1. Example 1 Use the coordinate mapping ( x, y ) → ( x + 8, y + 3) to translate ΔSAM to create ΔS’A’M’.

2. Dilations Objectives: 1.To use dilations to create similar figures 2.To perform dilations in the coordinate plane using coordinate notation

3. Dilations dilation A dilation is a type of transformation that enlarges or reduces a figure. scale factor center of dilation The dilation is described by a scale factor and a center of dilation.

4. The scale factor k is the ratio of the length of any side in the image to the length of its corresponding side in the preimage. Dilations

5. Example 2 What happens to any point ( x, y ) under a dilation centered at the origin with a scale factor of k ?

6. Dilations in the Coordinate Plane You can describe a dilation with respect to the origin with the notation ( x, y ) → ( kx, ky ), where k is the scale factor.

7. Dilations in the Coordinate Plane You can describe a dilation with respect to the origin with the notation ( x, y ) → ( kx, ky ), where k is the scale factor. Enlargement: Enlargement: k > 1.

8. Dilations in the Coordinate Plane You can describe a dilation with respect to the origin with the notation ( x, y ) → ( kx, ky ), where k is the scale factor. Reduction: Reduction: 0 < k < 1.

9. Example 3 Determine if ABCD and A’B’C’D’ are similar figures. If so, identify the scale factor of the dilation that maps ABCD onto A’B’C’D’ as well as the center of dilation. Is this a reduction or an enlargement?

10. Example 4 A graph shows  PQR with vertices P(2, 4), Q(8, 6), and R(6, 2), and segment ST with endpoints S(5, 10) and T(15, 5). At what coordinate would vertex U be placed to create ΔSUT, a triangle similar to ΔPQR?

11. Example 5 Figure J’K’L’M’N’ is a dilation of figure JKLMN. Find the coordinates of J’ and M’.

13. This Exploration of Tessellations will guide you through the following: Exploring Tessellations Definition of Tessellation Semi-Regular Tessellations Symmetry in Tessellations Regular Tessellations Create your own Tessellation View artistic tessellations by M.C. Escher Tessellations Around Us

14. What is a Tessellation? A Tessellation is a collection of shapes that fit together to cover a surface without overlapping or leaving gaps.

15. Tessellations in the World Around Us: Brick WallsFloor TilesCheckerboards HoneycombsTextile Patterns Art Can you think of some more?

16. Are you ready to learn more about Tessellations? Symmetry in Tessellations Regular Tessellations Semi-Regular Tessellations

17. Regular Tessellations Regular Tessellations consist of only one type of regular polygon. Do you remember what a regular polygon is? A regular polygon is a shape in which all of the sides and angles are equal. Some examples are shown here: TriangleSquarePentagonHexagonOctagon

18. Regular Tessellations Which regular polygons will fit together without overlapping or leaving gaps to create a Regular Tessellation? Maybe you can guess which ones will tessellate just by looking at them. But, if you need some help, CLICK on each of the Regular Polygons below to determine which ones will tessellate and which ones won’t: TriangleOctagonHexagonPentagonSquare

19. Does a Triangle Tessellate? Regular Tessellations The shapes fit together without overlapping or leaving gaps, so the answer is YES.

20. Does a Square Tessellate? Regular Tessellations The shapes fit together without overlapping or leaving gaps, so the answer is YES.

21. Does a Pentagon Tessellate? Regular Tessellations Gap The shapes DO NOT fit together because there is a gap. So the answer is NO.

22. Does a Hexagon Tessellate? Regular Tessellations The shapes fit together without overlapping or leaving gaps, so the answer is YES. Hexagon Tessellation in Nature

23. Does an Octagon Tessellate? Regular Tessellations The shapes DO NOT fit together because there are gaps. So the answer is NO. Gaps

24. Figures that Tessellate

25. Regular Tessellations As it turns out, the only regular polygons that tessellate are: TRIANGLES SQUARES HEXAGONS Summary of Regular Tessellations: Regular Tessellations consist of only one type of regular polygon. The only three regular polygons that will tessellate are the triangle, square, and hexagon.

26. Are you ready to learn more about Tessellations? Symmetry in Tessellations Regular Tessellations Semi-Regular Tessellations

27. Semi-Regular Tessellations Semi-Regular Tessellations consist of more than one type of regular polygon. (Remember that a regular polygon is a shape in which all of the sides and angles are equal.) How will two or more regular polygons fit together without overlapping or leaving gaps to create a Semi-Regular Tessellation? CLICK on each of the combinations below to see examples of these semi-regular tessellations. Hexagon & Triangle Octagon & Square Square & Triangle Hexagon, Square & Triangle

28. Semi-Regular Tessellations Hexagon & Triangle Can you think of other ways to arrange these hexagons and triangles?

29. Semi-Regular Tessellations Octagon & Square Many floor tiles have these tessellating patterns. Look familiar?

30. Semi-Regular Tessellations Square & Triangle

31. Semi-Regular Tessellations Hexagon, Square, & Triangle

32. Summary of Semi-Regular Tessellations: Semi-Regular Tessellations consist of more than one type of regular polygon. You can arrange any combination of regular polygons to create a semi-regular tessellation, just as long as there are no overlaps and no gaps. Semi-Regular Tessellations What other semi-regular tessellations can you think of?

33. Translation Reflection Glide Reflection Symmetry in Tessellations The four types of Symmetry in Tessellations are: Rotation

34. Symmetry in Tessellations Rotation To rotate an object means to turn it around. Every rotation has a center and an angle. A tessellation possesses rotational symmetry if it can be rotated through some angle and remain unchanged. Examples of objects with rotational symmetry include automobile wheels, flowers, and kaleidoscope patterns. CLICK HERECLICK HERE to view some examples of rotational symmetry. Back to Symmetry in Tessellations

35. Rotational Symmetry

37. Back to Rotations

38. Translation To translate an object means to move it without rotating or reflecting it. Every translation has a direction and a distance. A tessellation possesses translational symmetry if it can be translated (moved) by some distance and remain unchanged. A tessellation or pattern with translational symmetry is repeating, like a wallpaper or fabric pattern. Symmetry in Tessellations CLICK HERECLICK HERE to view some examples of translational symmetry. Back to Symmetry in Tessellations

39. Translational Symmetry Back to Translations

40. Reflection To reflect an object means to produce its mirror image. Every reflection has a mirror line. A tessellation possesses reflection symmetry if it can be mirrored about a line and remain unchanged. A reflection of an “R” is a backwards “R”. Symmetry in Tessellations CLICK HERECLICK HERE to view some examples of reflection symmetry. Back to Symmetry in Tessellations

41. Reflection Symmetry

42. Back to Reflections

43. Symmetry in Tessellations Glide Reflection A glide reflection combines a reflection with a translation along the direction of the mirror line. Glide reflections are the only type of symmetry that involve more than one step. A tessellation possesses glide reflection symmetry if it can be translated by some distance and mirrored about a line and remain unchanged. CLICK HERECLICK HERE to view some examples of glide reflection symmetry. Back to Symmetry in Tessellations

44. Glide Reflection Symmetry

45. Back to Glide Reflections

46. Symmetry in Tessellations Summary of Symmetry in Tessellations: The four types of Symmetry in Tessellations are: Rotation Translation Reflection Glide Reflection Each of these types of symmetry can be found in various tessellations in the world around us.

47. Exploring Tessellations We have explored tessellations by learning the definition of Tessellations, and discovering them in the world around us.

48. Exploring Tessellations We have also learned about Regular Tessellations, Semi- Regular Tessellations, and the four types of Symmetry in Tessellations.

49. Create Your Own Tessellation! Now that you’ve learned all about Tessellations, it’s time to create your own. You can create your own Tessellation by hand, or by using the computer. It’s your choice!

50. * He was born Maurits Cornelis Escher in 1898, in Leeuwarden, Holland. M.C. Escher developed the tessellating shape as an art form * Escher was a graphic artist, who specialized in woodcuts and lithographs. * His father wanted him to be an architect, but bad grades in school and a love of drawing and design led him to a career in the graphic arts.

51. His interest began in 1936, when he traveled to Spain and saw the tile patterns used in the Alhambra. Escher saw tile patterns that gave him ideas for his art work

52. Alhambra Palace * The Alhambra is a walled city and fortress in Granada, Spain. It was built during the last Islamic Dynasty (1238-1492). * The palace is lavishly decorated with stone and wood carvings and tile patterns on most of the ceilings, walls, and floors.

53. The Alhambra Palace is a famous example of Moorish architecture. It may be the most well known Muslim construction. Islamic art does not usually use representations of living beings, but uses geometric patterns, especially symmetric (repeating) patterns.

54. By “distorting” the basic shapes he changed them into animals, birds, and other figures. The effect can be both startling and beautiful.

55. Escher Horses

56. Lets make a simple tessellating shape

57. Begin with a simple geometric shape - the square

58. Change the shape of one side

59. Copy this line on the opposite side

60. Rotate the line and repeat it on the remaining edges

61. Erase the original shape

62. Add lines to the inside of the shapes to turn them into pictures.

63. Add color to enhance your picture.

64. By repeating your shape you create a tessellated picture