Symmetry:Chinese Lattice Designs

The AlhambraM. C. Escher

• Wallpaper Designs

• Hungarian Needlework

also

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.

• 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

p4mm

Belvedere

May 1958

Lithograph

Man with Cuboid, 1958,

Wood Engraving

Relativity, 1953

Woodcut Lithograph

Relativity, 1953

Woodcut Lithograph

Mobius Strip II (Red Ants), 1963

Ascending

And

Descending

1960

Lithograph

Drawing Hands, 1948, Lithograph

Creation

Keukenhof Garden