Math Concept Reader MCR g6 geometry in art

Geometry
in Art
Math Concept Reader
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Walk the Distance
by Jennifer Marrewa
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DIGITAL FINAL PROOF
by Matt Doeden
Geometry
in Art
Math Concept Reader
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ISBN 13: 978-0-15-360493-5
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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.
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
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.
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
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.
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
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.”
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,
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.”
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
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.
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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 they’ve discovered, wondering whether or not
any of their classmates have discovered fractals.
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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 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.”
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&'
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.
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.
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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.
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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. 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 is not just
something on the pages of your school books,” he says.
“Geometry is all around you, everywhere you look.”
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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 Wacław Sierpiński/Nol Aders/GNU Free Documentation
License/Wikipedia; p. 15 Private Collection, Giraudon/The Bridgeman Art Library.
Glossary
abstract 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
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&,
think and respond
1. Look at Popova’s Painterly Architectonics on page 3
and McClure’s 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.