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F L E X A G O N A R T

C R E A T I V E A C T I V I T I E S T H A T M A K E M A T H & S C I E N C E

F U N F O R K I D S !

A N D E R S H A N S O N A N D E L I S S A M A N N

C H

E C K

E R B O A R

D

H O W - T O LI B R A R Y

COOL ART WITH MATH & SCIENCE




A N D E R S H A N S O N A N D E L I S S A M A N N

C R E A T I V E A C T I V I T I E S T H A T M A K E M A T H & SCIENCE

F U N F O R K I D S !

C O O L A R T W I T H M A T H & S C I E N C E



V I S I T U S AT W W W . A B D O P U B L I S H I N G .C O M

Published by ABDO Publishing Company, a division of ABDO, P.O. Box 398166, Minneapolis,

Minnesota 55439. Copyright © 2014 by Abdo Consulting Group, Inc. I nternational copyrights

reserved in all countries. No part of this book may be reproduced in any form without written

permission from the publisher. Checkerboard Library™ is a trademark and logo of ABDO Publishing

Company.

Printed in the United States of America, North Mankato, Minnesota

062013

092013

Design and Production: Anders Hanson, Mighty Media, Inc.

Series Editor: Liz SalzmannPhoto Credits: Anders Hanson, Shutterstock

LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA

Hanson, Anders, 1980- Cool flexagon art : creative activities that make math & science fun for kids! / Anders Hanson andElissa Mann. pages cm. -- (Cool art with math & science)Includes index.

ISBN 978-1-61783-821-71. Polyhedra--Models--Juvenile literature. 2. Geometry, Solid--Models--Juvenile literature.3. Mathematical recreations--Juvenile literature. 4. Paper work--Juvenile literature. I. Mann, Elissa,1990- II. Title.QA491.H355 2013 516’.156--dc23 2013001902



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C O O L F L E X A G O N A R T

HIDDEN IN THE FOLDS

P O L Y G O N B A S I C S SIDES, VERTICES, AND ANGLES

I N T H E F O L D

TYPES OF FLEXAGONS

F O L D I N G T I P SMAKE IT EASIER!

P R O J E C T 1

MAKE A HEXA-TETRA FLEXAGON

P R O J E C T 2

MAKE A HEXA-HEXA FLEXAGON

3-D G E O M E T R Y

FROM SHAPES TO SOLIDS

P R O J E C T 3

MAKE A 3-D FLEXAGON

P R O J E C T 4

TRANSFORMER STAR

M A T H T E R M S

G L O S S A R Y

W E B S I T E S

I N D E X

C O N T E N T S




H I D D E N I N T H EF O L D S

4

Flexagons are toys made of folded paper. The paper is folded so some parts

are hidden. When the folds are flexed, they reveal the hidden parts!

Once you’ve mastered the basic flexagons, there’s even cooler stuff to make.

Build a transforming paper star. Or try your hand at a 3-D flexagon!



Most flexagons are flat. You can see the

front and back. But there are more faces.

They can’t be seen until the paper is flexed.

3-D flexagons have an extra dimension.

You can turn them inside out!

5



F

lexagons are made up of shapes

called polygons. A polygon

is a flat shape with straight sides.

Each point where two sides meet is

called a vertex. For any polygon, the

number of sides equals the number

of vertices. For example, a square

has four sides and four vertices.

A SQUARE

SIDE

VERTEX

ANGLE

When two lines meet at a vertex,

they form an angle. Angles are

measured in degrees.

A TRIANGLE

SIDE

VERTEX

ANGLE

P O L Y G O N B A S I C SS I D E S , V E R T I C E S , A N D A N G L E S

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C O N V E X O R C O N C A V E ?

All polygons are either convex or concave. Convex polygons are more

common. In a convex polygon, all of the vertices point away from the center

of the shape. In a concave polygon, at least one vertex points in toward the

center. Both shapes below are hexagons. They each have six sides.

CONVEX HEXAGON CONCAVE HEXAGON

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I N T H E F O L DT Y P E S O F F L E X A G O N S

Most flexagons have a square or hexagonal shape. Square flexagons are called tetra

flexagons. They are made of smaller folded squares. Hexagonal flexagons are called

hexa flexagons. They are made of folded triangles.

TETRA FLEXAGON HEXA FLEXAGON

C O M M O N F L E X A G O N T Y P E S

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Some flexagons have more hidden faces than others. The total number of faces a flexagon

has is indicated by a prefix added to its name. For example, a tri-tetra flexagon is a square

flexagon with three faces. A tetra-tetra flexagon is a square flexagon with four faces.

T R I - H E X A W H A T ? !

Think about the words unicycle, bicycle, andtricycle. Each word has a different prefix.

Uni means one, bi means two, and tri means

three. They are called number prefixes.

A number prefix indicates how much of

something there is.

A unicycle has one wheel and a bicycle

has two wheels. How many wheels does a

tricycle have? How many wheels would a

hexa-cycle have?

NUMBER

1

2

3

4

5

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PREFIX

UNI

BI

TRI

TETRA

PENTA

HEXA

HEPTA

OCTA

ENNEA

DECA

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F O L D I N G T I P SM A K E I T E A S I E R !

Making flexagons is fun! It’s even more fun when your projects turn out well. Use the tips

on these pages to help.

Use origami paper or lightweight card

stock. If the paper is too thin, the flexagon

may tear or wrinkle. If the paper is toothick, it will be difficult to fold and crease.

For glue, use flat-drying paste. It won’t

warp the paper. Scrapbooking glues, such

as Yes! Paste, work well. Don’t use toomuch, though. A little glue goes a long way.

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If you struggle with a project, take a break and come back to it later. It’s easier to solve

problems when you aren’t frustrated.

If you use a template, make the folds exactly

on the lines. The more accurate the folds,

the easier it will be to flex it.

After making a fold, run a fingernail firmly

along it. That’s called creasing. It makes sure

the fold is set. Wooden craft sticks workwell too.

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A

hexa-tetra flexagon is a square flexagon with

six faces. Fold it in half and open it like a book.

Hidden faces are revealed!

P R O J E C T

1

S T U F F Y O U´L LN E E D

SQUARE PIECEOF PAPER

PENCIL

SCISSORS

MARKERS

T E R M S

ROTATE

SQUARE

FACE

M A K E AH E X A - T E T R A

F L E X A G O N

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H O W T O M A K E I T

1 Fold the square paper in half. Crease it

and then unfold it. Fold it in half the other

direction. Crease it and then unfold it.

2 Bring one edge to meet the center crease.

Crease it and then unfold it. Rotate the

paper a quarter turn to the right.

3 Repeat step 2 three times. The paper

should be divided into 16 squares.

Draw along the creases with a pencil.

4 Cut out the four middle squares.

5 Label the squares on one side with the

letters AABCCBAABCCB. Start in the top

left square and go clockwise. Write one

letter in each square. Color the squares

that have the same letter the same color.

6 Flip the paper over from left to right.

Label the squares on one side with

the letters DDEFFEDDEFFE. Start at

the top left square and go clockwise.

Write one letter in each square. Color

the squares that have the same letterthe same color. Use different colors

than you used on the other side.

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5

6

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9

10

7 Flip the paper back over from left to

right. Fold the left edge to the center.

8 Fold the top edge down to the center.

9 Fold the right edge to the center.

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Unfold the bottom left fold. Pull thevisible D square up. Underneath will

be another D square. Pinch the D

squares together. Pull the D squares

through the center opening of the

paper. Twist them under the E square.

11 Fold the top half of the square back. The

middle fold should point upward. Open

the paper like a book. Rotate the square

a quarter turn. Repeat fold. Open.

Repeat to find every face of the flexagon.

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Ahexa-hexa flexagon is hexagonal. It has a total of

six faces. Four faces are hidden. You can write

secret messages or draw maps on them. Only your

friends will know how to find your messages!

P R O J E C T

2


PAPER

SCISSORS

COLORED

PENCILS

T E R M S

FACE

EQUILATERAL

TRIANGLE

HEXAGONAL

M A K E AH E X A - H E X A

F L E X A G O N

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1 Draw a row of 19 equilateral triangles.

Cut out the row of triangles. Fold and

crease the strip along each line.

2 Lay the strip flat with the long edge on

the bottom. Label the triangles with the

letters ABCABCABCABCABCABC. Label

one letter per triangle. Color the trianglesthat have the same letter the same color.

3 Flip the strip over. Keep the long edge on

the bottom. Label the triangles with the

letters DDEEFFDDEEFFDDEEFF. Label

one letter per triangle. Color the trianglesthat have the same letter the same color.

4 Turn the strip back over with the long edge

on the bottom. Fold the left two triangles

under and down. The B triangle should be

behind the C triangle next to it. Then foldthe A triangle under and down. It should

be behind the B triangle next to it. Keep

folding until you get to the end of the strip.

5 Now you should have a shorter, folded

strip. Lay it down with an A triangle on theleft. A C triangle should be on the right.

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5

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6 Fold the left half underneath

the right half. You should

now see four A triangles.

7 Fold the two triangles on the right down

so they are on top. You should

now see five A triangles.

8 Fold the bottom two triangles

up and underneath. Glue the

two blank triangles together.

9 Flex your flexagon! Pinch two of the

triangles together so that one point ofevery triangle is pointing up. Fold one

of the sides back to reveal a new face.

Continue to pinch and unfold to see

all the different faces of the flexagon.

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3-D G E O M E T R Y F R O M S H A P E S T O S O L I D S

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So far you’ve been working with flat shapes. Now let’s look at 3-D solids!

A polyhedron is a 3-D version of a polygon. A polyhedron has flat faces and straight edges.

Body (3-D) The body is the total space a

polyhedron takes up. It is measured in terms

of volume.

Faces (2-D) The surface of a polyhedron

is made of flat polygons. The edges of the

polygons join together. The polygons are called

faces. Faces are measured in terms of area.

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A point has no dimension. It has no shape that can be measured. A line has one dimension,

length. A flat surface has two dimensions, length and width. A 3-D object has three

dimensions, length, width, and height. How many dimensions do you have?

Edges (1-D) The frame, or skeleton, of a

polyhedron is made of straight edges. The

edges are measured in terms of length.

Vertices (0-D) The edges of a polyhedron

meet at points. The points are called vertices.

They have no dimension.

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T ime to add some dimension! This 3-D flexagon has

four faces. One of them is hidden from view. The

cool thing about this project is how it moves. It unfoldsoutward from its center as it rotates!

P R O J E C T

3


COPIER ORSCANNER

MARKERS OR

COLORED

PENCILS

SCISSORS

GLUE

FOAM BRUSH

T E R M S

3-D

DIMENSION

FACE

ROTATE

TRIANGLE

M A K E A 3 - D

F L E X A G O N

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1




1 Copy the template on the opposite

page. Cut away the gray parts. Leave

the black parts. Color each long row

of triangles a different color.

2 Crease the template along all of the lines.

3 Lay the template down with the

colored side up. The black

triangles should be on the left side.

Put glue on on all six black triangles.

(see opposite page for the next three photos)

4 Fold the side with the black

triangles back. Line up the colored

triangles over the black triangles.

Press them so the glue sticks.

5 Bring the ends together. Line the colored

triangles up over the black triangles. Press

them so the glue sticks. Let the glue dry.

6 To flex the flexagon, gently press on theoutside edge. A new color will appear.

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S T U F F



T his amazing star changes into an octagon! Just push

or pull the pieces. Watch it transform. Impress your

friends with the star’s shape-changing ability. Or havefun throwing it like a Frisbee!

P R O J E C T

4


8 SQUARES

OF PAPER

T E R M S

OCTAGON

PARALLELOGRAM

SQUARE

T R A N S F O R M E R

S T A R

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2




1 Start with a square sheet

of paper. Fold it in half.Crease it. Lay it down with

the fold on the left.

2 Fold up the bottom-left

corner to meet the right

side. Fold and crease.

3 Open the paper up. Bring the two

top corners down to the middle

of the paper. Fold and crease.

4 With one hand, start to bring

the left edge to the right edge.

With your other hand, push the

center of the bottom edge up. It

should be in between the sides

as you fold it. Crease the folds.

5 The folded shape is a

parallelogram. It has a pocket

on one side. It has a point

on the other side. Repeat

steps 1 through 4 with sevenmore sheets of paper.

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4

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6



6 Connect the pieces. Insert one

piece into the pocket of a secondpiece. Fold the tips of the first

piece down. They will be inside

the pocket of the second piece.

7 Insert a third piece into the pocket

of the second piece. Fold the

tips of the second piece inside

the pocket of the third piece.

8 Add the fourth, fifth, and sixth

pieces the same way. You

should have two pieces left.

9 Put the pocket of the seventh piece

around the point of the first piece.

Put the pocket of the eighth piece

around the point of the seventhpiece. The eighth piece’s point

should be in the sixth piece’s

pocket. Fold all the tips down.

10 Pull and push the sides to

change the shape.

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9

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M A T H T E R M S



M A T H T E R M S

3-D – having length,width, and height and

taking up space.DIMENSION – themeasurement of extensionin one direction. Length,width, and height areall dimensions.

EQUILATERAL TRIANGLE – a triangle with sides thatare all the same length.

FACE – a polygon thatforms one of the flat

surfaces of a 3-D shape.HEXAGONAL – havingsix sides and angles.

OCTAGON – a polygon witheight sides and eight angles.

PARALLELOGRAM – afour-sided shape in whichthe opposite sides are thesame length and parallel.

ROTATE – to turn on oraround on a center.

SQUARE – a shape withfour straight, equal sidesand four equal angles.

TRIANGLE – a shape withthree straight sides.

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G L O S S A R Y



G L O S S A R Y

ABILITY – the skill orpower to do something.

ACCURATE – exactor correct.

FLEX – to bend orstretch something.

FRUSTRATED – discouraged

or upset because somethingisn’t working or going well.

INSERT – to stick somethinginto something else.

ORIGAMI – the Japaneseart of paper folding.

OUTWARD – awayfrom the center.

PREFIX – a letter or group

of letters at the beginningof a word that affects themeaning of the word.

TEMPLATE – a shape youdraw or cut around to copy

it onto something else.VERSION – a different formor type from the original.

W E B S I T E STo learn more about math and science, visit ABDO Publishing Company on the World Wide

Web at www.abdopublishing.com. Web sites about creative ways for kids to experience math

and science are featured on our Book Links page. These links are routinely monitored and

updated to provide the most current information available.

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I N D E X



I N D E X

C

Concave

polygons, 7Convex polygons, 7

Creasing, 11

D

Dimension, 21

FFlexagons

making of, 10–11

materials for, 10

projects for,13–15, 17–19,23–25, 27–29

qualities of,4–5, 6, 9

templates for,11, 24–25

types of, 4, 5,8–9, 20–21

H

Hexa flexagons, 8

Hexa-hexaflexagons,making of, 17–19

Hexa-tetraflexagons,making of, 13–15

Hexagons, 8

MMaterials, 10

N

Number prefixes, 9

O

Octagons, 27

P

Polygons

qualities of, 6

types of, 7, 20

Polyhedrons, 20–21

S

Squares, 6, 8

TTemplates, 11, 24–25

Tetra flexagons, 8, 9

Three-dimensional(3-D) flexagons

projects for,23–25, 27–29

qualities of,4, 5, 20–21

types of, 20

Transformer star,project for, 27–29

Triangles, 8

W

Web sites, about

math andscience, 31

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