Download presentation

Presentation is loading. Please wait.

Published byMya Stout Modified about 1 year ago

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

3

4

5

6

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

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

21

22
Hall of Repose - The Alhambra

23

24
Resting Hall - The Alhambra

25
Collage of Alhambra Tilings

26
M. C. Escher, 1898 - 1972

27
Keukenhof Gardens

28

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

37

38

39

40
Pegasus - p1 105 Baarn, 1959 System I D

41
Pegasus - p1

42

43

44

45
p1 Birds Baarn 1959

46

47

48
p1 Birds Baarn 1967

49

50

51

52

53
2-Fold Rotation

54

55

56

57
p211

58
Doves, Ukkel, Winter 1937-38 p2

59

60

61

62
3-Fold Rotation

63

64

65

66

67

68
Reptiles, Ukkel, 1939

69
Escher’s Drawing – Unit Cell p3

70

71

72

73

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

84

85

86

87

88
Reptiles, Baarn, 1959 p4

89

90

91
Reptiles, Baarn, 1959

92
6-Fold Rotation

93

94

95

96

97
P6 Birds Baarn, August, 1954

98

99

100
P6 Birds, Baarn, August, 1954

101
Rotations

102
Reflection

103

104

105

106

107

108

109

110

111

112

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

114
Glide Reflection

115

116

117

118

119
p1g1 Toads

120

121

122

123

124
p1g1 Toads, Baarn, January, 1961

125
Unicorns Baarn, November, 1950

126

127

128
Swans Baarn, December, 1955

129

130

131
Swans Baarn, December, 1955

132

133

134

135
p2mm Baarn 1950

136

137

138

139

140
p2mg

141

142

143

144

145

146

147

148

149

150
p2gg Baarn 1963

151

152
p2gg

153

154

155

156

157
p4mm

158

159

160

161

162
p4gm

163

164

165

166

167

168

169
p3m1

170

171
P3m1

172
p3m1

173

174

175
p31m

176
Flukes Baarn 1959

177
p31m

178

179

180
P31m, Baarn, 1959

181
p31m

182

183

184
p6mm

185

186

187

188

189

190
c1m1

191

192

193

194

195

196

197

198

199

200

201

202
Keukenhof Garden

203
Seville

204

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google