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10
Tessellations
Special
Topic
GOAL Recognize and design tessellations.
Word Watch
tessellation, p. 516
regular tessellation, p. 516
A tessellation is a covering of a plane with congruent copies of the same
pattern so that there are no gaps or overlaps. A regular tessellation is
made from only one type of regular polygon. The figure below is a regular
tessellation made from regular triangles.
EXAMPLE
1
Identifying Polygons that Tessellate
Tell whether each type of polygon can form a regular tessellation.
a. Square
b. Regular pentagon
Solution
a. Start with a square. Make copies of
the square and fit them together so
that they cover the plane without
gaps or overlaps. One possible
arrangement is shown. So, a square
forms a regular tessellation.
b. Start with a regular pentagon. Make
copies of the pentagon. When you
try to fit them together, you find that
three pentagons can share a common
vertex, but there is a gap that is too
small to fit a fourth pentagon. So, a
regular pentagon does not form a
regular tessellation.
Other Tessellations You can also make
tessellations using one or more nonregular
polygons by translating, reflecting, or
rotating the figures to cover a plane. An
isosceles triangle is reflected and translated
to form the tessellation at the right.
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CHAPTER 10
SPECIAL TOPIC
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EXAMPLE
2
Making a Tessellation
Make a tessellation of the quadrilateral shown.
Describe the transformation(s) you use.
1 Mark the point at the middle
of one of the sides of the
quadrilateral. Then rotate
the quadrilateral 180
about that point.
2 Translate the new figure as
shown. The pattern that results
covers the plane without gaps or
overlaps. So, the quadrilateral
forms a tessellation.
Exercises
Tell whether the polygon can form a regular tessellation. If it can,
draw the tessellation.
1. Regular hexagon
2. Regular heptagon
Make a tessellation of the indicated polygon. Describe the
transformation(s) you use.
3. Parallelogram
4. Right triangle
5. Can you make a tessellation of the quadrilateral shown in Example 2
using only rotations? only translations? only reflections? Explain your
reasoning.
Tell whether the two polygons can be used to make a tessellation.
If they can, draw the tessellation.
6. A regular octagon and a square with all sides from both figures equal
in length
7. A regular triangle and a rhombus with all sides from both figures equal
in length
8. Critical Thinking In the tessellation shown at the top of page 516,
notice that six triangles fit around a common vertex because each
angle at the vertex measures 60, for a total of 360. Explain how this
observation applies to the square and pentagon in Example 1.
SPECIAL TOPIC
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