THE MATHEMATICS OF PATTERNS

Transcription

1 THE MATHEMATICS OF PATTERNS Isometries, Symmetries and Patterns By: Francis Joseph H. Campeña

2 WHAT IS MATHEMATICS?

3 SOME SAY. It is a study of numbers and arithmetic operations.

4 SOME SAY. It is a tool or a collection of skills that helps us answer question of HOW MANY or HOW MUCH.

5 SOME SAY. It is a science of logical reasoning, drawing conclusions from assumed premises or strategic reasoning.

6 What ever point of view we take, there is no denying the fact that MATHEMATICS IS UNIVERSAL

7 PATTERNS??? Mathematics maybe regarded as a study of patterns. WHATS THE NEXT SHAPE? 1 1 = = = 12,321 1,111 1,111 = 1,234,321 11,111 11,111 = 123,454, , ,111 =????

8 VARIOUS PATTERNS Logic Patterns

9 VARIOUS PATTERNS Number Patterns 1 1 = = = 12,321 1,111 1,111 = 1,234,321 11,111 11,111 = 123,454, , ,111 =???? 1,3,7,15,31,?

10 VARIOUS PATTERNS Geometric Patterns

11 PATTERNS IN NATURE Snow covers an orchard in the United States. "No orchard's the worse for the wintriest storm; but one thing about it, it mustn't get warm," wrote Robert Frost in his poem "Good-bye, and Keep Cold. Photograph by Richard Olsenius

12 PATTERNS IN NATURE Lakeside Reflection Photograph by Raymond Gehman A still lake reflects sky and trees in Canada.

13 PATTERNS IN NATURE Scales from butterfly wings radiate from a glass-shelled diatom. Photograph by Darlyne A. Murawski.

14 ISOMETRIES OF THE PLANE Translations Reflections Rotations A Review on

15 TRANSFORMATION A Transformation is a process which shifts points of the plane to possibly new locations on the plane.

16 TRANSLATION A translation (or a slide) moves a shape in a given direction by sliding it up, down, sideways, or diagonally.

17 REFLECTION A reflection (or a flip) can be thought of a getting a mirror image. It has a line of reflection or mirror line where the distance between the image and the mirror line is the same as that between the original figure and the mirror line.

18 ROTATION A rotation (or a turn) has a point about which the rotation is made and an angle that says how far to rotate. Birds -an Escher Artwork

19 DILATION A dilation is a transformation which changes the size of an object

20 RIGID TRANSFORMATIONS Transformations which leave the dimensions of the object and its image unchanged are called rigid transformations, or isometric transformations, or isometries.

21 FORMAL DEFINITION An isometry of the plane is a mapping that preserves distance (and therefore shape): d f x, f y = d(x, y) iso : Greek for the same metry/metria : Greek for measure

22 COMBINATION OF ISOMETRIES It is possible to combine isometries to produce other isometries. Reflect then Translate Translate then Reflect GLIDE REFLECTION!!!

23 GLIDE REFLECTION A reflection followed by a translation or vice versa is called a glide reflection

24 CHASLE S THEOREM Every motion of the plane is one of these transformations: a translation, a rotation, a reflection or a glide reflection

25 MULTIPLICATION TABLE FOR ISOMETRIES REFLECTION REFLECTION TRANSLATION ROTATION TRANSLATION or ROTATION GLIDE REFLECTION GLIDE REFLECTION GLIDE REFLECTION TRANSLATION or ROTATION TRANSLATION GLIDE REFLECTION TRANSLATION ROTATION REFLECTION or GLIDE REFLECTION ROTATION GLIDE REFLECTION ROTATION TRANSLATION or ROTATION GLIDE REFLECTION GLIDE REFLECTION TRANSLATION or ROTATION REFLECTION or GLIDE REFLECTION GLIDE REFLECTION TRANSLATION or ROTATION

26 POSSIBLE STUDENT ACTIVITY MATH DANCE? Yes, Mathematics applied to Dancing

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28 Sample Dance

29 SYMMETRIC PATTERNS A figure has symmetry if there is a non-trivial transformation that maps the figure onto itself.

30 SYMMETRIES OF A SQUARE

31 A design is a figure with at least one non-trivial symmetry. A pattern is a design that has a translation symmetry. A plane pattern has symmetry if there is an isometry of the plane that preserves it.

32 TYPES OF SYMMETRIC PATTERNS ROSETTE PATTERN FRIEZE PATTERN WALLPAPER PATTERN

33 ROSETTE PATTERN This pattern consist of taking a motif or an element and rotating and/or reflecting that element.

34 TYPES OF ROSETTE PATTERN Cyclic has n fold rotational symmetry and no reflectional symmetry. Dihedral has n fold rotational symmetry and reflectional symmetry.

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36 EXAMPLES Identify what type of Rosette Pattern each figure has. A B C

37 EXAMPLES Identify what type of Rosette Pattern each figure has. D E F

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39 Construction of Snowflakes SAMPLE ACTIVITY

40 Construction of Snowflakes SAMPLE ACTIVITY

41 Construction of Snowflakes SAMPLE ACTIVITY

42 Construction of Snowflakes SAMPLE ACTIVITY

43 Construction of Snowflakes SAMPLE ACTIVITY

44 Construction of Snowflakes SAMPLE ACTIVITY

45 Construction of Snowflakes SAMPLE ACTIVITY

46 Construction of Snowflakes SAMPLE ACTIVITY

47 STUDENT ACTIVITY? Describe the type of rosette pattern in their favorite snowflake creation. What are the rotation symmetries of their snowflake. Is there a reflection symmetries? If there are, what are describe their lines of symmetries? Construct a table that describes a sequential application of two symmetries in their snowflake.

48 FRIEZE PATTERNS This pattern is an infinitely long strip imprinted with a design given by a repeating pattern motif.

49 7 POSSIBLE FRIEZE PATTERNS PATTERN 1: Frieze pattern that has only the translation symmetry.

50 7 POSSIBLE FRIEZE PATTERNS PATTERN 2: Frieze pattern that has only the translation and vertical symmetry.

51 7 POSSIBLE FRIEZE PATTERNS PATTERN 3: Frieze pattern that has only the translation symmetry and glide reflection.

52 7 POSSIBLE FRIEZE PATTERNS PATTERN 4: Frieze pattern that has only the translation symmetry and rotation.

53 7 POSSIBLE FRIEZE PATTERNS PATTERN 5: Frieze pattern that has only the translation, horizontal symmetry and glide reflection.

54 7 POSSIBLE FRIEZE PATTERNS PATTERN 6: Frieze pattern that has only the translation, vertical symmetry, rotation and glide reflection.

55 7 POSSIBLE FRIEZE PATTERNS PATTERN 8: Frieze pattern that has only the translation, vertical, horizontal symmetry, rotation and glide reflection.

56 EXAMPLES OF FRIEZE PATTERN

57 EXAMPLES OF FRIEZE PATTERN

58 EXAMPLES OF FRIEZE PATTERN

59 CONSTRUCTING FRIEZE PATTERNS RECALL!!! Using Addition Table Modulo m An integer a is congruent to an integer b modulo a positive integer m if a b is divisible by m.

60 CONSTRUCTING FRIEZE PATTERN Using Addition Table Modulo m Addition Table Modulo 3

61 Reflect along the vertical line

62 Reflect along the horizontal line

63 Example of Frieze Pattern 8

64 CONSTRUCTING FRIEZE PATTERN Using Addition Table Modulo 4

65 Reflect along the vertical line

66 Reflect along the horizontal line

67 Example of Frieze Pattern 8

68 POSSIBLE STUDENT ACTIVITY Consider other addition modulo m and use this to create some frieze patterns. What if the addition table is extended? What pattern can be seen in the table?

69 Addition table modulo 4 with 7 rows.

70 Example of Frieze Pattern 8 using the addition table modulo 4 with 7 rows.

71 POSSIBLE STUDENT ACTIVITY What are other ways of representing the remainders in the addition table modulo m?

72 In the addition table modulo 9, what if the representation of the remainders are given by

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74 Frieze Pattern 8 Frieze Pattern 5

75 EXERCISE Identify the Frieze Pattern in the following designs. Pattern 3: TG

76 WALLPAPER PATTERN Whereas a frieze pattern can be mapped onto itself by a horizontal translation, a wallpaper pattern covers the plane and can be mapped onto itself by translation in more than one direction

77 1. Using the motif EXAMPLE 2. Reflect the motif vertically to obtain 3. Using the resulting figure rotate 120 and 240 to produce 4. Translate the resulting figure to obtain 5. Glide the resulting pattern downward to produce.

78 EXAMPLES

79 EXAMPLES

80 EXAMPLES

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