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.
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!
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.
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.
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!
All of these processes take an infinite number of steps, and there are incredibly interesting mathematical concepts, including fractional dimensions, hidden within their shapes. You can learn more about the Sierpinksi triangle (and lots of other amazing fractals) in the GLeaM article "Fractals: A Comprehensive Guide to Infinite Geometries!"
In fact, you may recognize the Sierpinski triangle from the GLeaM logo itself. I included a paintbrush with the fractal to further reveal the relationship between math and art—and to also recognize one of the most influential female mathematicians, Maryam Mirzakhani. I will reprint GLeaM's logo story here to tell a piece of Mirzakhani's story:
The great mathematician Maryam Mirzakhani's daughter often described her work as "painting," as she crouched on the floor drawing on large sheets of paper. Mirzakhani was the first woman to win the Fields Medal, an illustrious award that is often equated with the Nobel Prize for math. She was a major advocate for women in mathematics, and in 2017, she died from breast cancer at age 40. The paintbrush is a symbol of Mirzakhani, but also of the beauty of mathematics, closely intertwined with art in the way ideas transition marvelously into one another. Around the paintbrush is a fractal, the Sierpinski triangle, one of the most magnificent aspects of math, reminding us to pursue seemingly impossible dreams in surprising fractional dimensions.
Mirzakhani's deep and devoted creativity revolutionized her fields, including hyperbolic geometry, ergodic theory, Teichmüller theory, and symplectic geometry, and the expressiveness of her method shows how mathematicians are, in a sense, also artists.
Many famous mathematicians became artists throughout their lives. The Dutch graphic artist M.C. Escher studied tesselations and other mathematical objects from the papers of George Pólya, while researching plane symmetry groups, and wrote up his findings in a sketchbook, "Regelmatige vlakverdeling in asymmetrische congruente veelhoeken" ("Regular division of the plane with asymmetric congruent polygons"), which is largely viewed as mathematical research. He channeled that research directly into fantastical mathematical prints involving tessellations. Here are three examples of his work:
Lewis Carroll, the pen name of the author who wrote Alice's Adventures in Wonderland and Through the Looking Glass, was in "real life" a mathematician named Charles Dodgson. Many mathematical ideas and flights of fantasy are referenced in his books. I recently found The Annotated Alice in a used bookstore, which is full of notes by the popular mathematics writer Martin Gardner, who wrote a mathematical column for the Scientific American for 25 years and authored and edited over 100 books. The Annotated Alice was Gardner's most popular book, and it includes all sorts of mathematical allusions, as Charles Dodgson/Lewis Carroll brought his loves of children's literature and mathematics together. (Dodgson's illustrator, Sir John Tenniel, also used a lot of math in his drawings.)
Once you start to see the connections between art and math, you'll begin to notice how artists incorporate geometry and number theory in so many clever ways.
This famous woodblock print above, The Great Wave off Kanagawa by the Japanese artist Hokusai, incorporates a certain specific spiral. And so does the Mona Lisa:
And even the Parthenon:
This spiral comes from a famous sequence called the Fibonacci Sequence where each number is the sum of the previous two: 1,1,2,3,5,8,13,21... It turns out that the ratio of consecutive terms in this series approaches the golden ratio, φ (phi), which is approximately 1.61803. This number pops up all the time in shapes that look aesthetically pleasing, including proportions within the Parthenon and Mona Lisa. It seems that we simply gravitate towards this ratio when trying to create something beautiful, and perhaps, the very concept of beauty lies in mathematics.
Here, the spiral is called the golden spiral, and it is constructed from a series of spiraling squares, which create rectangles called golden rectangles whose proportions are precisely the golden ratio. There is a lot of debate about whether the prevalance of the golden ratio is legitimately scientifically important or if it is merely a construct that artists and graphic designers have propagated over time, but both its mathematical importance and its frequency in the art world can hardly be disputed.
You can learn lots more about the golden ratio and Fibonacci numbers in this GLeaM article: "Good as Gold: The Fibonacci Sequence, the Golden Ratio, and More"
In a broader sense, geometric proportions in general are critical for visual artists, and one of the best places to see this is in the Dutch artist Piet Mondrian's work, who is famed for his artwork with geometric, especially rectangular, shapes.
Indeed, music also springs directly from math, as the harmonies we find the most pleasing to the ear come in simple ratios. When the frequency of one note is exactly twice another, we have an octave, and when the frequency of one note is exactly 3/2 times another, we have a fifth. These values are often called Pythagorean Intervals, as they were developed by the same mathematician as the Pythagorean Theorem. There is so much more to talk about when it comes to math and music, but I'll have to leave that discussion for an article if you would like to see it. (In fact, Fourier waves (which are involved in the Riemann Hypothesis) are also intrinsically associated with music.)
You can find math everywhere in art—and art everywhere in math. I hope this article revealed the sheer magnitude of the connections between the two subjects. Indeed, the closer you look, the more they appear to mirror each other: Both artists and mathematicians revel in patterns, and both disciplines are taught initially from techniques and tools (and calculations) until you're able to forge a path on your own (whether that's by solving a problem or painting a tapestry).
As for the questions I originally posed at the top of this article, I think today's journey has moved us a step closer to understanding math intuitively—by delving into the artwork and visuals that make up the scaffolding of the field. In general, I think today's teachers are doing a much better job incorporating visuals into the coursework, but we are still a long way from truly teaching intuition and creativity within math—and it starts with revealing how fundamental art and creativity are to a subject that t can otherwise seem like simply flat numbers on a page. As you go through math classes, I urge you to continue looking for your own insights, asking your own questions, and sketching out everything you can; then, you'll be thinking like a mathematician!
Of course, this article was meant as a broad overview, and there are so many other topics on which I would have loved to delve into depth, so much so that this may end up being the topic of my future book! Until we reach that point (probably way into the future), I highly recommend you check out Beautiful Geometry by Eli Maor and Eugen Jost and Math Art: Truth, Beauty, and Equations by Stephen Ornes. Both are amazing reads, and I recently found out Eli Maor also has a book on music: Music by the Numbers: From Pythagoras to Schoenberg.
Feel free to discuss anything related to this article in the comments and forums or by emailing me; I would love to hear what you have to say! Have a great week everybody!
Algebraic Proof of the Rectangle Problem
If we reach for slightly more advanced tools, we can find the answer to the problem in the introduction of this article algebraically.
Given an m by n rectangle, we know the area is mn and the perimeter is 2(m+n). Setting these two quantities equal, we have the equation:
mn = 2(m+n) → mn - 2m - 2n = 0
We can factor this expression if we add 4 to both sides. (This is called Simon's Favorite Factoring Trick, and you can learn more about it here.)
mn - 2m - 2n + 4 = 4 → (m-2)(n-2) = 4
At this point, if we want integer answers, we can factor 4 in its only possible three ways as 1*4, 2*2, 4*1. These give the values for (m,n) as (3,6), (4,4), and (6,3), and if we consider the 6 by 3 rectangle to be the same as the 3 by 6 rectangle, we have the same two rectangles we found at the beginning.
Notice, however, that the visual insight helps us understand where (m-2)(n-2) = 4 comes from in a much more intuitive way. m-2 is simply the dimension of the rectangle by removing the outside squares that touch the perimeter along one direction, and n-2 is the same in the opposite direction, so they represent the dimensions of that "middle" rectangle we were dealing with in our visual.
If you can wield the algebra, this method is more straightforward and direct to the answer, but the visual method still grants us the most insight.