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 p6p6mm
p1 p4
p2 p6
p3 pm
p2mm p2gg
p4mm p2mg
p6mmp4gm
cm
c2mm
p3m1
p31m
pg
Wall Panel, Iran, 13th/14th cent(p4mm)
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
Church/Mosque in Cordoba
Church/Mosque in Cordoba
Pillars by M. C. Escher
Cordoba
Seville
Seville
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
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 in the plane. These images may be obtained by:
• Translations
• Rotations
• Reflections
• Glide Reflections
Unit Cell -- de Porcelain Fles
Translation
Translation
Pegasus - p1105
Baarn, 1959System I D
Pegasus - p1
Joseph L. Teeters
Ernest R. Ranucci
Suggestion
From
Ranucci
and
Teeters
Outline
Of
One
Pegasus
Why
Is
Red
Used?
A School in The Hague
Slightly
Modified
Pattern
Types
p1
Birds
Baarn
1959
p1 Birds
p1
Birds
Baarn
1967
p1
Birds
Baarn
1967
3-Fold Rotation
Reptiles, Ukkel, 1939
Suggestion
From
Rannuci
and
Teeters
One
Of
Escher’s
Sketches
Escher’s Drawing – Unit Cell
p3
Pattern Type
p3 Toads
Sketch for Reptiles
Reptiles, 1943 (Lithograph)
Metamorphosis II November 1939 - March 1940
Metamorphosis II November 1939 - March 1940
1967 - 1968
Woodcut
Metamorphosis III
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
2-Fold Rotation
Doves, Ukkel, Winter 1937-38
p2
p211 Doves
4-Fold Rotation
Reptiles, Baarn, 1959
p4
p4 Reptiles
Reptiles, Baarn, 1963
p4
6-Fold Rotation
Reptiles, Baarn, 1942
p6
Rotations
Reflection
p11m Cows
Glide Reflection
Glide Reflection
p1g1 Toads
p1g1 Toads
Flukes
Baarn
1959
p31m
p31m “flukes”
p3m1
Baarn
1952
p2mm
Baarn
1950
c1m1
Baarn
1953
p2gg
Baarn
1963
Baarn
1964
p4gm
Determine the Pattern Typeand Then Replicate This Design