Pre-Algebra

5.9

Tessellations

Identify each polygon.

1. polygon with 10 sides

2. polygon with 3 congruent sides

3. polygon with 4 congruent sidesand no right angles

decagon

equilateral triangle

rhombus

Warm Up

Learn to predict and verify patterns involving tessellations.

tessellation

regular tessellation

semiregular tessellation

Vocabulary

Fascinating designs can be made by repeating a figure or group of figures. These designs are often used in art and architecture.

A repeating pattern of plane figures that completely covers a plane with no gaps or overlaps is a tessellation.

In a regular tessellation, a regular polygon is repeated to fill a plane. The angles at each vertex add to 360°, so exactly three regular tessellations exist.

In a semiregular tessellation, two or more regular polygons are repeated to fill the plane and the vertices are all identical.

11 Understand the Problem

Find all the possible semiregular tessellations that use triangles and squares.

List the important information:

• The angles at each vertex add to 360°.

• All the angles in a square measure 90°.

• All the angles in an equilateral triangle measure 60°.

Example: Problem Solving Application

22 Make a Plan

Account for all possibilities: List all possible combinations of triangles and squares around a vertex that add to 360°. Then see which combinations can be used to create a semiregular tessellation.

6 triangles, 0 squares 6(60°) = 360° regular

3 triangles, 2 squares 3(60°) + 2(90°) = 360°

0 triangles, 4 squares 4(90°) = 360° regular

Example Continued

Solve33

There are two arrangements of three triangles and two squares around a vertex.

Example Continued

Repeat each arrangement around every vertex, if possible, to create a tessellation.

Solve33

Example Continued

There are exactly two semiregular tessellations that use triangles and squares.

Solve33

Example Continued

Look Back44

Every vertex in each arrangement is identical to the other vertices in that arrangement, so these are the only arrangements that produce semiregular tessellations.

Example Continued

Create a tessellation with quadrilateral EFGH.

There must be a copy of each angle of quadrilateral EFGH at every vertex.

Example: Creating a Tessellation

Identify each polygon.

1. polygon with 10 sides

2. polygon with 3 congruent sides

3. polygon with 4 congruent sidesand no right angles

decagon

equilateral triangle

rhombus

Warm Up

tessellation

regular tessellation

semiregular tessellation

Vocabulary

11 Understand the Problem

Find all the possible semiregular tessellations that use triangles and squares.

List the important information:

• The angles at each vertex add to 360°.

• All the angles in a square measure 90°.

• All the angles in an equilateral triangle measure 60°.

Example: Problem Solving Application

22 Make a Plan

Account for all possibilities: List all possible combinations of triangles and squares around a vertex that add to 360°. Then see which combinations can be used to create a semiregular tessellation.

6 triangles, 0 squares 6(60°) = 360° regular

3 triangles, 2 squares 3(60°) + 2(90°) = 360°

0 triangles, 4 squares 4(90°) = 360° regular

Example Continued

Look Back44

Every vertex in each arrangement is identical to the other vertices in that arrangement, so these are the only arrangements that produce semiregular tessellations.

Example Continued

Create a tessellation with quadrilateral EFGH.

There must be a copy of each angle of quadrilateral EFGH at every vertex.

Example: Creating a Tessellation

Use rotations to create a tessellation with the quadrilateral given below.

Step 1: Find the midpoint of a side.

Step 2: Make a new edge for half of the side.

Step 3: Rotate the new edge around the midpoint to form the edge of the other half of the side.Step 4: Repeat with the other sides.

Example: Creating a Tessellation by Transforming a

polygon

Step 5: Use the figure to make a tessellation.

Example Continued

Create a tessellation with quadrilateral IJKL.

There must be a copy of each angle of quadrilateral IJKL at every vertex.

I

J

K

L

Try This

Use rotations to create a tessellation with the quadrilateral given below.

Step 1: Find the midpoint of a side.

Step 2: Make a new edge for half of the side.

Step 3: Rotate the new edge around the midpoint to form the edge of the other half of the side.Step 4: Repeat with the other sides.

Example: Creating a Tessellation by Transforming a

polygon

Step 5: Use the figure to make a tessellation.

Example Continued

Use rotations to create a tessellation with the quadrilateral given below.

Step 1: Find the midpoint of a side.

Step 2: Make a new edge for half of the side.

Step 3: Rotate the new edge around the midpoint to form the edge of the other half of the side.Step 4: Repeat with the other sides.

Try This

Look Back44

Every vertex in each arrangement is identical to the other vertices in that arrangement, so these are the only arrangements that produce semiregular tessellations.

Example Continued

Create a tessellation with quadrilateral EFGH.

There must be a copy of each angle of quadrilateral EFGH at every vertex.

Example: Creating a Tessellation

Identify each polygon.

1. polygon with 10 sides

2. polygon with 3 congruent sides

3. polygon with 4 congruent sidesand no right angles

decagon

equilateral triangle

rhombus

Warm Up

tessellation

regular tessellation

semiregular tessellation

Vocabulary

Example: Problem Solving Application

11 Understand the Problem

Find all the possible semiregular tessellations that use triangles and squares.

List the important information:

• The angles at each vertex add to 360°.

• All the angles in a square measure 90°.

• All the angles in an equilateral triangle measure 60°.

22 Make a Plan

Account for all possibilities: List all possible combinations of triangles and squares around a vertex that add to 360°. Then see which combinations can be used to create a semiregular tessellation.

6 triangles, 0 squares 6(60°) = 360° regular

3 triangles, 2 squares 3(60°) + 2(90°) = 360°

0 triangles, 4 squares 4(90°) = 360° regular

Example Continued

Look Back44

Every vertex in each arrangement is identical to the other vertices in that arrangement, so these are the only arrangements that produce semiregular tessellations.

Example Continued

Create a tessellation with quadrilateral EFGH.

There must be a copy of each angle of quadrilateral EFGH at every vertex.

Example: Creating a Tessellation

Use rotations to create a tessellation with the quadrilateral given below.

Step 1: Find the midpoint of a side.

Step 2: Make a new edge for half of the side.

Step 3: Rotate the new edge around the midpoint to form the edge of the other half of the side.Step 4: Repeat with the other sides.

Example: Creating a Tessellation by Transforming a polygon

Step 5: Use the figure to make a tessellation.

Example Continued

Create a tessellation with quadrilateral IJKL.

There must be a copy of each angle of quadrilateral IJKL at every vertex.

I

J

K

L

Try This

Use rotations to create a tessellation with the quadrilateral given below.

Step 1: Find the midpoint of a side.

Step 2: Make a new edge for half of the side.

Step 3: Rotate the new edge around the midpoint to form the edge of the other half of the side.Step 4: Repeat with the other sides.

Example: Creating a Tessellation by Transforming a polygon

Step 5: Use the figure to make a tessellation.

Example Continued

Use rotations to create a tessellation with the quadrilateral given below.

Step 1: Find the midpoint of a side.

Step 2: Make a new edge for half of the side.

Step 3: Rotate the new edge around the midpoint to form the edge of the other half of the side.Step 4: Repeat with the other sides.

Try This

Step 5: Use the figure to make a tessellation.

Try This Continued

1. Find all possible semiregular tessellations that use squares and regular hexagons.

2. Explain why a regular tessellation with regular octagons is impossible. Each angle measure in a regular octagon is 135° and 135° is not a factor of 360°

none

Lesson Quiz: Part 1

3. Can a semiregular tessellation be formed using a regular 12-sided polygon and a regular hexagon? Explain.

No; a regular 12-sided polygon has angles that measure 150° and a regular hexagon has angles that measure 120°. No combinations of 120° and 150° add to 360°

Lesson Quiz: Part 2