Math Concept Reader MCR g6 geometry in art

Math Concept Reader

Geometry

in Art

Math Concept Reader

Walk the Distance

by Jennifer Marrewa DIGITAL FINAL PROOF

by Matt Doeden

Geometry

in Art Math Concept Reader

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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

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

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

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.”

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.”

The Montreal Biosphère 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 Biosphère 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

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 they’ve discovered, wondering whether or not any of their classmates have discovered fractals

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 park’s 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 Legorreta’s 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

“That’s an interesting park,” Luis answers

“I never thought that there would be so much geometry in art.”

Making

A 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 I’m glad you’ve 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

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

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