Lesson 15.8: Art Connection

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Whether Art or Math, Here's a New Place to See It

A Palace of His Own

Optical illusions, staircases leading nowhere, pictured reptiles that seem to crawl off the paper they're printed on—these describe some of the works of M.C. Escher. Called “Escher in the Palace,” the recently opened museum in The Hague is the first museum in the world dedicated to Escher's art.

Over 200 of Escher's original works—lithographs, woodcuts, engravings, drawings, and sketches—are housed in this museum. The artwork is displayed chronologically on the first two floors. Visitors can gaze at the optical illusions of Escher's “transformation prints,” like Sky and Water, in which bird images transform into fish, Metamorphosis, in which a city block transforms into a human shape, and Day and Night, in which flying birds transform into plowed fields far below. Also on view are many of Escher's “symmetry” prints as well as his “impossible” prints—in one, a waterfall flows upwards; in another, people walk up a sideways staircase.

On the top floor of the museum is a unique interactive display, a virtual-reality exhibit. Visitors put on headsets and move through scenes of Escher's design that appear as holograms, 3-D images formed by laser light. Here, for example, the reptiles seem to come alive as they appear to crawl out of the picture and then back into it.

Not a Mathematician

Escher, a Dutch graphic artist, is sometimes thought to have been a wonderful mathematician. But in school, Escher got poor grades in math as well as in his other subjects. Art was the only subject in which he excelled.

Although he never graduated from high school, Escher attended the Higher Technology School in Delft to study architecture. As a young man he traveled through Europe observing the art and architecture that motivated his work. In Granada, Spain, Escher visited the Alhambra, the intricately decorated Moorish palace. The tiling designs that covered the walls of the Alhambra were, said Escher, “…the richest source of inspiration that I have ever tapped.”

Mathematicians claimed that the triangle, square, and hexagon were the only polygons that could tessellate the plane. Escher distorted these basic shapes by turning them into various kinds of interwoven animals and objects that would also tessellate.

Escher once said, “It is…a pleasure to deliberately mix together objects of two and three dimensions, surface and spatial relationships, and to make fun of gravity.”

Word Wise

Arranged in order of occurrence: Our assignment was to make a chronological list of the events of the industrial revolution.

Unusual, one-of-a-kind: My cousin lives in a unique house built into the side of a mountain.

virtual reality:
A seemingly real “world” created by a computer: The virtual-reality ride made us feel like we were walking on the moon.

graphic artist:
Artist who creates drawings, paintings, and prints: The company had a graphic artist design advertisements for its product.

Elaborate, or very detailed: My mother's needlework design was very intricate.

Twist out of shape: Shawn used a computer program to distort these photographs.

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The prefix “uni-”, which begins the word unique, comes from the Latin word meaning “one.” In your group or with a partner, discuss the meanings of unique and these other “uni-” words:

  • unicorn
  • unicycle
  • union
  • unison
  • unit
  • unite
  • unity

Think about how some of the “uni-” words could be used together in a story. Then write a paragraph using at least three of the words. Share your paragraph with others in the class.

Data Hunt

You know that a tessellation is a repeating pattern that completely covers a plane without gaps or overlaps. But did you know that there are two kinds of tessellations, regular and semi-regular?

Regular tessellations are made up of all of the same regular polygons—all equilateral triangles, all squares, or all hexagons.

  1. Sketch a portion of each of the three regular tessellations. (You may want to use dot paper or triangle paper if you have it available.)
  2. Look at the adjacent vertices in each tessellation, the places where the corners of the adjacent polygons meet. Find the total number of degrees at each vertex.
  3. Write the total number of degrees at the vertices in each tessellation. This table will help.

    This table can help you keep track of the angle sizes.

    Regular Tessellations
    Kind of Tessellation Total Number of Degrees at Adjacent Vertices
    all equilateral triangles 60° + 60° + 60° + 60° + 60° + 60° = ?°
    all squares 90° + 90° + blank° + blank° = ?°
    all hexagons 120° + blank° + blank° = ?°

    Semi-regular tessellations are made up of two or more polygons. The sum of the adjacent vertices in a semi-regular tessellation is the same as the sum of the vertices in a regular tessellation!
  4. Create and sketch some semi-regular tessellations. Check that the sum of the adjacent vertex has the correct number of degrees! (If it does not, then your tessellation is not semi-regular.)