CK-12 Geometry
Page 45
Note the measures of the angles where two octagons and a square come together to surround a point. They are and and
One way to look at this tessellation is as a translation. Each octagon is repeatedly translated to the right and down. Or the tessellation could be seen as a reflection. Reflect each octagon in its vertical and horizontal edges. The tessellation can even be viewed as a glide reflection. Translate an octagon to the right, then reflect it vertically.
A tessellation based on more than one regular polygon is called a semi-regular tessellation. One semi-regular tessellation is shown in the example 3. You’ll have the opportunity to create others in the Lesson Exercises.
Tessellation DIY (“Do It Yourself”)
All people can create their own tessellations. Here is an example.
Note that the same basic “unit” can be used to make different tessellations.
Further Reading
M. C. Escher was a famous twentieth-century graphic artist who specialized in extremely original, provocative tessellations. You can read about him and see many examples of his art in M.C. Escher: His Life and Complete Graphic Work. New York: H.N. Abrams
Lesson Summary
Tessellations are at the intersection of geometry and design. Many—but not all—of the most common polygons will tessellate; some will not. Some of the regular polygons will tessellate by themselves. Semi-regular tessellations are made up of two (or more) of the regular polygons. There is no need to limit tessellations to regular polygons or even to polygons. Anyone can draw a tessellation, using whatever shape desired as long as it will, in fact, tessellate.
The repetitive patterns that make tessellations are related to transformations. For example, a tessellation may consist of a basic unit that is repeatedly translated or reflected.
Points to Consider
All tessellations show some kind of symmetry. Why? Because that is a natural result of creating a pattern through reflection or translation. We will examine symmetry more thoroughly in an upcoming lesson.
Tessellations can also be created through rotations. Just as we have seen composite transformations, there are also composite tessellations that use two or more transformations.
Look around in your daily life. Where do you see tessellations?
Review Questions
Will the given shape tessellate? If the answer is yes, make a drawing on grid paper to show the tessellation. (D1)
A square
A rectangle
A rhombus
A parallelogram
A trapezoid
A kite
A completely irregular quadrilateral
Which regular polygons will tessellate?
Use equilateral triangles and regular hexagons to draw a semiregular tessellation.
Review Answers
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Equilateral triangle, square, regular hexagon
One example is shown here
These are regular hexagons with equilateral triangles that fit in to exactly fill the space between the hexagons.
Symmetry
Learning Objectives
Understand the meaning of symmetry.
Determine all the symmetries for a given plane figure.
Draw or complete a figure with a given symmetry.
Identify planes of symmetry for three-dimensional figures.
Introduction
You know a lot about symmetry, even if you haven’t studied it before. Symmetry is found throughout our world—both the natural world and the human-made world that we live in. You may have studied symmetry in math classes, or even in other classes such as biology and art, where symmetry is a basic principle.
The transformations we developed in earlier work have counterparts in symmetry. We will focus here on three plane symmetries and a three-dimensional symmetry.
line symmetry
rotational symmetry
point symmetry
planes of symmetry
A plane of symmetry may be a new concept, as it applies to three-dimensional objects.
Line Symmetry
Line symmetry is very familiar. It could be called “left-right” symmetry.
A plane (two-dimensional) figure has a line of symmetry if the figure can be reflected over the line and the image of every point of the figure is a point on the original figure.
In effect, this says that the reflection is the original figure itself.
Another way to express this is to say that:
the line of symmetry divides the figure into two congruent halves
each half can be flipped (reflected) over the line
and when it is flipped each half is identical to the other half
Many figures have line symmetry, but some do not have line symmetry. Some figures have more than one line of symmetry.
In biology line symmetry is called bilateral symmetry. The plane representation of a leaf, for example, may have bilateral symmetry—it can be split down the middle into two halves that are reflections of each other.
Rotational Symmetry
A plane (two-dimensional) figure has rotational symmetry if the figure can be rotated and the image of every point of the figure is a point on the original figure.
In effect, this says that the rotated image is the original figure itself.
Another way to express this is to say: After being rotated, the figure looks exactly as it did before the rotation.
Note that the center of rotation is the “center” of the figure.
In biology rotational symmetry is called radial symmetry. The plane representation of a starfish, for example, may have radial symmetry—it can be turned (rotated) and it will look the same before, and after, being turned. The photographs below show how sea stars (commonly called starfish) demonstrate 5-fold radial symmetry.
Point Symmetry
We need to define some terms before point symmetry can be defined.
Reflection in a point: Points and are reflections of each other in point if and are collinear and .
In the diagram:
is the reflection of in point (and vice versa).
is the reflection of in point (and vice versa).
A plane (two-dimensional) figure has point symmetry if the reflection (in the center) of every point on the figure is also a point on the figure.
A figure with point symmetry looks the same right side up and upside down; it looks the same from the left and from the right.
The figures below have point symmetry.
Note that all segments connecting a point of the figure to its image intersect at a common point called the center.
Point symmetry is a special case of rotational symmetry.
If a figure has point symmetry it has rotational symmetry.
The converse is not true. If a figure has rotational symmetry it may, or may not, have point symmetry.
Many flowers have petals that are arranged in point symmetry. (Keep in mind that some flowers have petals. They do not have point symmetry. See the next example.)
Here is a figure that has rotational symmetry but not point symmetry.
Planes of Symmetry
Three-dimensional (3-D) figures also have symmetry. They can have line or point symmetry, just as two-dimensional figures can.
A 3-D figure can also have one or more planes of symmetry.
A plane of symmetry divides a 3-D figure into two parts that are reflections of each other in the plane.
The plane cuts through the cylinder exactly halfway up the cylinder. It is a plane of symmetry for the cylinder.
Notice that this cylinder has many more planes of symmetry. Every plane that is perpendicular to the top base of the cylinder and contains the center of the base is a plane of symmetry.
The plane in the diagram above is the only plane of symmetry of the cylinder, that is parallel to the base.
Example 1
How many planes of symmetry does the rectangular prism below have?
There are three planes of symmetry: one parallel to and halfway between each pair of parallel faces.
Lesson Summary
In this lesson we brought together our earlier concepts of transformations and our knowledge about different kinds of shapes and figures. These were combined to enable us to describe the symmetry of an object.
For two-dimensional figures we worked with:
line symmetry
rotational symmetry
point symmetry
For three-dimensional figures we defined one additional symmetry, which is a plane of symmetry.
Points to Consider
Symmetry seems to be the preferred format for objects in the real world. Think about animals and plants, and about microbes and planets. There is a reason why nearly all built objects are symmetric too. Think about buildings, and tires, and light bulbs, and much more.
As you go through your daily life, be alert and aware of the symmetry you encounter.
Review Question
True or false?
Every triangle has line symmetry.
Some triangles have line symmetry.
Every rectangle has line symmetry.
Every rectangle has exactly two lines of symmetry.
Every parallelogram has line symmetry.
Some parallelograms have line symmetry.
No rhombus has more than two lines of symmetry.
No right triangle has a line of symmetry.
Every regular polygon has more than two lines of symmetry.
Every sector of a circle has a line of symmetry.
Every parallelogram has rotational symmetry.
Every pentagon has rotational symmetry.
No pentagon has point symmetry.
Every plane that contains the center is a plane of symmetry of a sphere.
A football shape has a line of symmetry.
Add a line of symmetry the drawing.
Draw a quadrilateral that has two pairs of congruent sides and exactly one line of symmetry.
Which of the following pictures has point symmetry?
How many planes of symmetry does a cube have?
Review Answers
False
True
True
False
False
True
False
False
True
True
True
False
True
True
True
Any kite that is not a rhombus. Two examples are shown.
The four of hearts
Nine
Dilations
Learning Objectives
Use the language of dilations.
Calculate and apply scalar products.
Use scalar products to represent dilations.
Introduction
We begin the lesson with a review of dilations, which were introduced in an earlier chapter. Like the other transformations, dilations can be expressed using matrices. Before we can do that, though, you will learn about a second kind of multiplication with matrices called scalar multiplication.
Dilation Refresher
The image of point in a dilation centered at the origin, with a scale factor , is the point .
For , the dilation is an enlargement.
For , the dilation is a reduction.
Any linear feature of an image is times as long as the length in the original figure.
Areas in the image are times the corresponding area in the original figure.
Scalar Multiplication
In an earlier lesson you learned about matrix multiplication: multiplication of one matrix by another matrix. Scalar multiplication is the multiplication of a matrix by a real number. The product in scalar multiplication is a matrix. Each element of the original matrix is multiplied by the scalar (the real number) to produce the corresponding element in the scalar product.
Note that in scalar multiplication:
any matrix can be multiplied by any real number
the product is a matrix
the product has the same dimensions as the original matrix
Example 1
Let , and
What is the scalar product ?
Let’s continue an example from an earlier lesson.
Example 2
A company has two warehouses in their eastern region, where they store three models of their product. A matrix can represent the numbers of each model available in each warehouse.
Here the row number of an element represents the warehouse number, and the column number represents the model number.
Suppose that managers decide to increase the number of each model in each warehouse by . After the increase there will be three times as many of each model in each warehouse.
A scalar product can represent this situation very precisely. Let be the matrix that represents the distribution of items by model and warehouse after the increase.
For instance, after the increase there will be Model items in Warehouse .
Scalar Products for Dilations
Recall from the Dilation Refresher above:
The image of point in a dilation centered at the origin, with a scale factor , is the point .
This is exactly the tool we need in order to use matrices for dilations.
Example 3
The following rectangle is dilated with a scale factor of .
a) What is the polygon matrix for the rectangle?
b) Write a scalar product for the dilation.
c) What are the coordinates of the vertices of the image rectangle?
d) What is the perimeter of the image?
Perimeter of the original rectangle is .
Perimeter of the image is .
e) What is the area of the image?
Area of the original rectangle is .
Area of the image is .
Compositions with Dilations
Dilations can be one of the transformations in a composition, just as translations, reflections, and rotations can.
Example 4
We will use two transformations to move the circle below.
First we will dilate the circle with scale factor . Then, we will translate the new image right and up.
We can call this a translation-dilation.
a) What are the coordinates of the center of the final image circle?
b) What is the radius of the final image?
c) What is the circumference of the final image?
d) What is the area of the original circle?
e) What is the area of the final image circle?
If is a polygon matrix for a set of points in a coordinate plane, we could use matrix arithmetic to find , the matrix of the image of the polygon after the translation-dilation of this example 4.
Let’s use this translation-dilation to move the rectangle in example 3.
Dilation scalar is
Translation matrix is
The final image is the rectangle with vertices at and .
Lesson Summary
In this lesson we completed our study of transformations. Dilations complete the collection of transformations we have now learned about: translations, reflections, rotations, and dilations.
Scalar multiplication was defined. Differences of scalar multiplication compared to matrix multiplication were observed: any scalar can multiply any matrix, and the dimensions of a scalar product are the same as the dimensions of the matrix being multiplied.
Compositions involving dilations gave us another way to change and move polygons. All sorts of matrix operations—scalar multiplication, matrix multiplication, and matrix addition—can be used to find the image of a polygon in these compositions.
Points to Consider
All of our work with the matrices that represent polygons and translations in two-dimensional space (a coordinate plane) ha
s rather obvious parallels in three dimensions.
A matrix that represents points would have rows and columns rather than .
A dilation is still a scalar product.
A translation matrix for points would have rows and columns in which the rows are all the same.
And, of course, there seem to be many more than three dimensions!
Review Questions
A dilation has a scale factor of . How does the image of a polygon compare to the original polygon in this dilation?
The matrix represents the prices Marci’s company charges for deliveries in four zones. Explain what the scalar product could represent.
The matrices for three triangles are:
Describe how the triangles represented by and are related to each other.
Describe how the triangles represented by and are related to each other.
Write the product in matrix form.
Describe how the triangle represented by the product is related to the triangle represented by
In example 4 above, the circle is first dilated and then translated. Describe how to achieve the same result with a translation first and then a dilation.