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Using MAPLE to Construct Repeating Patterns and Several Tessellations Inspired by M. C. Escher Elliot A. Tanis Professor Emeritus of Mathematics Hope College.

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Presentation on theme: "Using MAPLE to Construct Repeating Patterns and Several Tessellations Inspired by M. C. Escher Elliot A. Tanis Professor Emeritus of Mathematics Hope College."— Presentation transcript:

1 Using MAPLE to Construct Repeating Patterns and Several Tessellations Inspired by M. C. Escher Elliot A. Tanis Professor Emeritus of Mathematics Hope College March 2, 2006

2 PARADE MAGAZINE, December 8, 2002

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7 BIKE BOX CHECKBOOK DECKED HEED HIDE BIKE BOX CHECKBOOK DECKED HEED HIDE BIKE BOX CHECKBOOK DECKED HEED HIDE BIKE BOX CHECKBOOK DECKED REFLECT ROTATEROTATE

8 A Computer Algebra System (CAS) such as MAPLE can be used to construct tessellations. The way in which tessellations are classified will be illustrated using examples from Chinese Lattice Designs, The Alhambra, Hungarian Needlework, and M. C. Escher's Tessellations. Some examples of the 17 plane symmetry groups will be shown.

9 A repeating pattern or a tessellation or a tiling of the plane is a covering of the plane by one or more figures with a repeating pattern of the figures that has no gaps and no overlapping of the figures. Equilateral triangles Squares Regular Hexagons Examples : Regular Polygons

10 Some examples of periodic or repeating patterns, sometimes called “wallpaper designs,” will be shown. There are 17 “plane symmetry groups” or types of patterns.

11 Examples of places where repeating patterns are found: Wallpaper Designs Chinese Lattice Designs Hungarian Needlework Islamic Art The Alhambra M. C. Escher’s Tessellations

12 Wallpaper Designs

13 Chinese Lattice Designs

14 Chinese Lattice Design

15 Chinese Garden

16 p1 p211p1m1 p2mg p2gg c2mm pg c1m1 p2mm p4 p4m p4gm p3p3m1 p31m p6 p6mm

17 p1p2pmpgcmp2mm pmgpggc2mmp4p4mmp4gm p3p3m1p31mp6p6mm

18 p1 p4 p2 p6 p3 pm p2mm p2gg p4mm p2mg p 6mm p4gm cm c2mm p3m1 p31m pg Journal of Chemical Education

19 Wall Panel, Iran, 13th/14th cent (p6mm)

20 Design at the Alhambra

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22 Hall of Repose - The Alhambra

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24 Resting Hall - The Alhambra

25 Collage of Alhambra Tilings

26 M. C. Escher, 1898 - 1972

27 Keukenhof Gardens

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29 Escher’s Drawings of Alhambra Repeating Patterns

30 Escher Sketches of designs in the Alhambra and La Mezquita (Cordoba)

31 Mathematical Reference: “The Plane Symmetry Groups: Their Recognition and Notation” by Doris Schattschneider, The Mathematical Monthly, June-July, 1978 Artistic Source: Maurits C. Escher (1898-1972) was a master at constructing tessellations

32 Visions of Symmetry Doris Schattschneider W.H. Freeman 1990

33 1981, 1982, 1984, 1992

34 A unit cell or “tile” is the smallest region in the plane having the property that the set of all of its images will fill the plane. These images may be obtained by: Translations : plottools[translate](tile,XD,YD) Rotations: plottools[rotate](M,Pi/2,[40,40]) Reflections: plottools[reflect](M,[[0,0],[40,40]]) Glide Reflections: translate & reflect

35 Unit Cell -- de Porcelain Fles

36 Translation

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40 Pegasus - p1 105 Baarn, 1959 System I D

41 Pegasus - p1

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45 p1 Birds Baarn 1959

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48 p1 Birds Baarn 1967

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53 2-Fold Rotation

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57 p211

58 Doves, Ukkel, Winter 1937-38 p2

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62 3-Fold Rotation

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68 Reptiles, Ukkel, 1939

69 Escher’s Drawing – Unit Cell p3

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74 One Of Escher’s Sketches

75 Sketch for Reptiles

76 Reptiles, 1943 (Lithograph)

77 Metamorphose, PO, Window 5

78 Metamorphose, Windows 6-9

79 Metamorphose, Windows 11-14

80 Air Mail Letters Baarn 1956

81 Air Mail Letters in PO

82 Post Office in The Hague Metamorphosis is 50 Meters Long

83 4-Fold Rotation

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88 Reptiles, Baarn, 1959 p4

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91 Reptiles, Baarn, 1959

92 6-Fold Rotation

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97 P6 Birds Baarn, August, 1954

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100 P6 Birds, Baarn, August, 1954

101 Rotations

102 Reflection

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113 Design from Ancient Egypt Handbook of Regular Patterns by Peter S. Stevens

114 Glide Reflection

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119 p1g1 Toads

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124 p1g1 Toads, Baarn, January, 1961

125 Unicorns Baarn, November, 1950

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128 Swans Baarn, December, 1955

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131 Swans Baarn, December, 1955

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135 p2mm Baarn 1950

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140 p2mg

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150 p2gg Baarn 1963

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152 p2gg

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157 p4mm

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162 p4gm

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169 p3m1

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176 Flukes Baarn 1959

177 p31m

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180 P31m, Baarn, 1959

181 p31m

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184 p6mm

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202 Keukenhof Garden

203 Seville

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