10.6 Geometric Symmetry and Tessellations
Recall: Two polygons were similar if the
angles were the same but the sides were
proportional.
We could think of taking the first polygon,
moving it, rotating it, and scaling it.
A rigid motion is an action of only
moving and rotating an object, without
changing its shape.
An object has symmetry if you get the
same object back after a rigid motion.
Pop quiz!!!
How many rigid motions can
you find which are symmetries
for this object?
To study the possible symmetries an
object can have, we need to know the
types of rigid motions possible.
Rigid Motions
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Section 10.6, Slide 8
To reflect an object, you need a mirror
(axis of reflection) that you are reflecting
across.
Notice that the distance of a point and its
mirror image are the same distance from
the line.
Also, lines connecting corresponding
points are perpendicular to the axis of
reflection.
Pop quiz!!!
The following is not a rigid motion:
1) reflection
2) translation
3) symmetry
4) rotation
5) glide reflection
Rigid Motions
• Example: Reflect polygon
ABCDE about the axis of
reflection l.
(continued on next slide)
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Section 10.6, Slide 12
Rigid Motions
2nd: Draw segments BB', CC', DD', and EE',
similar to AA'.
(continued on next slide)
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Section 10.6, Slide 13
Rigid Motions
3rd: Draw the reflected polygon by connecting
vertices A', B', C', D', and E'.
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Section 10.6, Slide 14
A translation is a rigid motion where the
object is moved without rotating it.
(Finish the drawing)
Every vertex is moved by a translation
vector. It is the same for all verticies.
Example: Translate the object by the
given translation vector.
Pop quiz!!!
If you translate a polygon with 5 verticies,
how many verticies does the new polygon
have?
Rigid Motions
• Example: Use the translation
vector and the axis of reflection
to produce a glide reflection of
the object shown.
(continued on next slide)
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Section 10.6, Slide 20
Rigid Motions
• Solution:
1st: Place a copy of the translation vector at
some point, say Y, on the object.
(continued on next slide)
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Section 10.6, Slide 21
Rigid Motions
2nd: Slide the object along the translation vector
so that the point Y coincides with the tip of the
translation vector.
(continued on next slide)
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Section 10.6, Slide 22
Rigid Motions
3rd: Reflect the object about the axis of
reflection to get the final object.
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Section 10.6, Slide 23
Rigid Motions
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Section 10.6, Slide 24
Notice that distance PA = PA'
Angle APA' = 90 degrees
Distance PC = PC'
Angle CPC' = 90 degrees
Recall: A rigid motion moves an object
without scaling it or distorting it.
The following are rigid motions:
- reflection
- translation
- glide reflection
- rotation
Symmetries
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Section 10.6, Slide 27
Symmetries
• Example: Symmetries of the star.
(continued on next slide)
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Section 10.6, Slide 28
Symmetries
• One Symmetry:
(continued on next slide)
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Section 10.6, Slide 29
Symmetries
• Another Symmetry:
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Section 10.6, Slide 30
What are the symmetries of this object?
Are there any symmetries from
- reflection
- translation
- glide reflection
- rotation
What are the symmetries of this object?
Are there any symmetries from
- reflection
- translation
- glide reflection
- rotation
Pop quiz!!!
The following is not a rigid motion:
1) reflection
2) translation
3) symmetry
4) rotation
5) glide reflection
A tessellation, or tiling, of the plane is a
pattern that covers the plane completely
by polygons.
It should have no holes and no
overlapping pieces.
A regular tessellation, or regular tiling,
is built from copies of a single regular
polygon.
Which regular polygons tessellate the
plane?
Recall the formula for angles in regular
polygons.
Pop quiz!!!
The size of an interior angle of a regular
hexagon in degrees is
1) 90
2) 120
3) 135
4) 180
5) 360
6) 540
For hexagons, n = 6.
The interior angles are then 120 degrees.
Notice that around a vertex of
any regular tessellation, we
must have an angle sum of
360°, and we also must have
three or more polygons of the
same shape and size.
Let's try with a regular pentagon.
n = 5 and the interior angle is 108 degrees.
th
The 4 added pentagon causes overlap!
Thus we cannot use pentagons to tile the
plane.
If the regular polygon has more than 6
sides, n > 6.
Each time, this formula will give an angle
bigger than 120 degrees.
Three of these n-gons will overlap.
So 7-gons, 8-gons, etc cannot tile the
plane.
Regular Tessellations
Only three regular tessellations exist:
equilateral triangles (n=3), squares (n=4), and
regular hexagons (n=6).
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Section 10.6, Slide 41
Pop quiz!!!
When tessellating the plane with a regular
triangle (equilateral triangle), how many
triangles meet at a vertex?
Plain Tessellations
• Example: An architect has tiles shaped like
equilateral triangles and squares. All tiles have
sides of the same length. Is it possible to
produce a tessellation using a combination that
contains both types of tiles?
(continued on next slide)
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Section 10.6, Slide 43
Tessellations
Now try different numbers of squares and
corresponding triangles.
(continued on next slide)
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Section 10.6, Slide 45
Tessellations
A tessellation with 2 squares and 3 equilateral
triangles is shown.
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Section 10.6, Slide 46
Can you tile a floor with octagons and
squares if the sides are the same length?
Octagon: n=8, interior angle = 135