Presentation on theme: "Professor Emeritus of Mathematics"— Presentation transcript:
1Professor Emeritus of Mathematics Using MAPLE to Construct Repeating Patterns and Several Tessellations Inspired by M. C. EscherElliot A. TanisProfessor Emeritus of MathematicsHope CollegeMarch 2, 2006
7REFLECT ROTATE p BIKE BOX CHECKBOOK DECKED HEED HIDE
8A 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.
9A 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 trianglesSquaresRegular HexagonsRegularPolygons
10Some examples of periodic or repeating patterns, sometimes called “wallpaper designs,” will be shown. There are 17 “plane symmetry groups” or types of patterns.
11Examples of places where repeating patterns are found: Wallpaper DesignsChinese Lattice DesignsHungarian NeedleworkIslamic ArtThe AlhambraM. C. Escher’s Tessellations
29Escher’s Drawings of Alhambra Repeating Patterns
30Escher Sketches of designs in the Alhambra and La Mezquita (Cordoba)
31Mathematical Reference: “The Plane Symmetry Groups: Their Recognition and Notation”by Doris Schattschneider,The Mathematical Monthly, June-July, 1978Artistic Source: Maurits C. Escher ( ) was a master at constructing tessellations
34Rotations: 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