Using MAPLE to Construct Repeating Patterns and

Several Tessellations Inspired by

M. C. EscherElliot A. Tanis

Professor Emeritus of Mathematics

Hope CollegeMarch 2, 2006

PARADE MAGAZINE, December 8, 2002

BIKE BOX CHECKBOOK DECKED

HEED HIDE

HEED HIDE

BIKE BOX CHECKBOOK DECKED

HEED HIDE

BIKE BOX CHECKBOOK DECKED

HEED HIDE

BIKE BOX CHECKBOOK DECKED

REFLECT

ROTATE

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

Wallpaper Designs

Chinese Lattice Designs

Chinese

Lattice

Design

Chinese Garden

p1p211 p1m1

p2mg p2gg c2mm

pg c1m1 p2mm

p4 p4m p4gm

p3 p3m1 p31m p6p6m

m

p1 p2

pm pg cm

p2mm

pmg

pgg c2mm

p4

p4mm

p4gm

p3 p3m1 p31m p6 p6mm

p1 p4

p2 p6

p3pm

p2mm p2gg

p4mm

p2mg

p6mmp4gm

cm

c2mm

p3m1

p31m

pgJournal 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, 1898 - 1972

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, 1978Artistic Source: Maurits C. Escher (1898-1972) was a master at constructing tessellations

Visions of

Symmetry

Doris

Schattschneider

W.H. Freeman

1990

1981, 1982,

1984, 1992

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 - p1105

Baarn, 1959

System ID

Pegasus - p1

p1

Birds

Baarn

1959

p1

Birds

Baarn

1967

2-Fold Rotation

2-Fold Rotation

p211

Doves, Ukkel, Winter 1937-38

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 HagueMetamorphosis 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

p1g1 Toads

p1g1 Toads, Baarn, January,

1961

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