Unit 4
The Sierpinski Tetrahedron
Building the Sierpinski tetrahedron
In this unit, students will use their experience with iterative processes to construct
the Sierpinski tetrahedron and investigate its properties. While the construction of
the Sierpinski tetrahedron is purely geometric and therefore quite intuitive, the
understanding of its properties is based on some important observations on how
length, area, and volume of an object are affected when the size of the objects is
changed. A detailed discussion of these concepts can be found in the Scaling and
Dimension CATE. The students will work in cooperative groups of four to build
successive stages of the Sierpinski tetrahedron.
Grade Level:
5-8
Materials:
Four 8-inch equilateral triangles per student
Six 4-inch tetrahedron templates per student (four of one color,
two of a different color)
16 2-inch tetrahedron templates per group of four students
Scissors
Transparent tape
Finding Patterns in the Sierpinski Tetrahedron handout
Finding Patterns in the Sierpinski Tetrahedron Transparency
1. Ask students, “How is the area of a square affected if the length of each side of
the square is reduced by 1/2?” [Most students will say the area is also reduced by
1/2.] After discussing their ideas, substitute the new lengths of the sides into
the formula for the area of a square. Simplify the equation and demonstrate that
the area of the smaller square is 1/4 the area of the larger square.
SSS
Sunshine State Standards
MA.A.2.3.1
MA.A.5.3.1
MA.B.12.1
MA.B.1.3.1
MA.B.1.3.3
MA.B.2.2.2
MA.C.1.3.1
MA.C.2.2.1
MA.C.2.3.1
MA.D.1.2.1
MA.D.1.2.2
MA.D.1.3.1
LA.A.2.2.8
SC.H.1.2.2
SC.H.1.2.5
SC.H.1.3.5
SC.H.2.3.1
The Sierpinski Triangle
a
1
a
2
A = a2
1
1
A =  a = a 2
2 
4
2
65
Unit 4
Developed through the National Science Foundation Project Pattern Exploration: Integrating Mathematics and Science for the Middle Grades, 1999
Ask students, “How is the area of an equilateral triangle affected if the length of
each side of the triangle is reduced by 1/2?” (The altitude is also reduced by 1/2.)
After discussing their ideas, substitute the new lengths of the side and altitude
into the formula for the area of a triangle. Simplify the equation and observe that
the area of the smaller triangle is 1/4 the area of the larger triangle.
1
h
2
h
1
a
2
a
A=
1
ah
2
A=
1  1  1  1  1  1  1 
a
h =
ah =
ah
2  2  2  2  4  4  2 
Students will see that whenever the sides of a two-dimensional figure are reduced
by 1/2, the area of the new figure is 1/4, or (1/2)2 as much as the area of the
original figure. Point out to students that the “area” of a geometric figure refers
to something two-dimensional. Thus, the exponent 2.
2. Repeat this procedure for volume. Ask students, “How is the volume of a cube
affected if the length of each edge of the cube is reduced by 1/2?” [Most
students will probably say the volume is reduced by a factor of 1/4.] After
discussing their ideas, substitute the new lengths of the edges into the formula
for the volume of a cube. Simplify the equation and demonstrate that the volume
of the smaller cube is 1/8 the volume of the larger cube.
1
e
2
3
1
1
V =  e = e3
2 
8
e
V = e3
Next ask students, “How is the volume of a tetrahedron affected if the length of
each edge of the tetrahedron is reduced by 1/2?” [Review the formula for the
volume of a tetrahedron.] After discussing students’ ideas, substitute the new
lengths of the edge and altitude into the formula for the volume of a tetrahedron.
Simplify the equation and demonstrate that the volume of the smaller tetrahedron
is 1/8 the volume of the larger tetrahedron.
The Sierpinski Triangle
66
Unit 4
Developed through the National Science Foundation Project Pattern Exploration: Integrating Mathematics and Science for the Middle Grades, 1999
1
k
2
k
1
e
2
e
V=
1
Bk
3
V=
1
B = area of base triangle of tetrahedron = eh
2
k = height of tetrahedron
1 1 
1
V =  eh k = ehk
3 2 
6
1  1   1   1   1 
e
h
k
3  2   2   2   2 
1 1 1 1 1
=           ehk
 3  2   2   2   2 
1 1
=    ehk 
 8  6

In conclusion, whenever the edges of a polyhedron are reduced by 1/2, the
volume of the new polyhedron is 1/8 or (1/2)3 as much as the volume of the
original polyhedron. Explain to the students that the volume of a polyhedra refers
to something three-dimensional. Thus, the exponent is 3.
3. Distribute four 8-inch equilateral triangles to each student. Ask students to
construct a tetrahedron. This tetrahedron is referred to as Stage 0.
4. Show the students a Stage 1 tetrahedron. Distribute four 4-inch tetrahedron
templates per student. Ask students to construct a Stage 1 tetrahedron. Secure
the top tetrahedron at only one vertex to form a hinge so that the tetrahedron
can be flipped open to enable the students to look inside the figure. Discuss with
students how the two figures they have constructed are alike. [The height of each
completed figure is the same; the tetrahedra that make up Stage 1 are similar to
the tetrahedron of Stage 0.]
5. Distribute sixteen 2-inch tetrahedron templates to each group of four students.
Each student should construct four tetrahedra. With the sixteen tetrahedra, ask
the students to assemble a Stage 2 tetrahedron. Allow the students time to
explore and experiment. Discuss with students how the three figures they have
constructed are alike. [The height of each completed figure is the same; the
tetrahedra that make up the stages are similar.]
If students view the tetrahedron from a different perspective, an interesting
phenomenon occurs. Have the students look directly onto the edge of the
tetrahedron. From this perspective all “holes“ become invisible! In other words,
the three-dimensional Sierpinski tetrahedron can be projected onto a twodimensional object—a square!
The Sierpinski Triangle
67
Unit 4
Developed through the National Science Foundation Project Pattern Exploration: Integrating Mathematics and Science for the Middle Grades, 1999
Finding Patterns in the Sierpinski Tetrahedron
6. Distribute the Finding Patterns in the Sierpinski Tetrahedron table. The purpose for
completing the table is to examine the changes occurring in the total length of
the edges, the total volume, and the total surface area. Stage 0 has a length of
one edge, e; an area of one face, A; and a volume, V. Together with the students,
complete the table for Stages 0, 1, and 2. Ask students to complete the table for
Stage 3. Upon completion of the row for Stage 3, discuss the answers with the
class to verify that they have filled in the row correctly and determined the
patterns. Together with the students, complete the table for Stage n by examining
the patterns of change that occur in each column.
Observe with students that as the stage number approaches infinity, the resulting
figure will have:
• no volume
• infinite length of edges
• surface area that remains constant.
This remarkable fact flies in the face of logic, common sense, and previous
experience!!
The Ins and Outs of the Sierpinski Tetrahedron
7. Ask students to refer to the Stage 1
tetrahedron. Have the students flip
the top tetrahedron open and
examine the inside. Ask students,
“What shape is the hole?”
[Responses will vary as this is a
three dimensional shape that is
difficult for students to visualize.]
Distribute two four-inch
tetrahedron templates of a different
color to each student. Using these
two templates, students will
construct the “hole” of the Stage 1
tetrahedron. Fold the templates
but do not tape the edges closed.
Holding the base of a tetrahedron in each hand, have the students rotate one of
the figures 90° and dovetail the faces together to form an octahedron. Work in
pairs so that one student can hold the figure while the other student tapes the
edges together. Have the students place the constructed octahedron in the hole
and close the tetrahedron.
Compare the volume of the Stage 1 tetrahedron to the volume of its interior.
According to the table, the total volume of Stage 1, 1/2V, is half the volume of
Stage 0, V. If Stage 1 has a volume of 1/2V, then the hole also has a volume of
1/2V, because together they have to equal the volume V of Stage 0. It is hard for
students to believe that the “inside” of Stage 1 and the “outside” of Stage 1 have
the same volume!
The Sierpinski Triangle
68
Unit 4
Developed through the National Science Foundation Project Pattern Exploration: Integrating Mathematics and Science for the Middle Grades, 1999
8. Because of the physical limitations of reducing the size of the tetrahedra, the
students will build a larger tetrahedron from the Stage 2 tetrahedra they have
previously assembled. Using four of the Stage 2 tetrahedra, students will
construct an enlarged version of Stage 3. Students enjoy building larger and
larger tetrahedra. By combining more than one class, the students can attempt to
build an enlarged version of Stage 4.
The Sierpinski Triangle
69
Unit 4
Developed through the National Science Foundation Project Pattern Exploration: Integrating Mathematics and Science for the Middle Grades, 1999
Stage
0
1
2
3
n
Number of
one edge
Length of
of edges
Total length
one face
Area of
Surface Area
Total
tetrahedron
Volume of one
Finding Patterns in the Sierpinski Tetrahedron
tetrahedra
Total volume
Unit 4
70
The Sierpinski Triangle
Developed through the National Science Foundation Project Pattern Exploration: Integrating Mathematics and Science for the Middle Grades, 1999
Answer Keys for
The Sierpinski Triangle Unit 4
The Sierpinski Triangle
71
Unit 4
Developed through the National Science Foundation Project Pattern Exploration: Integrating Mathematics and Science for the Middle Grades, 1999
Number of
one edge
Length of
6e
of edges
Total length
A
one face
Area of
4A
Surface Area
Total
V
tetrahedron
Volume of one
V
Finding Patterns in the Sierpinski Tetrahedron Answer Key
tetrahedra
e
Total volume
1
Stage
0
1
4 ⋅ V
 8
4n ⋅ 4 ⋅
1
A = 4A
4n
 1  1 V
 8  8
1
16 ⋅   V
 8
3
1
64 ⋅   V
 8
n
 1  1  1 V
 8  8  8
n
1
1
4n ⋅   V =   V
 8
 2
n
 1 V
 8
2
1
V
8
1
⋅ A = 4A
16
1
4 ⋅ 4 ⋅ ⋅ A = 4A
4
16 ⋅ 4 ⋅
1
A
4
 1 1 A
 4 4
1
4⋅6⋅ e
2
1
16 ⋅ 6 ⋅ e
4
1
e
2
 1 1 e
 2 2
n
 1 A
 4
 1   1   1  A 64 ⋅ 4 ⋅ 1 ⋅ A = 4 A
 4 4 4
64
1
64 ⋅ 6 ⋅ e
8
n
 1 1 1 e
 2 2 2
n
1
4n ⋅ 6 ⋅   e = 6 ⋅ 2n
 2
4
16
64
4n
 1 e
 2
1
2
3
n
Unit 4
72
The Sierpinski Triangle
Developed through the National Science Foundation Project Pattern Exploration: Integrating Mathematics and Science for the Middle Grades, 1999
Blackline Masters for
The Sierpinski Triangle Unit 4
The Sierpinski Triangle
73
Unit 4
Developed through the National Science Foundation Project Pattern Exploration: Integrating Mathematics and Science for the Middle Grades, 1999
Stage
0
1
2
3
n
Number of
one edge
Length of
of edges
Total length
one face
Area of
Surface Area
Total
tetrahedron
Volume of one
Finding Patterns in the Sierpinski Tetrahedron
tetrahedra
Total volume
Unit 4
74
The Sierpinski Triangle
Developed through the National Science Foundation Project Pattern Exploration: Integrating Mathematics and Science for the Middle Grades, 1999
8-inch Tetrahedron Template
The Sierpinski Triangle
75
Unit 4
Developed through the National Science Foundation Project Pattern Exploration: Integrating Mathematics and Science for the Middle Grades, 1999
4-inch Tetrahedron Template
The Sierpinski Triangle
76
Unit 4
Developed through the National Science Foundation Project Pattern Exploration: Integrating Mathematics and Science for the Middle Grades, 1999
2-inch Tetrahedron Template
The Sierpinski Triangle
77
Unit 4
Developed through the National Science Foundation Project Pattern Exploration: Integrating Mathematics and Science for the Middle Grades, 1999
Sierpinski Tetrahedron Stages
The Sierpinski Triangle
78
Unit 4
Developed through the National Science Foundation Project Pattern Exploration: Integrating Mathematics and Science for the Middle Grades, 1999
Sierpinski Tetrahedron
The Sierpinski Triangle
79
Unit 4
Developed through the National Science Foundation Project Pattern Exploration: Integrating Mathematics and Science for the Middle Grades, 1999
The Sierpinski Triangle
80
Unit 4
Developed through the National Science Foundation Project Pattern Exploration: Integrating Mathematics and Science for the Middle Grades, 1999