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Activity Set 9.3
Activity Set 9.3
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Models for Regular and Semiregular Polyhedra
285
MODELS FOR REGULAR AND SEMIREGULAR
POLYHEDRA
PURPOSE
To construct and use models in order to observe and examine the properties of regular and semiregular polyhedra.
MATERIALS
Scissors and patterns for constructing regular polyhedra on Material Cards 29 and 30. TwoCentimeter Grid Paper from Material Card 31.
M.C. Escher’s “Stars”
© 2008 The M.C.
Escher Company—
Baarn—Holland. All
rights reserved.
www.mcescher.com
INTRODUCTION
The term polyhedron comes from Greek and means “many faces.” A regular polyhedron is a
convex polyhedron whose faces are regular polygons and whose vertices are each surrounded by
the same number of these polygons. The regular polyhedra are also called Platonic solids because
the Greek philosopher Plato (427–347 b.c.e.) immortalized them in his writings. Euclid
(340 b.c.e.) proved that there were exactly five regular polyhedra. Can you identify the Platonic
solids in M.C. Escher’s wood engraving Stars?
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Geometric Figures
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Models for Regular and Semiregular Polyhedra
Platonic Solids
1. There are several common methods for constructing the five Platonic solids. The polyhedra
pictured here are see-through models similar to those in Escher’s wood engraving. They can
be made from drinking straws that are threaded or stuck together.
Another method is to cut out regular polygons and glue or tape them together. Material Cards
29 and 30 contain regular polygonal patterns for making the Platonic solids. Cut out the
patterns, use a ballpoint pen or sharp point to score the dashed lines, and assemble them by
folding on the dashed lines and taping the edges. For each Platonic solid, record the number
of faces (polygonal regions) and the name of the polygon used as the face.
Number of Faces
Type of Face
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
2. The faces of the icosahedron in the photo were made from colorful greeting cards. (The
triangular pattern for the faces are also shown. The vertices of these patterns were cut with
a paper punch and scissors, and the edges bent up along the dashed lines. Adjoining faces of
the polyhedra are held together by one rubber band.)
Icosahedron created by
a student at the University of New Hampshire
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Geometric Figures
a. If you were to make your Platonic solids from greeting cards as described on the previous page, how many rubber bands would you need to make each one? (One band is used
for each edge.)
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
b. You may enjoy planning colors for your Platonic solids so that no two faces with a common edge have the same color. For this condition to be satisfied, what is the least number
of colors needed for the octahedron? Explain.
3. Euler’s Formula
*a. In the following table, record the numbers of vertices, faces, and edges for each of the
Platonic solids you constructed in activity 1.
Vertices (V )
Faces (F )
Edges (E )
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
b. There is a relationship among the numbers of vertices, faces, and edges. Make and record
a conjecture about this relationship and express it in terms of the letters V, F, and E.
Conjecture:
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Models for Regular and Semiregular Polyhedra
*c. The following polyhedra are not Platonic solids. Count their vertices, faces, and edges
and record your results. Test your conjecture from part b on these polyhedra.
(1)
(2)
(3)
(4)
Vertices
Faces
Edges
4. Net Patterns for Cubes: A two-dimensional pattern that can be folded into a threedimensional shape without overlap is called a net for the three-dimensional shape. The shaded
areas in parts a–f below were formed by joining 6 squares along their edges. Three of the
following patterns are nets for a cube and three are not. (i) Identify the patterns in parts a–f
that are nets for a cube. (ii) Use Two-Centimeter Grid Paper (Material Card 31) to create at
least five more nets for a cube (there are 11 different nets in total that have 6 squares adjoined
along their edges that can be folded into a cube). You may wish to cut your patterns out to test
them. Sketch the additional nets for a cube in the provided grids.
a. Net or not?
b. Net or not?
c. Net or not?
d. Net or not?
e. Net or not?
f. Net or not?
g. Sketch a net
h. Sketch a net
i. Sketch a net
j. Sketch a net
k. Sketch a net
l. Sketch a net
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Geometric Figures
5. Archimedean Solids: A semiregular polyhedron is a polyhedron that has as faces two or
more regular polygons and the same arrangement of polygons about each vertex. There are
a total of 13 semiregular polyhedra shown below, and they are called Archimedean solids.
These were known to Archimedes (287–212 b.c.e.) who wrote a book on these solids, but the
book has been lost.
a.
b.
8 hexagons
12 squares
6 octagons
c.
8 hexagons
6 squares
f.
d.
20 hexagons
30 squares
12 decagons
g.
8 triangles
6 octagons
j.
20 triangles
12 pentagons
i.
20 triangles
12 decagons
l.
20 triangles
30 squares
12 pentagons
20 hexagons
12 pentagons
8 triangles
18 squares
h.
k.
8 triangles
6 squares
e.
32 triangles
6 squares
m.
80 triangles
12 pentagons
4 triangles
4 hexagons
The seven Archimedean solids shown in b, e, f, g, h, j, and l can
be obtained by truncating Platonic solids. For example, if each
vertex of a tetrahedron is cut off, as shown in the figure at the
right, we obtain a truncated tetrahedron whose faces are regular
hexagons and triangles, as shown in figure m above.
• Identify which Platonic solid is truncated to obtain each of
the semiregular solids in b, e, f, g, h, and j.
Truncated Tetrahedron
• Note: Vertices may also be cut off all of the way to the
midpoints of the edges.
• It may be helpful to look at the Platonic solids you constructed in activity 1.
6. Archimedean solids such as the one shown below (made by a student using this book) can be
created by cutting out the polygonal faces and taping them together.
a. How many pentagons and hexagons were needed to make this model?
b. What well-known sports ball has the same number of pentagon and hexagon faces?
An Archimedean solid
created by a student
at the University of
New Hampshire
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Activity Set 9.3
291
Models for Regular and Semiregular Polyhedra
JUST FOR FUN
INSTANT INSANITY
Instant Insanity is a popular puzzle that consists of four
cubes with their faces colored either red, yellow, blue, or
green. The object of the puzzle is to stack the cubes so that
each side of the stack (or column) has one of each of the four
colors.
Material Card 32 contains patterns for a set of four
cubes. Cut out and assemble the cubes and try to solve the
puzzle. Note; the arrangement of the cubes in the photo to
the right is not a solution to this puzzle.
Cube A
Cube B
Cube C
Robert E. Levin described the following method for
solving this puzzle.† He numbered the faces of the cubes,
using 1 for red, 2 for yellow, 3 for blue, and 4 for green, as
shown below. To solve the puzzle you must get the numbers
1, 2, 3, and 4 along each side of the stack. The sum of these
2
2
1
3
4
1
3
2
2
3
1
4
3
4
†R.
1
3
4
1
1
2
Cube B
Cube C
sum is 6. The bottom row of the table contains the sums of
opposite faces for each cube. For one combination of numbers
whose sum is 20, the sums have been circled. Notice that above
these circled numbers, the numbers 1, 2, 3, and 4 each occur
only twice. This tells you how to stack the cubes so that two
opposite sides of the stack will have a total sum of 20. (Both of
Sums of pairs
numbers is 10, and the sum of the numbers on opposite sides
of the stack must be 20.
Levin made the table at the bottom of this page showing
opposite pairs of numbers on each cube and their sums. For
example, 4 and 2 are on opposite faces of cube A and their
4
Cube A
Pairs of opposite faces
Cube D
1
2
4
Cube D
these sides will have all four colors.) The two remaining sides
of the stack must also have a sum of 20. Find four more numbers from the bottom row (one for each cube) such that their
sum is 20 and each of the numbers 1, 2, 3, and 4 occurs exactly
twice among the faces. The remaining two faces of each cube
will be its top and bottom faces when it is placed in the stack.
Cube A
Cube B
Cube C
Cube D
443
412
113
114
212
433
142
322
655
845
255
436
Levin, “Solving Instant Insanity,” Journal of Recreational Mathematics, pp. 189–192.
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Geometric Figures
Exercises and Activities 9.3
1. School Classroom: Suppose that your students are using sticks and gumdrops to create
models of the five Platonic solids. How will you help them decide how many sticks and how
many gumdrops they need and how to assemble their solids?
2. School Classroom: When asked to make a net for a cube, a drawing like net A is the usual
response. Design an activity for middle school students that will lead them to draw nets like
net B that are formed without using all squares. Describe your activity in such a way that
another person could follow your directions. Submit a few designs for a net by sketching
them on 2-Centimeter Grid Paper. 2-Centimeter Grid Paper is available for download at the
companion website.
Net A
Net B
3. School Classroom: Read the definitions of face, edge, and vertex in the Get Ready box at the
top of the Elementary School Text page at the beginning of this section. Review the answers
in the Check table for the number of faces, edges, and vertices for each shape and compare
them to the definitions. Make a list of at least three questions you anticipate that your students will ask about these ideas. Describe how you will answer those questions.
4. Math Concepts: Design a two-dimensional net for the following figure that is composed of
four cubes. Describe your procedure for constructing this net and submit a copy of the net
with your response.
5. Math Concepts: Open the Math Laboratory Investigation 9.3: Read Me—Pyramid
Patterns Instructions from the companion website and investigate the pyramid patterns
described in question 1 of the Starting Points for Investigations 9.3. Show your procedures
and explain your thinking.
6. Math Concepts: Check your conjecture for Euler’s formula (activity 3a in this activity set)
on the two-polygon Archimedean solids in activity 5 of this activity set (parts b, d, e, f, g, h,
i, j, and m) by listing and comparing the number of edges, vertices, and faces. You may wish
to do an Internet search on a site such as Math World (mathworld.wolfram.com) to find movable three-dimensional views of the solids pictured in activity 5.
7. NCTM Standards: Read over the Geometry Standards in the back pages of this book. Pick
one Expectation, from each grade level that the activities in this activity set address. State the
Expectations and the Standards they are under. Explain which activities address these
Expectations and how they do so.
www.mhhe.com/bbn
bennet t-burton-nelson website
Virtual Manipulatives
Interactive Chapter Applets
Puzzlers
Grid and Dot Paper
Color Transparencies
Extended Bibliography
Links and Readings
Geometer Sketchpad Modules