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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 March 2, 2006 Slide 2 PARADE MAGAZINE, December 8, 2002 Slide 3 Slide 4 Slide 5 Slide 6 Slide 7 BIKE BOX CHECKBOOK DECKED HEED HIDE BIKE BOX CHECKBOOK DECKED HEED HIDE BIKE BOX CHECKBOOK DECKED HEED HIDE BIKE BOX CHECKBOOK DECKED REFLECT ROTATEROTATE Slide 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. Slide 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 Slide 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. Slide 11 Examples of places where repeating patterns are found: Wallpaper Designs Chinese Lattice Designs Hungarian Needlework Islamic Art The Alhambra M. C. Eschers Tessellations Slide 12 Wallpaper Designs Slide 13 Chinese Lattice Designs Slide 14 Chinese Lattice Design Slide 15 Chinese Garden Slide 16 p1 p211p1m1 p2mg p2gg c2mm pg c1m1 p2mm p4 p4m p4gm p3p3m1 p31m p6 p6mm Slide 17 p1p2pmpgcmp2mm pmgpggc2mmp4p4mmp4gm p3p3m1p31mp6p6mm Slide 18 p1 p4 p2 p6 p3 pm p2mm p2gg p4mm p2mg p 6mm p4gm cm c2mm p3m1 p31m pg Journal of Chemical Education Slide 19 Wall Panel, Iran, 13th/14th cent (p6mm) Slide 20 Design at the Alhambra Slide 21 Slide 22 Hall of Repose - The Alhambra Slide 23 Slide 24 Resting Hall - The Alhambra Slide 25 Collage of Alhambra Tilings Slide 26 M. C. Escher, 1898 - 1972 Slide 27 Keukenhof Gardens Slide 28 Slide 29 Eschers Drawings of Alhambra Repeating Patterns Slide 30 Escher Sketches of designs in the Alhambra and La Mezquita (Cordoba) Slide 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 Slide 32 Visions of Symmetry Doris Schattschneider W.H. Freeman 1990 Slide 33 1981, 1982, 1984, 1992 Slide 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 Slide 35 Unit Cell -- de Porcelain Fles Slide 36 Translation Slide 37 Slide 38 Slide 39 Slide 40 Pegasus - p1 105 Baarn, 1959 System I D Slide 41 Pegasus - p1 Slide 42 Slide 43 Slide 44 Slide 45 p1 Birds Baarn 1959 Slide 46 Slide 47 Slide 48 p1 Birds Baarn 1967 Slide 49 Slide 50 Slide 51 Slide 52 Slide 53 2-Fold Rotation Slide 54 Slide 55 Slide 56 Slide 57 p211 Slide 58 Doves, Ukkel, Winter 1937-38 p2 Slide 59 Slide 60 Slide 61 Slide 62 3-Fold Rotation Slide 63 Slide 64 Slide 65 Slide 66 Slide 67 Slide 68 Reptiles, Ukkel, 1939 Slide 69 Eschers Drawing Unit Cell p3 Slide 70 Slide 71 Slide 72 Slide 73 Slide 74 One Of Eschers Sketches Slide 75 Sketch for Reptiles Slide 76 Reptiles, 1943 (Lithograph) Slide 77 Metamorphose, PO, Window 5 Slide 78 Metamorphose, Windows 6-9 Slide 79 Metamorphose, Windows 11-14 Slide 80 Air Mail Letters Baarn 1956 Slide 81 Air Mail Letters in PO Slide 82 Post Office in The Hague Metamorphosis is 50 Meters Long Slide 83 4-Fold Rotation Slide 84 Slide 85 Slide 86 Slide 87 Slide 88 Reptiles, Baarn, 1959 p4 Slide 89 Slide 90 Slide 91 Reptiles, Baarn, 1959 Slide 92 6-Fold Rotation Slide 93 Slide 94 Slide 95 Slide 96 Slide 97 P6 Birds Baarn, August, 1954 Slide 98 Slide 99 Slide 100 P6 Birds, Baarn, August, 1954 Slide 101 Rotations Slide 102 Reflection Slide 103 Slide 104 Slide 105 Slide 106 Slide 107 Slide 108 Slide 109 Slide 110 Slide 111 Slide 112 Slide 113 Design from Ancient Egypt Handbook of Regular Patterns by Peter S. Stevens Slide 114 Glide Reflection Slide 115 Slide 116 Slide 117 Slide 118 Slide 119 p1g1 Toads Slide 120 Slide 121 Slide 122 Slide 123 Slide 124 p1g1 Toads, Baarn, January, 1961 Slide 125 Unicorns Baarn, November, 1950 Slide 126 Slide 127 Slide 128 Swans Baarn, December, 1955 Slide 129 Slide 130 Slide 131 Swans Baarn, December, 1955 Slide 132 Slide 133 Slide 134 Slide 135 p2mm Baarn 1950 Slide 136 Slide 137 Slide 138 Slide 139 Slide 140 p2mg Slide 141 Slide 142 Slide 143 Slide 144 Slide 145 Slide 146 Slide 147 Slide 148 Slide 149 Slide 150 p2gg Baarn 1963 Slide 151 Slide 152 p2gg Slide 153 Slide 154 Slide 155 Slide 156 Slide 157 p4mm Slide 158 Slide 159 Slide 160 Slide 161 Slide 162 p4gm Slide 163 Slide 164 Slide 165 Slide 166 Slide 167 Slide 168 Slide 169 p3m1 Slide 170 Slide 171 P3m1 Slide 172 p3m1 Slide 173 Slide 174 Slide 175 p31m Slide 176 Flukes Baarn 1959 Slide 177 p31m Slide 178 Slide 179 Slide 180 P31m, Baarn, 1959 Slide 181 p31m Slide 182 Slide 183 Slide 184 p6mm Slide 185 Slide 186 Slide 187 Slide 188 Slide 189 Slide 190 c1m1 Slide 191 Slide 192 Slide 193 Slide 194 Slide 195 Slide 196 Slide 197 Slide 198 Slide 199 Slide 200 Slide 201 Slide 202 Keukenhof Garden Slide 203 Seville Slide 204