Using MAPLE to Construct Repeating Patterns and Several Tessellations Inspired by M. C. Escher

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

2. PARADE MAGAZINE, December 8, 2002

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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

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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, 1898 - 1972

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 (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. 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