Symmetry: Chinese Lattice Designs The Alhambra M. C. Escher also Wallpaper Designs Hungarian Needlework Elliot A. Tanis Hope College

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

p2gg p2mm p2mg p4mm p4gm p6mm p1 p4 p3m1 cm p6 p31m p2 c2mm p3 pm 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 - p1 105 D Baarn, 1959 System I

Pegasus - p1

Ernest R. Ranucci Joseph L. Teeters

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 Hague Metamorphosis 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 Type and Then Replicate This Design

p4mm

Belvedere May 1958 Lithograph

Man with Cuboid, 1958, Wood Engraving

Relativity, 1953 Woodcut Lithograph

Relativity, 1953 Lithograph Woodcut

Mobius Strip II (Red Ants), 1963

Ascending And Descending 1960 Lithograph

Drawing Hands, 1948, Lithograph

Creation

Keukenhof Garden