Mathematical Masterpieces Geometry Article for Students

Rows of fish change shape as the spaces between them transform into flying geese. A drawing of a house shows staircases, windows, and columns that appear to pop out of and into the page at the same time. You take a closer look—what, exactly, is going on?

Whether or not you know his name, you’ve probably seen works by M.C. Escher before. The Dutch artist is famous for his detailed prints that include surprising shapes, intricate patterns, and perspectives that fool the eye. This unique imagery has made Escher’s work popular around the world. His art adorns T-shirts, tote bags, puzzles, and posters decades after his death in 1972.

Escher was fascinated by geometry, even though he never formally studied it. His artistic experiments with tessellations—shapes that combine to seamlessly cover a surface—have even caught the interest of professional mathematicians. One of them is Doris Schattschneider, a retired math professor and Escher expert. “Escher never felt he could do mathematics,” she says. “Yet in his own way, he discovered some important and original things.”

Rows of fish change shape as the spaces between them turn into flying geese. A drawing of a house shows columns that appear to pop in and out of the page at the same time. You take a closer look at the artwork in front of you. What, exactly, is going on?

Whether or not you know his name, you’ve probably seen art by M.C. Escher before. The Dutch artist is famous for his prints showing surprising shapes, detailed patterns, and perspectives that fool the eye. This unique style has made Escher’s work popular around the world. His art decorates T-shirts and posters long after his death in 1972.

Escher was fascinated by geometry, even though he never formally studied it. He often experimented with tessellations—shapes that combine to cover a surface with no gaps in between. This work has caught the interest of professional mathematicians like Doris Schattschneider. She’s a retired math professor and Escher expert. “Escher never felt he could do mathematics,” she says. “Yet in his own way, he discovered some important and original things.”

“I often seem to have more in common with mathematicians than with my fellow artists.” —M.C. Escher

Growing up in the Netherlands, Escher showed a natural interest in geometry. According to Schattschneider, young Escher would carefully select pieces of cheese for his sandwiches and arrange them as a tessellation on the bread. But at school, Escher didn’t excel at math. “I was an extremely poor pupil in arithmetic and algebra,” he later wrote.

Encouraged by an art teacher, Escher learned printmaking in high school. He went to art school to study architecture, but quickly switched to a program in graphic arts. On a trip through southern Spain in 1922, Escher visited the Alhambra, a medieval Islamic palace in the city of Granada. Colorful painted tiles arranged in repeating geometric patterns decorated many of the palace walls.

“Escher was fascinated by it,” says Federico Guidiceandrea, one of the curators of a recent Escher exhibit in New York City. The artist returned in 1936 with his wife, Jetta, and the couple spent hours sketching the mosaic designs.

Escher grew up in the Netherlands and showed a natural interest in geometry. When he was young, says Schattschneider, Escher would carefully select pieces of cheese for his sandwiches. Then he would arrange them as a tessellation on the bread. But at school, Escher didn’t excel at math. “I was an extremely poor pupil in arithmetic and algebra,” he later wrote.

A high school art teacher encouraged Escher to learn printmaking. Then Escher went to art school to study architecture. But he quickly switched to a program in visual art. In 1922, Escher traveled through southern Spain. He visited the Alhambra, a medieval Islamic palace in the city of Granada. Many of the palace walls are decorated with colorful painted tiles. The tiles are arranged in repeating geometric patterns.

“Escher was fascinated by it,” says Federico Giudiceandrea. He’s one of the curators of a recent Escher exhibit in New York City. Escher returned to the Alhambra in 1936 with his wife, Jetta. The couple spent hours sketching the tile designs.

In Fish (No. 20), below, Escher made one drawing of a fish and arranged it over and over to seamlessly cover an area. Each identical fish perfectly matches up with all the fish touching it.

Inspired by the Alhambra, Escher began inventing his own tessellations using recognizable shapes, such as fish, birds, and lizards. Over the next few years, he came up with dozens of these patterns, many of which he incorporated into his art.

Escher knew he couldn’t use just any shapes if he wanted to repeat them without gaps in the pattern. He often started by sketching simple shapes he knew would work, such as triangles or parallelograms. Then he adjusted one edge into the natural shape he wanted, making symmetrical changes on another edge so that the tiles would still align.

Escher took careful notes about this process and whether he needed to flip, rotate, or slide the shapes to fit them into a pattern. Over time, he developed a set of rules that allowed him to predict whether a new shape would work as a tile.

Inspired by the Alhambra, Escher began inventing his own tessellations. He decided to use recognizable shapes, such as fish, birds, and lizards. Over the next few years, he came up with dozens of these patterns. He incorporated many of them into his art.

Escher knew he couldn’t use just any shapes if he wanted to repeat them without gaps in the pattern. He often started by sketching simple shapes he knew would work, such as triangles or parallelograms. Then he adjusted one edge into the natural shape he wanted. He made symmetrical changes on another edge so that the tiles would still align.

Escher took careful notes about this process. He wrote down whether he needed to flip, rotate, or slide the shapes to fit them into a pattern. Over time, he developed a set of rules for tessellations. These allowed him to predict whether a new shape would work as a tile.

Unbeknownst to Escher, mathematicians were also working out the rules of tessellations. But it wasn’t until the 1950s that they became aware of each other’s work. By then, Escher had figured out many of the same principles mathematicians had. Although he didn’t have the mathematical vocabulary to describe it, “he could guess from pictures what the rules were, and then work in his own way to apply them,” says Schattschneider.

Escher’s tessellations appear in some of his most famous works, such as the 1943 print Reptiles (shown above). In it, crocodile-like creatures pop out of a piece of paper where they fit together neatly as 2-D tiles. Not everyone who sees the print appreciates the mathematical principles behind it, says Guidiceandrea. But Escher’s obsession with geometry helped give his work its lasting appeal.

“Escher went very deep into every detail,” Guidiceandrea says. “He started like a mathematician, but then he used the tessellation to tell a story with his art.”

Escher didn’t know it, but mathematicians were also working out the rules of tessellations. They didn’t find out about each other’s work until the 1950s. By then, Escher had figured out many of the same ideas that mathematicians had. He didn’t know the mathematical terms to describe his ideas, says Schattschneider. But “he could guess from pictures what the rules were, and then work in his own way to apply them,” she says.

Escher’s tessellations appear in some of his most famous works. One of them is the 1943 print Reptiles (shown above). In it, crocodile-like creatures pop out of a piece of paper where they fit together neatly as 2-D tiles. Not everyone who sees the print appreciates the math behind it, says Giudiceandrea. But Escher’s obsession with geometry helped make his work as popular as it is.

“Escher went very deep into every detail,” says Giudiceandrea. “He started like a mathematician. But then he used the tessellation to tell a story with his art.”

From 1926 to 1971, Escher filled five notebooks with hand-drawn tessellations that he used as reference and inspiration for his finished artwork. Use the artwork below to analyze the transformations Escher used in these patterns. Record your work and answers on our answer sheet.

Escher often used animals and other recognizable shapes as tiles in his tessellations. He created the fish pattern on the left in 1942. What kind of transformation is shown by the tiles marked A and B?

How is tile C related to A?

To which tile(s) is tile D related by rotation?

Escher created this unicorn tessellation in 1950. To which other labeled tile(s) is tile E related by translation?

Tiles E and H are related by more than one transformation. Describe the series of transformations required to map tile E onto tile H.

In tesselations like this one, Escher challenged himself to use the fewest colors possible while ensuring that tiles of the same color didn’t touch. Explain the series of transformations that describe how tile G is related to the other red unicorns.

In tessellations like this one, Escher challenged himself to use the fewest colors possible while ensuring that tiles of the same color didn’t touch. Explain the series of transformations that describe how tile G is related to the other red unicorns.