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
PARADE MAGAZINE, December 8, 2002
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, 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, 1978 Artistic 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
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 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 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
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