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