Math Concept Reader
Geometry in Art Math Concept Reader us66xs_lay_070605ad_sb.indd 1 6/5/07 2:44:47 PM Walk the Distance by Jennifer Marrewa ca66xs_lay_070109af_kb.indd 4 1/9/07 10:12:46 PM 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. Requests for permission to make copies of any part of the work should be addressed to Permissions Department, Gareth Stevens, Inc., 1 Reader’s Digest Road, Pleasantville, NY 10570. HARCOURT and the Harcourt Logo are trademarks of Harcourt, Inc., registered in the United States of America and/or other jurisdictions. Printed in the United States of America ISBN 13: 978-0-15-360493-5 ISBN 10: 0-15-360493-X If you have received these materials as examination copies free of charge, Harcourt School Publishers retains title to the materials and they may not be resold. Resale of examination copies is strictly prohibited and is illegal. Possession of this publication in print format does not entitle users to convert this publication, or any portion of it, into electronic format. 1 2 3 4 5 6 7 8 9 10 175 16 15 14 13 12 11 10 09 08 07 us66xs_lay_070605ad_sb.indd 1 6/5/07 2:28:03 PM ' Geometry All 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, Luis’s math teacher, Mr. Perez, notices Luis’s 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 day’s 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. We’re going to assemble an art gallery that shows geometry in the real world.” Painterly Architectonics shows how Lyubov Popova used plane figures in her art. The class walks to the school’s 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 center’s 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 class’s art gallery project. Simon decides that his favorite Popova painting is Painterly Architectonics. The painting includes many colorful triangles, quadrilaterals, and other plane figures. us66xs_lay_070605ad_sb.indd 3 6/5/07 2:28:10 PM 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 artist’s use of geometry. Be sure the copyright information for the image appears on the printout.” With the help of the school’s media specialist, Simon prints a copy of Popova’s Painterly Architectonics for the class’s 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 class’s art gallery. us66xs_lay_070605ad_sb.indd 4 6/5/07 2:28:13 PM Theo van Doesburg painted Contra-Composition of Dissonances, XVI in 1925. Chen looks through a stack of art books at one of the media center’s 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. us66xs_lay_070605ad_sb.indd 5 6/5/07 2:28:16 PM Leonardo da Vinci’s 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 Vinci’s 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 history’s greatest and most influential artists. Our art gallery would not be complete without one of his works.” us66xs_lay_070605ad_sb.indd 6 6/5/07 2:28:18 PM , Geometry in other 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. “We’re 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. We’re 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 center’s history section, where he finds a book about ancient Egypt. The book’s 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.” us66xs_lay_070605ad_sb.indd 8 6/5/07 2:28:21 PM [...]... 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... 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... 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 Pershing Square would be perfect for the art gallery,” Kim tells Luis, showing him the photograph on the computer screen “That’s an interesting park,” Luis answers “I never thought that there would be so much geometry in art. ” Mexican architect Ricardo Legorreta designed Pershing Square in downtown Los Angeles, California 11 us66xs_lay_070605ad_sb.indd 11 6/5/07 2:28:30 PM Chapter 3: Making A gallery... sculpture includes a combination of rectangles, triangles, and line segments The Pentagon in Arlington, Virginia, is the headquarters of the United States military 13 us66xs_lay_070605ad_sb.indd 13 6/5/07 2:28:32 PM 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 rhombusshaped boxes He colors in. .. these examples is, each of them uses geometry, ” Mr Perez says 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... 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 It’s 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. .. us66xs_lay_070605ad_sb.indd 14 6/5/07 2:28:34 PM 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... us66xs_lay_070605ad_sb.indd 9 6/5/07 2:28:24 PM 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. .. The students print out one of the complex pieces of fractal art they’ve discovered, wondering whether or not any of their classmates have discovered fractals 10 us66xs_lay_070605ad_sb.indd 10 6/5/07 2:28:27 PM 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 . 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,. kinds of art that use geometry? ” “That is an excellent question, Luis,” says Mr. Perez. “Drawings and paintings are just one form of art. We’re making an art gallery, not just a painting gallery 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