9780153604935_geometry in art

Geometry in Art

Math Concept Reader

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Walk the Distanceby Jennifer Marrewa

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DIGITAL FINAL PROOF

by Matt Doeden

Geometry in Art

Math Concept Reader

Copyright by Gareth Stevens, Inc. All rights reserved.

Developed for Harcourt, Inc., by Gareth Stevens, Inc. This edition published by Harcourt, Inc., by agreement with Gareth Stevens, Inc. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the copyright holder.

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ISBN 13: 978-0-15-360493-5ISBN 10: 0-15-360493-X

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'GeometryAll Around

Chapter 1:

Every day at lunch in the school cafeteria, Luis looks at cartoon books and books of Japanese drawings, called manga. Manga is a style of Japanese illustration that is often rich in action. One day, Luiss math teacher, Mr. Perez, notices Luiss manga book.

This is fantastic, Luis, he says. Look at how the artist used plane figures to create this scene. This would be a great example of geometric figures to show during our math lesson this afternoon.

After lunch, Mr. Perez begins the days math lesson. He tells the students about how plane figures appear in the real world. He shows the class a photograph of a tall building, and explains how each side of the building is shaped like a rectangle. Then he invites Luis to show the class his manga book.

Luis stands and shows his classmates some of the illustrations in his book. The pages include many plane figures, such as triangles, parallelograms, and circles.

Art is a great place to explore geometry in the real world, Mr. Perez explains. So, I have a class project for us to work on together. Were going to assemble an art gallery that shows geometry in the real world.

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Painterly Architectonics shows how Lyubov Popova used plane figures in her art.

The class walks to the schools media center. There, Mr. Perez tells the students to search in art books, magazines, and on the Internet for examples of geometry that is used in different art forms. He encourages them to look for a wide variety of examples.

Simon and Maria decide to begin their search on the Internet, so they head to one of the media centers computer workstations. There, they discover the work of a Russian artist named Lyubov Popova. Born in 1889, Popova discovered a love of art when she was young. She started taking art lessons when she was 11 years old. She spent the early 1900s traveling throughout Europe painting, studying, and teaching art. Her work incorporates, or uses, plane figures and seems perfect for the classs art gallery project.

Simon decides that his favorite Popova painting is Painterly Architectonics. The painting includes many colorful triangles, quadrilaterals, and other plane figures.






Peter Hugo McClure combines triangles and quadrilaterals in 3x36 Permutations.

Simon calls Mr. Perez over to see the image on the computer screen. Excellent work, you two, Mr. Perez says. Each of you should print a copy of a painting that shows an artists use of geometry. Be sure the copyright information for the image appears on the printout.

With the help of the schools media specialist, Simon prints a copy of Popovas Painterly Architectonics for the classs art gallery. Because he likes the painting so much, he prints a second copy to tape to the inside of his math notebook.

Maria, meanwhile, has discovered the work of an Italian-born artist named Peter Hugo McClure. His piece,

3x36 Permutations, is a patchwork of triangles and quadrilaterals. Maria is amazed at how plane figures can be combined to create such beautiful and amazing patterns. She is sure that 3x36 Permutations is a perfect example of geometry in art, so she prints it out for the classs art gallery.






Theo van Doesburg painted Contra-Composition of Dissonances, XVI in 1925.

Chen looks through a stack of art books at one of the media centers tables. When he opens a book about abstract art, Chen discovers all kinds of paintings with plane figures. As he flips through the pages, Chen decides that he likes an abstract artist named Theo van Doesburg. Like many abstract artists, the Dutch-born van Doesburg focused on geometry and colors, not natural forms.

Chen likes how van Doesburg used diagonal lines and line segments to create a series of rectangles. Chen decides that his favorite van Doesburg painting is Contra-Composition of Dissonances, XVI. The lines and colors in the painting show how plane figures can come together to make beautiful and interesting art.

Chen carries the book to the media specialist, who helps Chen make a photocopy of Contra-Composition of Dissonances, XVI. Chen notes the name of the book, its author, and its copyright information in his math journal.






Leonardo da Vincis Rhombicuboctahedron is one example of how he used geometry in his work.

Eva is looking at a book about classic artists. One of the artists featured in the book is Leonardo da Vinci. Da Vinci, who was born in Italy in 1452, was a scientist, mathematician, musician, and writer. He also was interested in many other areas of study. He painted, drew, created sculptures, and even designed buildings. Da Vinci is especially famous for the way he included bold plane figures in many of his works of art.

Eva finds an illustration that she thinks would be perfect for the class project. Da Vincis Rhombicuboctahedron is a drawing of a complex, three-dimensional form that is made up of a series of squares and equilateral triangles. These simple plane figures come together to create an intricate, three-dimensional object called a rhombicuboctahedron. Da Vinci drew the figure in 1509 for a book by artist Jacobo de Barbari, who used a rhombicuboctahedron in his 1495 painting, Paciolo.

This drawing has lots of geometric figures, Eva tells

Mr. Perez as he passes by. What an interesting image, Eva, Mr. Perez says. Leonardo da Vinci is one of historys greatest and most influential artists. Our art gallery would not be complete without one of

his works.






,Geometry inother forms of art

Chapter 2:

Mr. Perez gathers the class together so he can show the students all of the images they have collected so far. Were off to a great start, everyone. We already have a wonderful group of paintings that shows geometry in the real world.

Do we have to find just paintings and drawings? Luis asks. Or are there other kinds of art that use geometry?

That is an excellent question, Luis, says Mr. Perez. Drawings and paintings are just one form of art. Were making an art gallery, not just a painting gallery. Can you think of any other types of art that might use geometry?

As the students return to their search, they talk among themselves, wondering what other types of art might fit into their gallery. With an idea in mind, Luis heads off to a different section of the media center.





The Great Pyramids of Giza in Egypt have stood for more than 5,000 years.

While many students are searching through art books, Luis walks to the media centers history section, where he finds a book about ancient Egypt. The books cover shows a photo of the Great Pyramids of Giza. Luis thinks that the pyramids are an excellent example of geometry in the real world.

Luis reads that the Egyptians constructed the huge pyramids 5,000 years ago. Each pyramid has four equilateral triangular faces and a square base. Luis is amazed that a geometric solid could seem so beautiful and mysterious. The Great Pyramids have stood for more than 50 centuries, enduring wind, sandstorms, earthquakes, and pollution.

What have you found, Luis? Mr. Perez asks.It is a picture of the Great Pyramids of Giza in Egypt,

Luis answers. The pyramids are a perfect example of geometry in the real world, but some people may not think that they are art.

I think we agree that the Great Pyramids are art, Mr. Perez says. The Great Pyramids are an example of architecture, which is another form of art. The image of the pyramids will be an excellent addition to our gallery.







The Montreal Biosphre in Canada is a geodesic sphere that stands 200 feet tall.

Luis shows Lisa his picture of the Great Pyramids. Lisa likes it so much that she decides to find another piece of architecture for the gallery. Searching the Internet, she comes across plane figures that are used in designing geodesic spheres and geodesic domes. A geodesic sphere is made up of many connected triangles. A dome has the same structure, but is made from half of a sphere.

Lisa discovers that in the 1950s, a famous American architect named Richard Buckminster Fuller helped to develop the first geodesic dome. He wanted to build structures that were very light, but also very strong. Geodesic spheres and domes were one answer. They are now common features in many buildings around the world.

Lisa prints a picture of her favorite geodesic sphere, the Montreal Biosphre. This 200-foot-tall structure was originally built for the 1967 World Exposition in Montreal, Canada. The sphere was renovated in the 1990s and now contains a museum dedicated to water and the environment.






10

Some artists use computers to create complex fractal art.

At another computer workstation, Joseph and Tyrone are using the Internet to discover a new kind of art called fractals. They learn that a fractal has an endlessly repeating pattern that contains shapes that are like the whole, but of different sizes throughout. A snowflake is an example of a fractal that appears in nature.

Many artists use computers to create stunning artwork using fractals. Special computer software helps them use mathematical formulas to create works of art. Artists learn, however, that they can make a fractal called a Sierpinski triangle with nothing but a pencil and paper. The Sierpinski triangle is just a series of triangles that get smaller and smaller, but could go on into infinity.

The students print out one of the complex pieces of fractal art theyve discovered, wondering whether or not any of their classmates have discovered fractals.






11

Mexican architect Ricardo Legorreta designed Pershing Square in downtown Los Angeles, California.

Other students are discovering that geometry is also important in sculpture. Kim finds a Web site about a park in Los Angeles called Pershing Square. The park is filled with sculptures and is surrounded by beautiful architecture.

One of the parks features is especially interesting. The park has a bell tower called a campanile. This campanile rises to a height of 120 feet. The main part of the campanile includes a triangular prism and a rectangular prism. Large stone spheres are set onto platforms all around the campanile. A famous Mexican architect named Ricardo Legorreta designed Pershing Square. Legorretas use of bright colors throughout the park is a tribute to his Mexican heritage.

An image of Pershing Square would be perfect for the art gallery, Kim tells Luis, showing him the photograph on the computer screen.

Thats an interesting park, Luis answers. I never thought that there would be so much geometry in art.






&'

MakingA gallery

Chapter 3:

Soon, the bell rings. The students gather their notebooks, photocopies, and computer printouts, and follow Mr. Perez back to class.

We saw a lot of excellent examples of geometry in art today, Mr. Perez tells the students as he collects the photocopies and printouts. You found so many different types of art. We need to remember that art is protected by copyright, though. Im glad youve all included references to where you found your images.

The students watch closely as Mr. Perez pins the printouts to the board. Next, they take turns explaining how geometry is important in each piece. Joseph tells the class how artists use complex formulas to create fractal art. Luis talks about how even the ancient Egyptians used geometry in their art and architecture. Chen explains how abstract artists focus on plane figures and other features, rather than natural objects, in their images.

As different as each of these examples is, each of them uses geometry, Mr. Perez says.






1

The Pentagon in Arlington, Virginia, is the headquarters of the United States military.

The next day, the students are still talking about geometry in art. Chen tells the class that he saw a story about the Pentagon on television last night. The Pentagon, located in Arlington, Virginia, is the headquarters of the United States military. In fact, the Pentagon got its name because the building is the same shape as a regular pentagon.

Mr. Perez tells his class that they will create their own works of art in class today. They will make items for the art gallery using pencils, markers, paper, clay, and whatever else they can find in the classroom.

Within minutes, the students have spread out across the room and are designing different forms of art. In one corner, students are constructing a sculpture of solid figures made from clay. Meanwhile, Kim is creating a metal sculpture out of a series of bent paperclips. Her paperclip sculpture includes a combination of rectangles, triangles, and line segments.






1

This is one example of a Sierpinski triangle.

Chen uses markers and a ruler to make an abstract drawing. He draws a series of diagonal lines to make a pattern of rhombus-shaped boxes. He colors in some of the boxes in blue, red, and yellow, just as Theo van Doesburg did in Contra-Composition of Dissonances, XVI.

Joseph and Tyrone quickly get to work drawing their own fractal. They find a large sheet of paper and use rulers, protractors, and pencils to measure and draw an image that is almost like the Sierpinski triangle. As they draw, they discover that all of the triangles in their image are similar. That means that each triangle has the same shape as every other triangle in the drawing, but not necessarily the same size. They carefully measure each side of every triangle and color in some of the triangles with colored pencils.

Other students are busy creating illustrations with paper and pencil. Lisa is still interested in architecture, so she works on a design for a futuristic building that uses all sorts of plane figures. Luis, meanwhile, is drawing a picture that uses the style of Japanese manga.






1

Composition, 1927 by Patrick Henry Bruce is an example of abstract art that uses plane and solid figures.

Even Mr. Perez joins in on the creative process. He draws an abstract picture that shows different colored plane shapes and figures. He tells the students that his drawing is in the style of the painting Composition, 1927, by American artist Patrick Henry Bruce.

The students are working hard, and by the end of class they have created dozens of different works of art. They have incorporated squares, parallelograms, triangles, circles, and other plane and solid figures into their artwork.

It looks like our geometry book exploded in here! says Luis. Making this art gallery helped me see the different ways geometry appears around us. Its fun to have a hands-on approach to math.

Smiling, Mr. Perez stands up to congratulate the class on their hard work and creativity. Geometry is not just something on the pages of your school books, he says. Geometry is all around you, everywhere you look.







1

Picture credits: cover Walter, Bieri/epa/Corbis; p. 3 Tretyakov Gallery, Moscow, Russia/The Bridgeman Art Library; p. 4 Private Collection/The Bridgeman Art Library; p. 5 Haags Gemeentemuseum, The Hague, Netherlands/The Bridgeman Art Library; p. 6 Wikipedia; p. 8 Kazuyoshi Nomachi/Corbis; p. 9 Winston Fraser/Alamy; p. 10 Fractal Art by Ken Keller, www.fractalartgallery.com; p. 11 Robert Landau/CORBIS; p. 13 CORBIS; p. 14 Wacaw Sierpiski/Nol Aders/GNU Free Documentation License/Wikipedia; p. 15 Private Collection, Giraudon/The Bridgeman Art Library.

Glossaryabstract art an art form in which artists focus on color and form rather than on objects found in the natural world

architecture the style in which buildings are designed

campanile a bell tower that stands unattached from another building

copyright the right to publish or sell an image, book, song, or other unique work

fractal a figure with repeating patterns containing shapes that are like the whole, but of different sizes throughout

geodesic dome half of a sphere made up of smaller, connected plane figures

manga a style of Japanese illustration that is often rich in action

permutation a selection of different items in which the order is important

sculpture a work of art shaped out of stone, clay, wood, metal, or some other material

Sierpinski triangle a type of fractal that features a series of triangles that get smaller and smaller into infinity













&,

think and respond1. Look at Popovas Painterly Architectonics on page 3

and McClures 3x36 Permutations on page 4. Name two geometric figures that appear in both works.

2. Look at the image of the Great Pyramids of Giza on page 8. Describe the features of this solid figure. Name the number of faces, edges, and vertices for this solid figure.

3. Look at the Sierpinski triangle on page 14. Suppose you could magnify just one of the triangles inside the larger triangle. How would one of the smaller triangles compare to one of the larger triangles in the image?

4. Create your own work of art that uses geometric figures. Explain which geometric figures you included in your image.






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