Every year, the Joint Mathematics Meeting brings more than 5,000 math lovers together. It’s the largest math meeting in the world. In January 2019, mathematicians flocked to the meeting in Baltimore, Md., to learn about new ideas and talk about their work. Many even — believe it or not — came to admire the latest in mathematical art.
The meeting included an entire art exhibition. Visitors marveled at sculptures made from metal, wood, porcelain and folded paper. One was based on a supersized Rubik’s cube. Many included triangles, hexagons or other shapes, arranged in strange and surprising sizes and colors. The collection also included drawings and paintings inspired by the study of numbers, curves and patterns.
Math artist Robert Fathauer, who lives near Phoenix, Ariz., organizes the art exhibit at the annual math meeting. Every year, he notes, more and more artists submit their work. Their creations explore new and creative ways to turn math into art. Fathauer himself makes pottery sculptures inspired by mathematical patterns, including those found on the frilly edges of corals or kale. He finds inspiration by looking at how math shows up in the everyday world.
“I'm always kicking around ideas in my mind,” he says.
Art and math may seem like a strange pairing. People usually experience art through their senses. They see a painting or listen to music. If this art moves them, they will have an emotional response. Solving math problems is usually viewed as something you think about — not feel.
But connections between the two fields reach far back in time. Sculptors and architects in some ancient civilizations included numbers and math ideas into their works. Patterns in Islamic art that was made thousands of years ago show symmetries that mathematicians continue to study today.
“A mathematician, like a painter or a poet, is a maker of patterns,” wrote British mathematician G.H. Hardy in 1940. If a mathematician’s patterns are more permanent, he continued, “it is because they are made with ideas.”
Modern artists continue to look to math for inspiration, and in new ways. Fathauer, for instance, created a screen print inspired by his hometown. It uses three shapes — a scorpion, a snake and a bird — to cover a flat surface. The design is a nod to tessellations (Tess-uh-LAY-shuns). They are patterns that use interlocking geometric shapes to “tile,” or fill in, a two-dimensional plane. The artist lives in Phoenix (represented by a bird), where one can encounter scorpions and diamondback rattlesnakes.
The field of math art is growing. The exhibition at the 2004 Joint Mathematics Meeting, for example, included only 10 artists. By 2019, that number had grown to 94. “We get more submissions than we have room for,” Fathauer observes.
Galleries, art museums and festivals around the world have staged math art shows. In 1998, mathematicians launched the Bridges Organization, which celebrates the connections between math and art, music, architecture and culture. Every year, Bridges holds a meeting to celebrate those intersections. The Bridges conferences include art shows, talks about research on math and art, poetry readings and even performances of plays and dance inspired by art. The math art community now includes hundreds, maybe even thousands, of people around the world.
Strange shapes and four-dimensional tourists
Liz Paley is an artist in Durham, N.C. She exhibited a ceramic sculpture at the 2019 Joint Mathematics Meeting that represents a math shape known as the Klein bottle. (It’s named for the German mathematician who first described it.) This shape is special because if you trace its curves, you’ll travel over the entire surface and end up where you started. There’s no clear difference between its “inside” and “outside.” In this way, the Klein bottle only has one side.
Paley says she’s been designing Klein bottles for years. She learned about the shape almost by accident. While teaching a pottery class, a few years ago, she made a pot that she didn’t like. “One of my students asked, ‘Are you making a Klein bottle?’” Paley had never heard of it. So she began studying what makes this shape so unusual. Afterward, she started making them on purpose, experimenting with different methods and glazes. “I’ve been making them ever since,” she notes.
A Klein bottle is like a Möbius strip, which you can make by twisting a strip of paper once and then taping the ends together to make a loop. Sculpting a Klein bottle out of pottery, says Fathauer, is a tricky process: “People don’t appreciate how hard it is to do that kind of work,” he says.
Paley’s creation is about as close as we can come to a true Klein bottle. That’s because a true Klein bottle has to actually pass through itself, which is interesting to think about — but impossible to achieve in physical reality.
Her sculpture won the 2019 prize for best sculpture, textile or other material at the Joint Mathematics Meeting. She made her Klein bottle out of stoneware, the type of pottery often used to make dishes.
Math artists have used other materials for Klein bottles, as well. Bathsheba Grossman, a math sculptor who lives in Somerville, Mass., has 3-D printed her Klein bottle out of metal. She even went one step farther, adding a bottle opener to the design. So her work is a kind of math joke you can hold in your hand: It’s a Klein bottle opener.
Looking for math and finding art
Math artists tell similar stories about how they started creating. In many cases, a person who was talented in math and art had to choose one or the other as a career. But they never truly left the other behind.
Grossman went to college to study math, but while she was there she took sculpture classes. They led her to think about new ways to use 3-D printing to create shapes with surprising symmetries. Now, she’s designed hundeds of weird, geometry-inspired surfaces, and is even writing a computer program to design math sculptures.
Henry Segerman is a mathematician and artist at Oklahoma State University in Stillwater. When he was in high school, in England, he was good at math and art. But he had to choose. “I went in the math direction back then,” he says. He thought pursuing a career in art seemed “fickle” because it’s so difficult to succeed as an artist.
Still, Segerman’s math studies led him into the visual areas of math, such as geometry. “I was lucky in a way,” he says. “I was better at visualizing things than other people, and I focused on that.” But art wouldn’t leave him alone. In college and graduate school, he continued to create. Now, he creates scuptures that can cross from one dimension to another. One of his pieces, for example, is a patterned sphere. But when you shine a light from the inside out, the shadows project a square grid. This model demonstrates “projective geometry.” It’s a field of math that focuses on how different shapes and dimensions relate to one another.
In 2015, Segerman and some math art friends created a virtual-reality artwork. Participants can don a pair of VR goggles to float around and through four-dimensional shapes. Art makes it possible to interact with these shapes, which would be impossible to create in our three-dimensional world. As beautiful as it is to see, Segerman’s work also offers a new view on mathematical ideas.
Pieces of pi
Artists also find inspiration in other areas of math — including numbers themselves. John Sims is one. This math artist lives in Sarasota, Fla. His works are inspired by a combination of math, African-American culture, politics and other sources. On March 14, 2019, a gallery in Sarasota began showing some of his works inspired by the number pi.
Pi is the number found by dividing the circumference of a circle by its diameter. Because this number is the same for every circle, mathematicians call it a constant. As a decimal, pi is a number that goes on forever without repeating. Here’s how it begins, in case you want to memorize it:
The first few digits are 3.14. Sims’ art opening was chosen deliberately. March 14 is Pi Day because it’s written (in the United States, at least) as 3-14.
Since the early 2000s, Sims has organized at least 15 exhibits of math-based art. His works have been displayed in many more. He trained as a mathematician at Wesleyan University in Middletown, Conn. But when he moved to Florida, he began thinking about how to create visual versions of pi.
He began making quilts — big quilts, eight feet (2.4 meters) on a side. (He chose quilts in part because quilt-making has been an important part of the lives and history of African-Americans in the United States.) The quilts are each made of a grid of smaller squares of color. Each color corresponds to a digit. In one quilt, for example, a “3” might be represented by a black square, and “1” by green, and so on.
The center square in each quilt represents 3, the first digit in pi. Then the squares around that square line up in a color sequence that matches the next few digits of pi. And so it goes, with the colors spiraling out to the edges in a pi-inspired pattern. A viewer who doesn’t know about pi would only see a pretty color pattern. But knowing about pi helps a viewer connect art to math, Sims says.
He’s also designed quilts based on the Pythagorean (Pih-THAG-or-EE-un) theorem. To date, he has created 13 quilts. Some only use black and white. One, called “American Pi,” relies on the patriotic colors red, white and blue.
Math art, says Sims, has the ability to connect people to math in ways they hadn’t expected. Making art with pi, he says, is a natural way to do that.
“It’s part of our popular mindset,” he says.
An award-winning science writer, Stephen Ornes has been a regular contributor to Science News for Students since 2008. This piece — his 589th story for us — was inspired by his latest book: Math Art: Truth, beauty, equations (Sterling Press, New York, 2019, 192 pp).
3-D Short for three-dimensional. This term is an adjective for something that has features that can be described in three dimensions — height, width and length.
3-D printing A means of producing physical items — including toys, foods and even body parts — using a machine that takes instructions from a computer program. That program tells the machine how and where to lay down successive layers of some raw material (the “ink”) to create a three-dimensional object.
annual Adjective for something that happens every year.
ceramic A hard but brittle material made by firing clay or some other non-metal-based mineral at a high temperature. Bricks, porcelain and other types of earthenware are examples of ceramics. Many high-performance ceramics are used in industry where materials must withstand harsh conditions.
circumference The size of a circle or other geometric object by measuring the distance all of the way along its outer edge.
computer program A set of instructions that a computer uses to perform some analysis or computation. The writing of these instructions is known as computer programming.
constant Continuous or uninterrupted. (in mathematics) A number that is known and unchanging, usually based on some mathematical definition. For example, π (pi) is a constant equal to 3.14. . . and defined as the circumference of a circle divided by its diameter.
coral Marine animals that often produce a hard and stony exoskeleton and tend to live on reefs (the exoskeletons of dead ancestor corals).
culture (n. in social science) The sum total of typical behaviors and social practices of a related group of people (such as a tribe or nation). Their culture includes their beliefs, values and the symbols that they accept and/or use. Culture is passed on from generation to generation through learning. Scientists once thought culture to be exclusive to humans. Now they recognize some other animals show signs of culture as well, including dolphins and primates.
diameter The length of a straight line that runs through the center of a circle or spherical object, starting at the edge on one side and ending at the edge on the far side.
digit A structure, like a finger or toe, at the end of the limbs of many vertebrates. (in math) An individual numeral (from 0 to 9) used to represent a number or some part of a number.
dimension Descriptive features of something that can be measured, such as length, width or time.
edge (n network mathematics) A connection or link between two people or things.
fickle A term used to describe an attitude, opinion or loyalty that is changeable, often for no obvious reason, such as on a whim.
field An area of study, as in: Her field of research was biology. Also a term to describe a real-world environment in which some research is conducted, such as at sea, in a forest, on a mountaintop or on a city street. It is the opposite of an artificial setting, such as a research laboratory.
geometry The mathematical study of shapes, especially points, lines, planes, curves and surfaces.
glaze A material used to give a smooth, glossy coating to materials, especially porcelain. When pigments are included in this material, it can impart color to the surface as well.
graduate school A university program that offers advanced degrees, such as a Master’s or PhD degree. It’s called graduate school because it is started only after someone has already graduated from college (usually with a four-year degree).
grid (in mathematics or mapping) A network of lines that cross each other at regular intervals, forming boxes or rectangles, or an orderly field of dots that mark where each pair of lines intersect, or cross one another.
hexagon A geometric shape that has six equal sides. It takes its name from the Greek word for six.
high school A designation for grades nine through 12 in the U.S. system of compulsory public education. High-school graduates may apply to colleges for further, advanced education.
Klein bottle A surface first described by mathematician Felix Klein in 1882. Mathematically, it’s described as being one-sided. This is illustrated by the fact that you can run your finger over every part of its surface (apparently inside and out) without ever lifting your finger from the object or going over an edge.
mindset In psychology, the belief about and attitude toward a situation that influences behavior. For instance, holding a mindset that stress may be beneficial can help improve performance under pressure.
Möbius strip A famous “surface” in mathematics that has two sides but appears to have just one (because you can trace your finger from the outside to the inside and then continue on, ending up on the outside again). Such a strip can be made by cutting a long, thin strip of paper, putting a half twist in it, and then attaching the ends together.
model A simulation of a real-world event (usually using a computer) that has been developed to predict one or more likely outcomes. Or an individual that is meant to display how something would work in or look on others.
muse Someone who inspires a writer, painter or other type of artist.
physical (adj.) A term for things that exist in the real world, as opposed to in memories or the imagination. It can also refer to properties of materials that are due to their size and non-chemical interactions (such as when one block slams with force into another).
pi (in mathematics) Usually written using the Greek letter π. It is a constant equal to 3.14. . . and defined as the circumference of a circle divided by its diameter.
plane (in mathematics) A flat 2-dimensional surface, meaning it has length and width but no depth. A plane also extends infinitely in all directions.
politics (adj. political) The activities of people charged with governing towns, states, nations or other groups of people. It can involve deliberations over whether to create or change laws, the setting of policies for governed communities, and attempts to resolve conflicts between people or groups that want to change rules or taxes or the interpretation of laws. The people who take on these tasks as a job (profession) are known as politicians.
porcelain A hard, brittle material made by treating clay to a long heat treatment. The process, first perfected in Asia, came to be known as “china.” When treated before heating with a glaze, its surface can become impermeable, making it a good material for holding foods or liquids.
Pythagorean theorem A mathematical theorem, or rule, stating that the sum of the squares of the lengths of the sides of a right triangle is equal to the square of the length of the hypotenuse.
square (in geometry) A rectangle with four sides of equal length.
stoneware Pottery made by firing clay that contains silica and feldspar at a very high temperature. What develops is a dense, opaque and very hard type of material that frequently is used in the mugs and plates used for serving food.
tesselation A two-dimensional mosaic created by fitting together one (or perhaps a few) general shapes such that each shape shares each of its edges with some neighbor. There can be no overlaps of the shapes nor even minimal gaps between the shapes making up this pattern.
textile Cloth or fabric that can be woven of nonwoven (such as when fibers are pressed and bonded together).
virtual Being almost like something. An object or concept that is virtually real would be almost true or real — but not quite. The term often is used to refer to something that has been modeled — by or accomplished by — a computer using numbers, not by using real-world parts. So a virtual motor would be one that could be seen on a computer screen and tested by computer programming (but it wouldn’t be a three-dimensional device made from metal).
virtual reality (or VR) A three-dimensional simulation of the real world that seems very realistic and allows people to interact with it. To do so, people usually wear a special helmet or glasses with sensors.
Journal: R.W. Fathauer. A survey of recent mathematical art exhibitions. Journal of Mathematics and the Arts. Vol. 1, November 20, 2007, p. 181. doi: 10.1080/17513470701689167.
Journal: S. Happersett. The Cartesian MathArt Hive Exhibition, The Bowery Poetry Club, New York, 2009-2010. Journal of Mathematics and the Arts. Vol. 4, September 10, 2010, p. 163. doi: 10.1080/17513472.2010.490756.