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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. PARADE MAGAZINE, December 8, 2002. BIKE BOX CHECKBOOK DECKED HEED HIDE. HEED HIDE
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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
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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.
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
Some examples of periodic or repeating patterns, sometimes called “wallpaper designs,” will be shown. There are 17 “plane symmetry groups” or types of patterns.
Examples of places where repeating patterns are found: • Wallpaper Designs • Chinese Lattice Designs • Hungarian Needlework • Islamic Art • The Alhambra • M. C. Escher’s Tessellations
Chinese Lattice Design
p1 p211 p1m1 pg c1m1 p2mm p2gg p4gm p2mg p4m c2mm p4 p3 p3m1 p6 p31m p6mm
p1 p2 pm pg cm p2mm pmg pgg c2mm p4 p4mm p4gm p3 p3m1 p31m p6 p6mm
p2gg p2mm p2mg p4mm p4gm p6mm p1 p4 p3m1 cm p6 p31m p2 c2mm p3 pm pg Journal of Chemical Education
Collage of Alhambra Tilings
Escher Sketches of designs in the Alhambra and La Mezquita (Cordoba)
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
Visions of Symmetry Doris Schattschneider W.H. Freeman 1990
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
Pegasus - p1 105 D Baarn, 1959 System I
p1 Birds Baarn 1959
p1 Birds Baarn 1967