Professor Emeritus of Mathematics

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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

REFLECT ROTATE p BIKE BOX CHECKBOOK DECKED HEED HIDE

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. Examples: Equilateral triangles Squares Regular Hexagons 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

Wallpaper Designs

Chinese Lattice Designs

Chinese Lattice Design

Chinese Garden

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

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

Design at the Alhambra

Design at the Alhambra

Hall of Repose - The Alhambra

Hall of Repose - The Alhambra

Resting Hall - The Alhambra

Collage of Alhambra Tilings

M. C. Escher,

Keukenhof Gardens

Keukenhof Gardens

Escher’s Drawings of Alhambra Repeating Patterns

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 ( ) was a master at constructing tessellations

Visions of Symmetry Doris Schattschneider W.H. Freeman 1990

1981, 1982, 1984, 1992

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

Unit Cell -- de Porcelain Fles

Translation

Translation

Translation

Translation

Pegasus - p1 105 D Baarn, 1959 System I

Pegasus - p1

p1 Birds Baarn 1959

p1 Birds Baarn 1967

2-Fold Rotation

2-Fold Rotation

p211

Doves, Ukkel, Winter p2

3-Fold Rotation

3-Fold Rotation

Reptiles, Ukkel, 1939

Escher’s Drawing – Unit Cell
p3

One Of Escher’s Sketches

Sketch for Reptiles

Reptiles, 1943 (Lithograph)

Metamorphose, PO, Window 5

Metamorphose, Windows 6-9

Metamorphose, Windows 11-14

Air Mail Letters Baarn 1956

Air Mail Letters in PO

Post Office in The Hague Metamorphosis is 50 Meters Long

4-Fold Rotation

4-Fold Rotation

Reptiles, Baarn, 1959 p4

Reptiles, Baarn, 1959

6-Fold Rotation

6-Fold Rotation

P6 Birds Baarn, August, 1954

P6 Birds, Baarn, August, 1954

Rotations

Reflection

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

Glide Reflection

Glide Reflection

Unicorns Baarn, November, 1950

Swans Baarn, December, 1955

Swans Baarn, December, 1955

p2mm Baarn 1950

p2mg

p2mg

p2mg

p2mg

p2mg

p2mg

p2gg Baarn 1963

p2gg

p4mm

p4mm

p4mm

p4mm

p4gm

p4gm

p4gm

p4gm

p4gm

p3m1

p3m1

P3m1

p3m1

p3m1

p31m

Flukes Baarn 1959

p31m

p31m

p31m

P31m, Baarn, 1959

p31m

p31m

p6mm

p6mm

p6mm

p6mm

p6mm

c1m1

c1m1

c1m1

c1m1

c1m1

c1m1

Keukenhof Garden

Seville

Seville