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LESSON
17-3
Challenge
Regular and Semiregular Tessellations
A tessellation is a pattern of congruent figures that completely covers
a plane without gaps or overlaps. A tessellation that consists of exactly
one type of regular polygon, with each polygon congruent to all the
others, is called a regular tessellation. Any point where the polygons
share a common vertex is a vertex point of the tessellation. The figure
at right, for instance, shows a regular tessellation of equilateral triangles
with one vertex point labeled A.
For Exercises 1–3, refer to the regular tessellation above.
1. What is the measure of each interior angle of each equilateral triangle?
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2. How many equilateral triangles meet at each vertex point?
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3. What is the sum of the measures of all the angles at each
vertex point?
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4. Based on your answers to Exercises 1–3, explain why it is not possible to make
a regular tessellation that consists of regular pentagons.
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A semiregular tessellation is a tessellation of two or more types of
regular polygons with congruent sides that has the same arrangement
of polygons at each vertex point. It is important to specify the
arrangement. The figure at right, for instance, is a semiregular
tessellation of equilateral triangles and squares. To identify a tessellation,
start with the polygon with the least number of sides and then list the
number of sides of each polygon as you move counterclockwise about the vertex.
Use 3 to represent a triangle and 4 to represent a square. Specify the arrangement
at each vertex point as 3-3-3-4-4 for this tessellation.
5. Show calculations to verify that the sum of the measures of the angles at each
vertex point of the 3-3-3-4-4 tessellation is 360°.
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6. Explain why the tessellation with code 3-3-3-4-4 is different from the tessellation
with code 3-3-4-3-4.
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7. Trace and cut out several copies of the regular polygons in the
figure at right. Working with these copies, find all possible regular
and semiregular tessellations. On a separate sheet of paper,
sketch each tessellation, record its code, and show calculations
to verify that the sum of the angle measures at each vertex is 360°.
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© Houghton Mifflin Harcourt Publishing Company
422
Holt McDougal Coordinate Algebra
in a regular nonagon measures 140°,
which does not divide evenly into 360°.
So the nonagon cannot be used in a
regular tessellation. If two nonagons are
used in a semiregular tessellation, then
the measures of the angles of the
polygons that are not nonagons must be
360° 280° 80°. There is no polygon or
combination of polygons with angles that
measure 80°. If one nonagon is used in a
semiregular tessellation, then the
measures of the angles of the polygon or
combination of polygons that are not
nonagons must be 360° 140° 220°.
No polygon can have an angle measure
greater than 180°, so a combination of
polygons must make up 220°. A triangle
has 60° angles, so that leaves 160°. The
formula for interior angle measure of a
regular polygon shows that an 18-gon
has 160° angles. This tiling works, but
there could be more. A square would
leave 130°, a pentagon would leave
112°, a hexagon would leave 100°, and
an octagon would leave 85°. No polygon
or combination of polygons makes up
any of those angle measures. So there is
only one possible tiling using a nonagon.
3. six triangles; four triangles and one
hexagon; three triangles and two
squares; two triangles and two hexagons;
two triangles, one square, and one
dodecagon; one triangle and two
dodecagons; one triangle, two squares,
and one hexagon; four squares; one
square and two octagons; one square,
one hexagon, and one dodecagon; two
pentagons and one decagon; three
hexagons
6.
Review for Mastery
1. translation symmetry
2. translation symmetry and glide reflection
symmetry
3.
4.
5. regular
7. no
8. yes
Challenge
1. 60°
3. 6 u 60°
2. 6
360°
4. Explanations will vary
5. 60° 60° 60° 90° 90°
360°
6. Although both codes indicate that 3
equilateral triangles and 2 squares meet
at a vertex point of the tessellation, each
code specifies a different arrangement.
4.
5. Although the angles at each vertex add
to 360°, the polygons do not tessellate.
They overlap.
6. neither
7. There are only three regular tessellations.
There are exactly eight semiregular
tessellations.
Problem Solving
1. translation, reflection, rotation
2. translation
© Houghton Mifflin Harcourt Publishing Company
A98
Holt McDougal Coordinate Algebra