Using MAPLE to Construct Repeating Patterns and Several Tessellations Inspired by M. C. Escher

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

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

pm pg cm

pgg c2mm

p3 p3m1 p31m p6 p6mm

p2mm p2gg

p6mmp4gm

pgJournal of Chemical Education

Wall Panel, Iran, 13th/14th cent (p6mm)

Design at the Alhambra

Hall of Repose - The Alhambra

Resting Hall - The Alhambra

Collage of

Alhambra

Tilings

M. C. Escher, 1898 - 1972

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

Schattschneider

W.H. Freeman

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

Pegasus - p1105

Baarn, 1959

System ID

Pegasus - p1

2-Fold Rotation

Doves, Ukkel, Winter 1937-38

3-Fold Rotation

Reptiles, Ukkel, 1939

Escher’s Drawing – Unit Cell

Escher’s

Sketches

Sketch for Reptiles

Reptiles, 1943 (Lithograph)

Metamorphose, PO, Window 5

Metamorphose, Windows 6-9

Metamorphose, Windows 11-14

Air Mail

Letters

Air Mail Letters in PO

Post Office in The HagueMetamorphosis is 50 Meters Long

4-Fold Rotation

Reptiles, Baarn, 1959

6-Fold Rotation

P6 Birds

Baarn, August,

P6 Birds, Baarn, August,

Rotations

Reflection

Design from Ancient Egypt

Handbook of Regular Patterns by Peter S. Stevens

Glide Reflection

p1g1 Toads

p1g1 Toads, Baarn, January,

Unicorns

Baarn, November, 1950

Baarn, December, 1955

Swans Baarn, December,

Flukes

P31m, Baarn, 1959

Keukenhof Garden

Seville

Using MAPLE to Construct Repeating Patterns and Several Tessellations Inspired by M. C. Escher

Documents

Transcript of Using MAPLE to Construct Repeating Patterns and Several Tessellations Inspired by M. C. Escher

Tessellations - Mrs. Wingard's Websitemelaniewingard.weebly.com/uploads/3/7/5/5/37554047/... · Tessellations A tessellation is a repeating pattern of polygons that covers a plane

Dihedral Escherization - University of Waterloo · Key words: Tilings, tessellations, Escher, aperiodic, Pen-rose tilings 1 Introduction The Dutch artist M.C. Escher had a singular

Making Tessellations

Tessellations Alexander Alvarez. A tessellation is a repeating pattern of figures that completely covers a plane without gaps or overlaps. A tessellation.

TESSELLATIONS Tessellation: Tiling a planecs6710/slides/Tessellationsx2.pdf · M. C. Escher Dutch graphic artist – (1898 – 1972) known for his often mathematically inspired woodcuts,

Tessellations 1

Using MAPLE to Construct Repeating Patterns and Several Tessellations Inspired by M. C. Escher Elliot A. Tanis Professor Emeritus of Mathematics Hope College.

5-5 M.C. Escher Day and Night Lesson Plan · 2015-09-25 · mathematical art technique called tessellations. ... Microsoft Word - 5-5 M.C. Escher Day and Night Lesson Plan.docx

NAPOLEON, ESCHER AND NAPOLEON, ESCHER … · Escher's theorem is aform of converse of Napoleon's theorem, and both theorems can be proved using tessellations, a method that would

Art Y5 Week 2 : Escher and tessellations...M.C.Esher or Maurits Cornelis Escher was born in 1898 and died in 1972. He was from the Netherlands. Escher didn’t do very well at school

Creating Escher-Type Tessellations Escher-type Tessellations 4 tkawas@mathwire.com. Step 4: Use Escher EyesStep 4: • Look at your shape from all angles. • Slide, flip and turn

Tessellations Complete

What is a Tessellation? A tessellation is a pattern of repeating figures that fit together with NO overlapping or empty spaces. Tessellations are formed.

CREATIVITY AND MOTIVATION FOR G TASKS DESIGNING IN ... - … · graphic artist Escher (see Figure 2). More about the author can be found in [9]. We can also see tessellations just

Tessellations. M. C. Escher 1. This initial hexagon shape can generate the entire tessellation through a series of translations. Visualize.

M.C. Escher (1898-1972) The Art of Tessellations.

This Exploration of Tessellations will guide you through the following: Exploring Tessellations Definition of Tessellation Semi-Regular Tessellations.

Creating Repeating Patterns with Color Symmetryddunham/dunisama10.pdfColor Symmetry: History and Theory Others created patterns with color symmetry before Escher, but he was quite

Tessellations. Featuring Artist M.C. Escher Zwanen Swans.

NAPOLEON, ESCHER AND NAPOLEON, ESCHER TESSEUATIONSmacosa.dima.unige.it/om/voci/triango/escher/escher2.pdf · Napoleon and Escher both have theorems about triangles named after them.