tessellation
AND
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 1
Chapter 9 Geometry
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 2
WHAT YOU WILL LEARN • Transformational geometry, symmetry, and tessellations • The Mobius Strip, Klein bottle, and maps • Non-Euclidian geometry and fractal geometry
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 3
Section 5 Transformational Geometry, Symmetry, and Tessellations
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 4
Definitions
The act of moving a geometric figure from some starting position to some ending position without altering its shape or size is called a rigid motion (or transformation).
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 5
Reflection
A reflection is a rigid motion that moves a geometric figure to a new position such that the figure in the new position is a mirror image of the figure in the starting position. In two dimensions, the figure and its mirror image are equidistant from a line called the reflection line or the axis of reflection.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 6
Construct the reflection of triangle ABC about the line l. A C
A C
B
l
2 units
B
2 units
B’
l
C’ A’
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 7
Translation
A translation (or glide) is a rigid motion that moves a geometric figure by sliding it along a straight line segment in the plane. The direction and length of the line segment completely determine the translation.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 8
Example
Given the parallelogram and translation vector, v, construct the translated parallelogram.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 9
Example (continued)
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 10
Rotation
A rotation is a rigid motion performed by rotating a geometric figure in the plane about a specific point, called the rotation point or the center of rotation. The angle through which the object is rotated is called the angle of rotation.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 11
Example
Given the rectangle and rotation point, P, construct rectangles that result from rotations of 90º and 180º.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 12
Example (continued)
90º Rotation 180º Rotation
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 13
Glide Reflection
A glide reflection is a rigid motion formed by performing a translation (or glide) followed by a reflection.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 14
Example
Construct a glide reflection of triangle ABC using translation vector v, and reflection line l.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 15
Symmetry
A symmetry of a geometric figure is a rigid motion that moves a figure back onto itself. That is, the beginning position and ending position of the figure must be identical.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 16
Example
Reflection about Line Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 17
Tessellations
A tessellation (or tiling) is a pattern consisting of the repeated use of the same geometric figures to entirely cover a plane, leaving no gaps. The geometric figures used are called the tessellating shapes of the tessellation.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 18
Example
The simplest tessellations use one single regular polygon.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 19
Example (continued)
Other examples of tessellations:
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 20
Create Your Own Tessellation
http://www.tessellations.org/methods-diypapercut.shtml
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 21
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 1
Chapter 9 Geometry
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 2
WHAT YOU WILL LEARN • Transformational geometry, symmetry, and tessellations • The Mobius Strip, Klein bottle, and maps • Non-Euclidian geometry and fractal geometry
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 3
Section 5 Transformational Geometry, Symmetry, and Tessellations
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 4
Definitions
The act of moving a geometric figure from some starting position to some ending position without altering its shape or size is called a rigid motion (or transformation).
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 5
Reflection
A reflection is a rigid motion that moves a geometric figure to a new position such that the figure in the new position is a mirror image of the figure in the starting position. In two dimensions, the figure and its mirror image are equidistant from a line called the reflection line or the axis of reflection.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 6
Construct the reflection of triangle ABC about the line l. A C
A C
B
l
2 units
B
2 units
B’
l
C’ A’
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 7
Translation
A translation (or glide) is a rigid motion that moves a geometric figure by sliding it along a straight line segment in the plane. The direction and length of the line segment completely determine the translation.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 8
Example
Given the parallelogram and translation vector, v, construct the translated parallelogram.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 9
Example (continued)
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 10
Rotation
A rotation is a rigid motion performed by rotating a geometric figure in the plane about a specific point, called the rotation point or the center of rotation. The angle through which the object is rotated is called the angle of rotation.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 11
Example
Given the rectangle and rotation point, P, construct rectangles that result from rotations of 90º and 180º.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 12
Example (continued)
90º Rotation 180º Rotation
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 13
Glide Reflection
A glide reflection is a rigid motion formed by performing a translation (or glide) followed by a reflection.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 14
Example
Construct a glide reflection of triangle ABC using translation vector v, and reflection line l.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 15
Symmetry
A symmetry of a geometric figure is a rigid motion that moves a figure back onto itself. That is, the beginning position and ending position of the figure must be identical.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 16
Example
Reflection about Line Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 17
Tessellations
A tessellation (or tiling) is a pattern consisting of the repeated use of the same geometric figures to entirely cover a plane, leaving no gaps. The geometric figures used are called the tessellating shapes of the tessellation.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 18
Example
The simplest tessellations use one single regular polygon.
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 19
Example (continued)
Other examples of tessellations:
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 20
Create Your Own Tessellation
http://www.tessellations.org/methods-diypapercut.shtml
Copyright © 2009 Pearson Education, Inc.
Chapter 9 Section 5 - Slide 21
