1. Tessellations
5.9
Pre-Algebra

2. Warm Up Identify each polygon. 1. polygon with 10 sides
2. polygon with 3 congruent sides
3. polygon with 4 congruent sides
and no right angles
decagon
equilateral triangle
rhombus

3. Learn to predict and verify patterns involving tessellations.

4. Vocabulary
tessellation
regular tessellation
semiregular tessellation

5. 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.

6. 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.

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

8. Example: Problem Solving Application
Find all the possible semiregular tessellations that use triangles and squares. 1
Understand the Problem
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°.

9. Example Continued 2
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

10. Example Continued
Solve 3
There are two arrangements of three triangles and two squares around a vertex.

11. Example Continued
Solve 3
Repeat each arrangement around every vertex, if possible, to create a tessellation.

12. Example Continued
Solve 3
There are exactly two semiregular tessellations that use triangles and squares.

13. Example Continued
Look Back 4
Every vertex in each arrangement is identical to the other vertices in that arrangement, so these are the only arrangements that produce semiregular tessellations.

14. Example: Creating a Tessellation
Create a tessellation with quadrilateral EFGH.
There must be a copy of each angle of quadrilateral EFGH at every vertex.

15. Warm Up Identify each polygon. 1. polygon with 10 sides
2. polygon with 3 congruent sides
3. polygon with 4 congruent sides
and no right angles
decagon
equilateral triangle
rhombus

16. Vocabulary
tessellation
regular tessellation
semiregular tessellation

17. Example: Problem Solving Application
Find all the possible semiregular tessellations that use triangles and squares. 1
Understand the Problem
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°.

18. Example Continued 2
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

19. Example Continued
Look Back 4
Every vertex in each arrangement is identical to the other vertices in that arrangement, so these are the only arrangements that produce semiregular tessellations.

20. Example: Creating a Tessellation
Create a tessellation with quadrilateral EFGH.
There must be a copy of each angle of quadrilateral EFGH at every vertex.

21. Example: Creating a Tessellation by Transforming a polygon
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.

22. Example Continued
Step 5: Use the figure to make a tessellation.

23. Try This Create a tessellation with quadrilateral IJKL. J K L I
There must be a copy of each angle of quadrilateral IJKL at every vertex.

24. Example: Creating a Tessellation by Transforming a polygon
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.

25. Example Continued
Step 5: Use the figure to make a tessellation.

26. 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.

27. Example Continued
Look Back 4
Every vertex in each arrangement is identical to the other vertices in that arrangement, so these are the only arrangements that produce semiregular tessellations.

28. Example: Creating a Tessellation
Create a tessellation with quadrilateral EFGH.
There must be a copy of each angle of quadrilateral EFGH at every vertex.

29. Warm Up Identify each polygon. 1. polygon with 10 sides
2. polygon with 3 congruent sides
3. polygon with 4 congruent sides
and no right angles
decagon
equilateral triangle
rhombus

30. Vocabulary
tessellation
regular tessellation
semiregular tessellation

31. Example: Problem Solving Application
Find all the possible semiregular tessellations that use triangles and squares. 1
Understand the Problem
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°.

32. Example Continued 2
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

33. Example Continued
Look Back 4
Every vertex in each arrangement is identical to the other vertices in that arrangement, so these are the only arrangements that produce semiregular tessellations.

34. Example: Creating a Tessellation
Create a tessellation with quadrilateral EFGH.
There must be a copy of each angle of quadrilateral EFGH at every vertex.

35. Example: Creating a Tessellation by Transforming a polygon
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.

36. Example Continued
Step 5: Use the figure to make a tessellation.

37. Try This Create a tessellation with quadrilateral IJKL. J K L I
There must be a copy of each angle of quadrilateral IJKL at every vertex.

38. Example: Creating a Tessellation by Transforming a polygon
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.

39. Example Continued
Step 5: Use the figure to make a tessellation.

40. 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.

41. Try This Continued
Step 5: Use the figure to make a tessellation.

42. Lesson Quiz: Part 1
1. Find all possible semiregular tessellations that use squares and regular hexagons.
2. Explain why a regular tessellation with regular octagons is impossible.
none
Each angle measure in a regular octagon is 135° and 135° is not a factor of 360°

43. Lesson Quiz: Part 2
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°