Geometric Symmetry and
Tessellations
Chapter 10 sec. 6
What is a polygon?
• Is a simple, closed plane figure consisting
only of line segments, called edges, such
that no tow consecutive edges lie on the
same line. It is regular if all its edges are
the same length and all of its angles have
the same measure.
What types of shapes can you have for
a polygon?
Regular:
sides are
same
length
Nonregular
polygon;
sides are not
the same
length
What about a circle?
•Is it a polygon?
•No, it is not made of line
segments.
What is symmetry?
•Define it later. But here is
something to think about.
•In mathematics, we begin with an
intuitive concept, and define it
precisely so that we can measure
it and calculate with it.
Rigid Motion
• A rigid motion is the action of taking a
geometric object in the plane and moving
it in some fashion to another position in
the plane without changing its shape or
size.
2 objects
Arrowhead
Star
Example
(a)
(b)
• The arrowhead has been flipped over
and returned to its resting place.
Property:
▫ Every rigid motion is essentially a
reflection, a translation, a glide
reflection, or a rotation.
What does essentially really mean?
• We are only interested in the
beginning and ending positions of the
object.
Reflection
• Is a rigid motion in which we move an
object so that the ending position is a
mirror image of the object in its starting
position.
Reflection example
Axis of reflection
Translation
• Is a rigid motion in which we move a
geometric object by sliding it along a line
segment in the lane. The direction and
length of the line segment completely
determine the translation.
Translate an object
Translation A
vector
B
A’
B’
Glide reflection
•Is a rigid motion formed by
performing a translation
(the glide) followed by a
reflection.
Steps to the Glide reflection
• 1. Place a copy of the translation vector
at some point A, on the object.
A
B
Axis of reflection
• 2. Slide the object along the translation vector
so that the point A coincides with the tip of
the translation vector.
A
B
Axis of reflection
A’
B’
• 3. Reflect the object about the axis of
reflection to get the final object.
C
C”
A
B
A”
B”
This is the final effect the glide reflection has in
moving the original polygon.
A
B
A”
B”
Rotation
• We perform a rotation by first selecting
a point, called the center of the rotation,
and them while holding this point fixed,
we rotate the plane about this point
through an angle called the angle of
rotation.
Example
• Say you have a piece of paper. If you stick
a pin in the plane at the center of
rotation and then rotate the plane, the
plane will turn about the pin, and all
points in the plane will move except the
point where the pin is placed.
A
90 degrees
A’
A
Symmetry
• A symmetry of a geometric object is
rigid motion such that the beginning
position and the ending position of
the object by the motion are exactly
the same.
Symmetry in Nature
• 2 professors at the University of
New Mexico, Randy Thornhill and
Steven Gangestad, have found that
female scorpion flies are attracted to
males with symmetrical wings.
• Biologist believe that animals with a
higher degree of symmetry have greater
genetic diversity, which enables them to
withstand environmental stress better
and makes them more resistant to
parasites. Lower symmetry goes hand in
hand with lower survival rates and fewer
offspring.
Tessellations
• (or tiling) of the plane is a pattern made
up entirely of polygons that completely
covers the plane. The pattern must have
no holes or gaps, and polygons cannot
overlap except at their edges.
Def.
• Tessellation The root of tessellation is, tessera,
the old Ionic (Greek) root for four. Tessera is
the name of the square chips of stone or glass
that are used to form a mosaic. Tessela is the
dimunitive form, and is used to describe smaller
tessera. Tiles, bricks and larger similar items
were called testa, which is preserved in the name
for the hard outer shell of seeds. The completed
project, then, became a tessellation.
Regular Tessellation
• Consist of regular polygons of the same
size and shape such that all vertices of
the polygons touch other polygons only
at their vertices.
Some tessellation patterns
n-sided polygon
• Each interior angle has measure
( n  2 ) x180 
n
.
• We can use this to determine which
other regular polygons tessellate the
plane.
What does that mean?
• Means that the angle sum around a
vertex in a tessellation must add up to
360 .
120
120
120
Regular
Tessellation
Is there a tessellation of the plane using
pentagon?
•How many sides does a
pentagon have?
•5
•Let’s use the formula
5  2  x180 
5

540 
 108 
5
•We will not have enough
pentagons to completely
surround the vertex.
• There is no regular tessellation of
the plane using pentagons.
108
Can you find the nonregular tessellations?
• Is it possible to find a nonregular
tessellation with using a square and
triangle?
• Yes
• Is it possible to find a nonregular
tessellation with using an octagon?
• yes