Tessellations
Bill Zahner
Lori Jordan
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Printed: June 19, 2016
AUTHORS
Bill Zahner
Lori Jordan
www.ck12.org
C HAPTER
Chapter 1. Tessellations
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Tessellations
Here you’ll learn what a tessellation is and how to tell whether or not a figure will tessellate.
What if you were given a hexagon and asked to tile it over a plane such that it would fill the plane with no overlaps
and no gaps? Could you do this? After completing this Concept, you’ll be able to determine if a figure tessellates.
Watch This
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/137550
CK-12 Foundation: Chapter12TessallationsA
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/10290
Teachertubemath: Create a Tessellation
Guidance
You have probably seen tessellations before, even though you did not call them that. Examples of tessellations are: a
tile floor, a brick or block wall, a checker or chess board, and a fabric pattern. A tessellation is a tiling over a plane
with one or more figures such that the figures fill the plane with no overlaps and no gaps. Here are a few examples.
Notice the hexagon (cubes, first tessellation) and the quadrilaterals fit together perfectly. If we keep adding more,
they will entirely cover the plane with no gaps or overlaps. The tessellation pattern could be colored creatively to
make interesting and/or attractive patterns. To tessellate a shape it must be able to exactly surround a point, or the
sum of the angles around each point in a tessellation must be 360◦ . Therefore, every quadrilateral and hexagon will
tessellate. For a shape to be tessellated, the angles around every point must add up to 360◦ . A regular pentagon does
not tessellate by itself. But, if we add in another shape, a rhombus, for example, then the two shapes together will
tessellate.
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Tessellations can also be much more complicated. Here are a couple of examples.
Example A
Tessellate the quadrilateral below.
To tessellate any image you will need to reflect and rotate the image so that the sides all fit together. First, start by
matching up each side with itself around the quadrilateral.
This is the final tessellation. You can continue to tessellate this shape forever.
Now, continue to fill in around the figures with either the original or the rotation.
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Chapter 1. Tessellations
Example B
Does a regular pentagon tessellate?
First, recall that there are (5 − 2)180◦ = 540◦ in a pentagon and each angle is 540◦ ÷ 5 = 108◦ . From this, we know
that a regular pentagon will not tessellate by itself because 108◦ × 3 = 324◦ and 108◦ × 4 = 432◦ .
Example C
How many squares will fit around one point?
First, recall how many degrees are in a circle, and then figure out how many degrees are in each angle of a square.
There are 360◦ in a circle and 90◦ in each interior angle of a square, so 360
90 = 4 squares will fit around one point.
Watch this video for help with the Examples above.
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/137551
CK-12 Foundation: Chapter12TessallationsB
Concept Problem Revisited
You could tessellate a regular hexagon over a plane with no overlaps or gaps because each of its interior angles is
120◦ . Three hexagons whose angles sum to 360◦ surround each point in the tessellation.
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Guided Practice
1. How many regular hexagons will fit around one point?
2. Does a regular octagon tessellate?
3. Tessellations can also be much more complicated. Check out http://www.mathsisfun.com/geometry/tessellation.
html to see other tessellations and play with the Tessellation Artist, which has a link at the bottom of the page.
Answers:
1. First, recall how many degrees are in a circle, and then figure out how many degrees are in each angle of a regular
hexagon. There are 360◦ in a circle and 120◦ in each interior angle of a hexagon, so 360
120 = 3 hexagons will fit around
one point.
2. First, recall that there are 1080◦ in a pentagon. Each angle in a regular pentagon is 1080◦ ÷ 8 = 135◦ . From this,
we know that a regular octagon will not tessellate by itself because 135◦ does not go evenly into 360◦ .
Explore More
Will the given shapes tessellate? If so, how many do you need to fit around a single point?
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
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A regular heptagon
A rectangle
A rhombus
A parallelogram
A trapezoid
A kite
A regular nonagon
A regular decagon
A completely irregular quadrilateral
In general, which regular polygons will tessellate?
Use equilateral triangles and regular hexagons to draw a tessellation.
The blue shapes are regular octagons. Determine what type of polygon the white shapes are. Be as specific as
you can.
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Chapter 1. Tessellations
13. Draw a tessellation using regular hexagons.
14. Draw a tessellation using octagons and squares.
15. Make a tessellation of an irregular quadrilateral using the directions from Example A.
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 12.7.
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