Tessellations
Just Plane Artistic
ACTIVITY
5.3
SUGGESTED LEARNING STRATEGIES: Activating Prior
Knowledge, Close Reading, Graphic Organizer, Think/
Pair/Share, Create Representations, Use Manipulatives,
Quickwrite, Self/Peer Revision
The interconnection between mathematics and art is exemplified by
tessellations. Decorative floors, murals, and fabrics often display linked
geometric designs. Beautiful geometric patterns may adorn the walls
and ceilings of palaces, mosques, and temples. This lesson explores the
mathematical properties of tessellations.
Many classrooms have floors and/or ceilings that are covered with
square tiles. In this section, you will explore other regular polygons that
could be used to completely cover flat surfaces.
1. Demonstrate that you can cover the top of your desk with equilateral
triangles in such a way that there are no gaps and no triangles
overlap. Use actual equilateral triangles cut out of index cards or
pattern blocks to help with this problem.
2. Describe the characteristics of an equilateral triangle that make it
possible to cover the top of your desk.
My Notes
ACADEMIC VOCABULARY
Covering a flat surface
with one or more types of
shapes so that there are no
gaps or overlaps is called a
tessellation.
MATH TERMS
Other words used to describe
tessellations include tilings
and mosaics. When one or more
figures completely cover a plane
they are said to tessellate the
plane.
© 2010 College Board. All rights reserved.
You probably did not have enough triangles to actually cover the top of
your desk, but you should see that given enough triangles, it would be
possible. For this reason we say that equilateral triangles will tessellate
by themselves.
3. Explore which other regular polygons tessellate the plane (a pure
tessellation). Complete the table below.
Regular
Polygon
Does It
Tessellate?
Triangle
yes
Square
yes
If Not, Describe Why Not
MATH TERMS
Tessellating a plane with
multiple copies of a single shape
is called a pure tessellation.
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Dodecagon
Unit 5 • Coordinate Geometry and Transformations
371
ACTIVITY 5.3
continued
Tessellations
Just Plane Artistic
My Notes
SUGGESTED LEARNING STRATEGIES: Graphic Organizer,
Think/Pair/Share, Create Representations, Look for a
Pattern, Quickwrite, Self/Peer Revision
4. List each regular polygon that tessellates the plane and the measure of
each interior angle.
Regular Polygon
Measure of the Interior Angle
5. List each regular polygon that does not tessellate the plane and the
measure of each interior angle.
Measure of the Interior Angle
6. Describe what you notice about the measures of the interior angles of
the polygons that tessellate the plane compared to the measures of the
interior angles of the polygons that do not tessellate.
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© 2010 College Board. All rights reserved.
Regular Polygon
Tessellations
ACTIVITY 5.3
continued
Just Plane Artistic
SUGGESTED LEARNING STRATEGIES: Predict and Confirm,
Think-Pair-Share, Identify a Subtask, Look for a Pattern,
Use Manipulatives, Quickwrite, Self/Peer Revision
My Notes
7. Do you think there are any other regular polygons that will tessellate
by themselves? Provide an argument to support your conclusion.
In the previous items, you investigated regular tessellations. They are
called regular because they consist of only one type of regular polygon
arranged in the same way around every point where the polygons meet.
In this section, you will explore pure tessellations using non-regular
polygons.
8. Consider the rectangle shown at the right. Could you create a
tessellation using multiple copies of it? Explain why or why not.
© 2010 College Board. All rights reserved.
9. Use an index card to draw a non-rectangular quadrilateral. Cut out
your quadrilateral. On a separate sheet of unlined paper, trace
multiple copies of your quadrilateral to determine if it will tessellate
the plane. Draw a sketch to show the tiling or to illustrate where the
quadrilaterals overlap, or have gaps. Compare your drawing to others
in your class.
10. Based on what you have observed about tessellating quadrilaterals, do
you believe that every triangle can be made to tessellate? Provide an
argument to support your conclusion.
11. Make a conjecture concerning angle measures of non-regular
polygons that tessellate. Explain your reasoning.
Unit 5 • Coordinate Geometry and Transformations
373
ACTIVITY 5.3
continued
Tessellations
Just Plane Artistic
SUGGESTED LEARNING STRATEGIES: Group Presentation,
Think-Pair-Share, Identify a Subtask, Work Backward,
Use Manipulatives, Self/Peer Revision
My Notes
The following is an example of how you can make a figure that can be
used to create a pure irregular translation tessellation.
A
CONNECT TO ART
D
M.C. Escher (1898–1973), from
the Netherlands, was an artist
who is credited with popularizing
unique irregular tessellations.
His artwork illustrates a deep, yet
playful, understanding of
geometric relationships.
Fig. 1
B
C
A
D
Fig. 2
B
C
A
D
B
C
A
D
Fig. 3
B
C
Fig. 4
12. Follow these steps to create your own unique translation tessellation.
a) Start with a rhombus on an index card with points labeled A, B, C,
and D as shown above (leave margins on all sides).
b) Draw a simple curve with endpoints at A and B (Figure 1).
c) Draw a simple curve from A to D (Figure 2).
d) Cut out along the two curves and along segment DC and segment
BC (Figure 2).
f) Translate your cut figure down, matching points A and B to points
D and C on the second index card. Trace the curve from D to C
(Figure 3).
g) Translate your cut figure up and to the right, matching points A
and D to points B and C on the second index card. Trace the curve
from B to C (Figure 4).
h) Cut out the second index card along all four curves. This is your
irregular tile to tessellate the plane (Figure 4).
i) Trace your shape to tessellate a plane (complete sheet of paper),
translating it to match the edges and leaving no gaps (Figure 4).
In the first part of this unit, you investigated tessellations formed by only
one type of polygon. To tessellate a plane, all of the polygon angles around
a vertex point have to sum to 360°. In this section, you will explore
tessellations that contain more than one type of polygon.
13. Using cutouts or templates of equilateral triangles and squares, find
combinations that fit around a vertex point without gaps or overlaps.
Sketch your results on unlined paper.
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SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
e) Trace the entire shape on a second index card and label points A,
B, C, and D as shown (Figure 2).
Tessellations
ACTIVITY 5.3
continued
Just Plane Artistic
SUGGESTED LEARNING STRATEGIES: Graphic Organizer,
Predict and Confirm, Group Presentation, Think-Pair-Share,
Create Representations, Work Backward, Use Manipulatives,
Quickwrite, Self/Peer Revision
My Notes
14. The measure of each interior angle of an equilateral triangle is 60°
and the measure of each interior angle of a square is 90°. Use this
information to verify that each combination you found in Item 13
can fit around a vertex point without gaps or overlaps.
15. Use polygon templates to discover more arrangements of regular
polygons that can be arranged around a single vertex without gaps
or overlaps. Sketch each arrangement on a separate sheet of paper.
© 2010 College Board. All rights reserved.
16. For each arrangement you found in Item 15, use the list of interior
angle measures of regular polygons, in the table below, to verify that
the sum of the angles around a vertex point is 360°.
Number
of Sides
Measure of Each
Interior Angle
3
60°
4
90°
5
108°
6
120°
7
4°
128 __
7
8
135°
9
140°
10
144°
11
3°
147 ___
11
12
150°
In Item 15 you discovered several arrangements of regular polygons that fit
around a vertex point. Only 18 such arrangements exist. Four of them use
the same polygons arranged around the vertex point in a different order.
Unit 5 • Coordinate Geometry and Transformations
375
ACTIVITY 5.3
continued
Tessellations
Just Plane Artistic
SUGGESTED LEARNING STRATEGIES: Visualization, Group
Presentation, Think-Pair-Share, Work Backwards, Use
Manipulatives, Self/Peer Revision
My Notes
Tessellations are named by listing the number of sides of each polygon
around a vertex point, in order. For example, the polygon arrangements
discovered in Item 13 could be named as follows:
Start Here
Start Here
3-3-3-4-4
3-3-4-3-4
17. Record the name of each arrangement you created in Item 15.
18. A semi-regular tessellation contains more than one type of regular
polygon arranged in the same way around every vertex point. Only
eight of the 18 arrangements of regular polygons repeat to form a
semi-regular tessellation. Use your polygon cutouts to help you find
all eight of the arrangements. On a separate sheet of paper sketch
enough of the tessellation to establish the pattern.
CHECK YOUR UNDERSTANDING
Write your answers on notebook paper or grid
paper.
1. Is it possible to
tessellate a plane with
this obtuse scalene
triangle? Explain.
2. Describe the given
tessellation as pure
or not, regular,
semi-regular, or
neither, and
describe
it by its vertices.
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SpringBoard® Mathematics with Meaning™ Geometry
3. What are characteristics of the polygons in the
tessellation in Item 2 that guarantee there will
be no overlapping or gaps?
4. MATHEMATICAL Describe at least three
R E F L E C T I O N different tessellations you
have seen outside of a textbook or the web.
State whether each tessellation is pure or not.
Is it regular or semi-regular, or neither?
Explain why.
© 2010 College Board. All rights reserved.
19. Using the definition of a semi-regular tessellation in Item 18, explain
why the tessellation to the left is not a semi-regular tessellation.