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**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

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**PARADE MAGAZINE, December 8, 2002**

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**REFLECT ROTATE p BIKE BOX CHECKBOOK DECKED HEED HIDE**

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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.

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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

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Some examples of periodic or repeating patterns, sometimes called “wallpaper designs,” will be shown. There are 17 “plane symmetry groups” or types of patterns.

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**Examples of places where repeating patterns are found:**

Wallpaper Designs Chinese Lattice Designs Hungarian Needlework Islamic Art The Alhambra M. C. Escher’s Tessellations

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Wallpaper Designs

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**Chinese Lattice Designs**

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Chinese Lattice Design

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Chinese Garden

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p1 p211 p1m1 pg c1m1 p2mm p2gg p4gm p2mg p4m c2mm p4 p3 p3m1 p6 p31m p6mm

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

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**p2gg p2mm p2mg p4mm p4gm p6mm p1 p4 p3m1 cm p6 p31m p2 c2mm p3 pm pg**

Journal of Chemical Education

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

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Design at the Alhambra

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Design at the Alhambra

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**Hall of Repose - The Alhambra**

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**Hall of Repose - The Alhambra**

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**Resting Hall - The Alhambra**

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Collage of Alhambra Tilings

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M. C. Escher,

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Keukenhof Gardens

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Keukenhof Gardens

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**Escher’s Drawings of Alhambra Repeating Patterns**

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**Escher Sketches of designs in the Alhambra and La Mezquita (Cordoba)**

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**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

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Visions of Symmetry Doris Schattschneider W.H. Freeman 1990

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1981, 1982, 1984, 1992

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**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

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**Unit Cell -- de Porcelain Fles**

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Translation

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Translation

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Translation

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Translation

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Pegasus - p1 105 D Baarn, 1959 System I

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Pegasus - p1

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p1 Birds Baarn 1959

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p1 Birds Baarn 1967

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2-Fold Rotation

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2-Fold Rotation

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p211

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Doves, Ukkel, Winter p2

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3-Fold Rotation

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3-Fold Rotation

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Reptiles, Ukkel, 1939

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**Escher’s Drawing – Unit Cell**

p3

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One Of Escher’s Sketches

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Sketch for Reptiles

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**Reptiles, 1943 (Lithograph)**

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**Metamorphose, PO, Window 5**

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**Metamorphose, Windows 6-9**

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**Metamorphose, Windows 11-14**

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Air Mail Letters Baarn 1956

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Air Mail Letters in PO

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**Post Office in The Hague Metamorphosis is 50 Meters Long**

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4-Fold Rotation

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4-Fold Rotation

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Reptiles, Baarn, 1959 p4

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Reptiles, Baarn, 1959

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6-Fold Rotation

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6-Fold Rotation

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P6 Birds Baarn, August, 1954

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P6 Birds, Baarn, August, 1954

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Rotations

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Reflection

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**Design from Ancient Egypt**

Handbook of Regular Patterns by Peter S. Stevens

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Glide Reflection

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Glide Reflection

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p1g1 Toads

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p1g1 Toads, Baarn, January, 1961

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Unicorns Baarn, November, 1950

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Swans Baarn, December, 1955

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Swans Baarn, December, 1955

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p2mm Baarn 1950

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p2mg

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p2mg

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p2mg

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p2mg

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p2mg

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p2mg

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p2gg Baarn 1963

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p2gg

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p4mm

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p4mm

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p4mm

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p4mm

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p4gm

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p4gm

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p4gm

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p4gm

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p4gm

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p3m1

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p3m1

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P3m1

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p3m1

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p3m1

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p31m

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Flukes Baarn 1959

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p31m

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p31m

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p31m

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P31m, Baarn, 1959

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p31m

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p31m

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p6mm

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p6mm

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p6mm

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p6mm

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p6mm

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c1m1

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c1m1

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c1m1

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c1m1

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c1m1

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c1m1

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Keukenhof Garden

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Seville

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Seville

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“Waterfall,” 1961 by M.C. Escher. Tessellations are designs featuring animals, shapes, birds, etc, which can fill the page, without over-lapping, to form.

“Waterfall,” 1961 by M.C. Escher. Tessellations are designs featuring animals, shapes, birds, etc, which can fill the page, without over-lapping, to form.

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