1. Professor Emeritus of Mathematics
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

7. REFLECT ROTATE p BIKE BOX CHECKBOOK DECKED HEED HIDE

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.
Examples:
Equilateral triangles
Squares
Regular Hexagons
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
p211
p1m1 pg
c1m1
p2mm
p2gg
p4gm
p2mg
p4m
c2mm p4 p3
p3m1 p6
p31m
p6mm

17. p1 p2 pm pg cm
p2mm
pmg
pgg
c2mm p4
p4mm
p4gm p3
p3m1
p31m p6
p6mm

18. p2gg p2mm p2mg p4mm p4gm p6mm p1 p4 p3m1 cm p6 p31m p2 c2mm p3 pm pg
Journal of Chemical Education

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

20. Design at the Alhambra

21. Design at the Alhambra

22. Hall of Repose - The Alhambra

23. Hall of Repose - The Alhambra

24. Resting Hall - The Alhambra

25. Collage of
Alhambra
Tilings

26. M. C. Escher,

27. Keukenhof Gardens

28. Keukenhof Gardens

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 ( ) was a master at constructing tessellations

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

33. 1981, 1982, 1984, 1992

34. Rotations: plottools[rotate](M,Pi/2,[40,40])
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. Translation

38. Translation

39. Translation

40. Pegasus - p1
105 D
Baarn, 1959
System I

41. Pegasus - p1

45. p1
Birds
Baarn
1959

48. p1
Birds
Baarn
1967

53. 2-Fold Rotation

54. 2-Fold Rotation

57. p211

58. Doves, Ukkel, Winter p2

62. 3-Fold Rotation

63. 3-Fold Rotation

68. Reptiles, Ukkel, 1939

69. Escher’s Drawing – Unit Cellp3

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. 4-Fold Rotation

88. Reptiles, Baarn, 1959 p4

91. Reptiles, Baarn, 1959

92. 6-Fold Rotation

93. 6-Fold Rotation

97. P6 Birds
Baarn, August, 1954

100. P6 Birds, Baarn, August, 1954

101. Rotations

102. Reflection

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

114. Glide Reflection

115. Glide Reflection

119. p1g1 Toads

124. p1g1 Toads, Baarn, January, 1961

125. Unicorns
Baarn, November, 1950

128. Swans
Baarn, December, 1955

131. Swans Baarn, December, 1955

135. p2mm
Baarn
1950

140. p2mg

141. p2mg

142. p2mg

145. p2mg

146. p2mg

147. p2mg

150. p2gg
Baarn
1963

152. p2gg

157. p4mm

158. p4mm

159. p4mm

160. p4mm

162. p4gm

163. p4gm

164. p4gm

165. p4gm

167. p4gm

169. p3m1

170. p3m1

171. P3m1

172. p3m1

173. p3m1

175. p31m

176. Flukes
Baarn
1959

177. p31m

178. p31m

179. p31m

180. P31m, Baarn, 1959

181. p31m

182. p31m

184. p6mm

185. p6mm

186. p6mm

187. p6mm

188. p6mm

190. c1m1

191. c1m1

192. c1m1

194. c1m1

195. c1m1

196. c1m1

202. Keukenhof Garden

203. Seville

204. Seville