Miles of Tiles

Austin TX
2013/04/09

It probably is not possible to make it through contemporary middle and high school math without seeing M.C. Escher’s work. A genius, for sure, Escher created mind-bending art, stretching and twisting classic geometric shapes to fit his desires.

A significant portion of Escher’s artwork was based on tessellations. A tessellation is a tiling of a plane: a fitting together of geometric shapes to cover a surface. Some shapes, like squares and hexagons, tile a surface all by themselves. The word “tessellation” comes to us from “tessella,” the Latin word for the little bits of stuff used in making mosaics.^[0]

I will never forget the images my algebra teacher, Mrs. Molloy, had in her classroom. The posters were images of planet earth projected onto different shapes like a cube, an ellipsoid and a torus. My first classroom as a high school math teacher was at Metro Arts, a school for visual and performing arts in Phoenix, Arizona. I hung a series of black and white Escher images above the chalkboard.^[1]. I hoped Escher’s images would be similarly etched in the minds of my students.

Earlier this year, I had my pre-algebra students do a tessellation project. We were in the midst of studying triangles, quadrilaterals, and common transformations of these shapes. M.C. Escher was there at the end, waiting for us. I think he knew we were coming for him. We viewed a brief history of tiling, the study of tessellations, and Escher’s work. Then, I set them to creating their own Escher-esque tessellations.

As you have read in my posts, I love background. I love learning about the broader context within which people lived and worked. The people who make math are people: people with families, homes, hopes, fears, and dreams. Some, like Escher, traveled.

Tiles at Alhambra (Wikipedia)

In 1922, at the ripe age of 24, Maurits Cornelis Escher visited Alhambra, a fortress in Spain from the 9th Century.^[2] Muslim rulers made Alhambra their home for centuries. Then, in the early 1500s, Charles V, Holy Roman Emperor, built a palace within the fortress walls.^[3] And, Alhambra has tiles! In one example, you can see the tessellation of the plane with altered equilateral triangles. Clear evidence of the influencing force Alhambra had on Escher. Follow this link to view more photos of tile work at Alhambra.

Escher was born in the Netherlands in 1898. Before traveling Europe, Escher embarked on becoming an architect. He fought recurring illness, failed courses, and altered his path to become the graphic artist we know. After Alhambra, Escher met and married an Italian woman. They lived in Rome until 1935.

Reptiles (Escher via Wikipedia)

As for many Europeans and Americans, the 1930s weren’t kind for Italians. In 1935 Italy, Mussolini had risen to power, fascism was unleashed, and World War II was on its way. Escher and his family moved around Europe until settling back in the Netherlands.

Amidst the political and economic tumult, Escher was drawing. Thousands of drawings.

Take a close look under those reptiles. You will see hexagons. And, somehow Escher made those reptiles crawl right off the page! If you thought “genius” was too strong a word, I invite you to reconsider.

Over the last several weeks, as if they have been whistling at me as I walk through rooms, I have been noticing tile. Tiles in bathrooms. Tiles in restaurants. Tiles in metro stations. Tiles in churches. Tiles in the Louvre. Tiles in cafés. Tiles made of clay. Tiles made of stone. Tiles made of glass. Miles of tiles.

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I thought of the graphpaper-like sections of cement around Austin, some of the same ones that inspired this whole shindig. I thought, “What if I create an Escher-esque tile and re-tessellate one of those stretches of cement?” I started on paper with a square and went to drafting possible tile shapes.

 

Then, I thought I should take a chance and tessellate some pavement without the aid of the grid. So, I went to work on a stencil.^[4]

You get a little better sense of scale with these:

With a big box of chalk and stencil in hand, it was time. Chalk at last.

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Here are four of my fave photos:

 

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[0] As I am guessing is frequently the case, this Latin word comes from a Greek word. In Greek, the word for square is “tessera.” While hexagons will always have a special place in my heart, the square tile must have gone into circulation soon after the first clay tile finished baking in the sun. Or, was it stone cut from the earth? Wikipedia’s Tessellation page has more on the history and mathematics of tiling.

[1] The year was 1999, Y2K was fast approaching, and life with chalk was lovely. I think that room had close to 30 linear feet of chalkboard. Even though, my clothes always ended the day all chalked up and the occasional deafening squeak was ever lurking, I loved the chalk.

[2] I have enjoyed learning about Escher’s life and work from The Official M.C. Escher Website.

[3] Be careful, the Alhambra de Grenada website has some breathtaking images. It also has pages of information on the history of the site. Spain is a destination on my list.

[4] I followed the same steps I asked my students to follow:

1. Choose a basic shape and draw it on paper
2. Cut a piece from one side, move it to another, tape in place
3. Use card stock (or cardboard) and make a stencil from the paper template
4. Tessellate large poster board (or sidewalk)
5. Decorate

These steps came from tessellations.org. They call it Slice Method 1.

And, yes, John, that is your classroom. Thanks for the workspace Khabele.