and so in the eyes of most of us, it becomes the equivalent of watching paint dry. While the paintings of the great masters are readily available, the math of the great masters is locked away. Edward Frenkel, Love and Math

This is one of my favorite quotes, and it illustrates, at least metaphorically, how similar math and art are. One of my major missions and philosophies for GLeaM is to showcase the creativity and accessibility of math, aspects that are often otherwise shoved to the side, and by walking through the proofs and ideas in my articles, I hope you've begun to see how much beauty lies in numbers. So without further ado, let's look at some mathy art!

Plate 4 Figurative Numbers from Beautiful Geometry by Eli Maor and Eugen Jost.

Take this visual from Beautiful Geometry by Eli Maor and Eugen Jost. Here you see a medley of triangular and square numbers.

A square number is any number of the form n*n = n^2, and you'll notice that you can take any square number (1,4,9,16,25, etc) and arrange that number of dots into a square. The bottom right visual, for example, is 7*7 = 7^2 = 49, and it becomes a square with side length 7.

A triangular number is any number of the form 1+2+3+...+n, and the sequence of triangular numbers begins with 1,3,6,10... You'll see that any number of this form can be arranged into a triangle:

Using Maor and Jost's compelling visuals, especially the one in the top-left corner, we're able to convince ourselves effectively that two consecutive triangular numbers add up to a square, and all the rearrangements of these dots grant us even more fascinating insight into the way that these two types of numbers interact—without even having to derive a more concise formula for triangular numbers!

In particular, from the bottom-right visual, we learn that a square number n^2 is the sum of n consecutive odd integers, while noticing that the triangular numbers are the sums of every other consecutive odd integer. Also, from the top-middle visual, we learn that the sum of eight identical triangular numbers is exactly one less than a square number.

(With a little algebra, you can prove that the formula for a triangular number is n(n+1)/2 because 1+2+3+...+n rearranges into (1+n)+(2+(n-1))+(3+(n-2)) + ..., which becomes (n/2)(n+1). See if you can expand on this and convince yourself of this. You can use these formulas to convince yourself of several of the facts we listed above, including the fact that two consecutive triangular numbers add up to a square number: n(n+1)/2 + (n+1)(n+2)/2 = (n+1)^2, but I would argue the visuals show a lot more than the formulas ever can.)

Of course, visuals are inherently associated with geometry too. One of my favorite figures is the nine-point circle.

Plate 41: French Connections from Beautiful Geometry by Eli Maor and Eugen Jost.

It turns out that the midpoints of the sides of a triangle, the feet of all the altitudes of the triangle, and the midpoints of the segments from the point where all the altitudes meet (the orthocenter) to the feet of those altitudes all lie on the same circle, called the nine-point circle! This theorem was published by two French mathematicians, Charles Julien Brianchon, best known for Brianchon's Theorem, and Jean-Victor Poncelet, best known for developing much of a field of math known as modern projective geometry.

Every triangle can be inscribed in one and only one circle called the circumcircle. Above, that circle is dark blue, and its center, called the circumcenter, is the yellow dot furthest to the bottom-left. The mathematicians also figured out that the center of the nine-point circle is halfway between the orthocenter and the circumcenter and its radius is one-half that of the circumcircle. Also, a third triangle center, the centroid, lies on the line (dubbed the Euler Line) that contains the orthocenter, circumcenter, and the nine-point circle center. All these connections are undoubtedly fascinating, and they are all revealed by this one beautiful visual.

Another one of my favorite theorems, Pick's Theorem, connects number theory, algebra, and geometry through art. Given a grid of lattice points (meaning points with integer coordinates), there is an amazing connection between the number of lattice points that are on the boundary of a polygon, the number of interior lattice points within a polygon, and the area of that polygon, as long as the polygon's vertices are also lattice points.

Photo courtesy of Wikimedia Commons.

It turns out that the number of interior points plus one-half the number of boundary points minus 1 is equal to the area of the polygon:

A = I + (B/2) - 1

The proof of this statement is incredible, and it involves many very intuitive visual steps. I would love to write an article about it in the future, so let me know if you would like to see that! For now, even if the formula seems arbitrary, I hope you can see how awe-inspiring it is that we can find the area of a polygon (a two-dimensional concept) just by visually counting points (a zero-dimensional concept). This is also often a fun and useful fact for math competitions.

If you're still skeptical and see many of the examples so far as more diagrams than true art, let's take this one step further. Besides simply serving as a visual tool for greater insight, mathematicians have been analyzing and creating artforms for centuries. One example of this is a fractal, any shape that has infinite self-symmetry, meaning that zooming in does not change the structure of the object.

This particular fractal is called a Sierpinski triangle, and it can be constructed in many ways. We can begin with an equilateral triangle, and then we cut out the middle triangle of the overall triangle, and then we cut out the middles of those triangles, and so on:

And we can get the Sierpinski triangle from a square which we repeatedly reduce to 1/4 of its original on the top. It turns out you can use any other shape in a similar way too to get the Sierpinski triangle!