Pattern, the visual mathematics: A search for logical connections in design through mathematics, science, and multicultural arts

Rochester Institute of Technology

RIT Scholar Works

Theses

Thesis/Dissertation Collections

6-1-1996

Pattern, the visual mathematics: A search for logical

connections in design through mathematics,

science, and multicultural arts

Seung-eun Lee

Follow this and additional works at:

http://scholarworks.rit.edu/theses

This Thesis is brought to you for free and open access by the Thesis/Dissertation Collections at RIT Scholar Works. It has been accepted for inclusion

in Theses by an authorized administrator of RIT Scholar Works. For more information, please contact

[email protected].

Recommended Citation

PicrdoTrKwe, r't.t/ry

'jX7ipu.lv z&ietQStd(XtrmaiKc'i-J: ffocub.The(i^piicaiom o". crWx<M Wfflrtu't. fifesspwn,Jiii*iaiin^jfli dtfifi,tmijpivxeiu3ili'-rCiluvjtv-tzxi patcrr?;i t.driftersvcy.

Symmetr, yo

Thetheoreticalconcepts ofsymmetry dealwith

grouptheoryandfiguretransformations.Figure transformations,orsymmetryoperations referto themovement and repetition of anone-, two,and

three-dimensionalspace.

Themotif ibelj urwlu. vaed U> cfaau patem.Afci

w the ivxttk arwlmetwung

J\ one variet.

Ilhowtuoirwti)

,^gjingnuirtmgiltjpg.v/htch art familiar rhtzrtiofIcingir-the\X''tJ. have been-.tapedfcjbrwJerfg I iivCTiitt-;.cfBattemtiatkindni WeiLon Imowledg*teemto ihare tki lame&asK pattum-farming prKtszet OutersatztheWmdnies o/natureand

.^rr.pcreand vafidWtxempB'f1 and mouj: iri termiof Semiociw Hut poM' vx*.-reaudto (tv.peoplean OpportunityCo get abroader under-standingaboutt)vzmeQ.fi and (wft'.nu

. ofcrou-toktadiMue.

d-:_ivi;i10mthefi>e

MathematicalpatternsSometimesreflectvisuaipattern',that tfre eyefinds particularlyaesthetic.Onefamous example i Kuchmathematicalpatternisthe golden ratio.Accordingto the

Greeks,thegoldenratiotstheidealproportionfoi'the sidesof : rectanglethattheeyefindsthemost pleasing The<)olci*?nrati* is found innature Thechamberedshellgrowsto termalogamf)

piral,a mathematical curvethatspiralsouti dependentonthegoldenraffb.'

Thevalue ofthegoldenratiois(1 +05) '2,an mr approximatelyequal to 1.67 S Itisthewhendividinga .-twopiecessothat theratio ofthelongerpieceto the'JnjJV'ta

.v+1

-flrWcmUlhihbe<|.m'" I'"* Incommunities,varioustaboos i-jmtintobeingregarding his wayn||(fe.Togetherwiththe concept ofrevering heaven,.,

rifin.il ritesinorder spiritstograntthis wish,andhemadethedoud Image Itssymbol, cloudshapes were carved on sacritidal uten silsinthehopesthaithedesires theysymbolizedwould reach heaven.Thedoudmotifs appearIn descriptionsofthe

ofheaventhatare scat tered throughandenllegends,

nd serve as symbolic represen-ons of(his idealrealm.

Sbrjo, Wand,SO0BX. B, Intuitionor some jreientreligionscience, r.iinnolthe mj(>cvv>e<n tcih.wc grM-j.iti.jidiespiral .i,robotizes artr.tr> in thelife gtvmg <d duos.

-. analytical t1

C

u l t u

g

Patternarrangementippeai meaningfulpatternsolhuminbehavioiSine* IhttUscveryand e; of patternedhumaibehavior isI logy,itwould set

behaviorpatterns cotintently |fate assetfori erycultureinIheworldii knowntodecoratealleastsomep

A

search

for logical

connections

in design

through mathematics,

science and multicultural arts

A Thesis Report

Submitted

to the

Faculty

of

The College

of

Imaging

Arts

and

Sciences

In

Candidacy

for

the

Degree

of

Master

of

Fine Arts

Seung-Eun Lee

Department

of

Graphic

Design

CONTENTS

TITLE PAGE

J

CONTENTS

5

APPROVALS

7

INTRODUCTION

11

ACKNOWLEDGEMENT

13

RESEARCH

15

METHODOLOGY

19

SYNTHESIS

34

IMPLEMENTATION

37

EVALUATION

42

CONCLUSION

42

FUTURE PLAN

43

I,

hereby grant permission to

the Wallace Memorial library of RIT to reproduce my thesis whole or part.

Any reproduction will not be for commercial use or profit.

I also prefer to be contacted each time a request for production is made.

1can be reached at the address following:

175 Boniface Drive

Rochester, NY 14620

U. S. A.

Tel

716) 271 2701

85-16 Chungdam-dong

Kangnam-gu

Seoul. 135-110

S. Korea

Tel

+011 82 2) 544 3313

e-mail

SEL2147/ilritvax. rit.isc. edu

SLee288918/ilaol.com

web-sile

http://www.espritdomain.com

APPROVALS

Advisor:

Professor

R.

Roger Remington

_

Date

Associate Advisor :

Professor Robert Keough

Date

Associate Advisor :

Dr. Richard D. Zakia

Date

Department Chairperson:

Professor Mary Ann Begland

Date

We

see patterns everywhere around usthat

have

a

relationship

with

both

the

natural and man-made world.

These

patterns provide visualenrichment and enjoyment.

It

is

alsotrue thatpatterns are sometimes regarded as

forms

of

decoration

such as wallpaper or

textile in

thecontext of art

history.

However,

patterns are morethan

just

a

form

of

decoration. Through many

eras,

different

cultures

have developed

patternsas a system of

recognition,

and

based

onmathematical system.

Patterns

are

formed

by

repeating

a motif as one of

the

fundamental

regions.

Some

ofthemotifs

have

a symbolic

meaning

connectedto

culture andregion.

Living

in

themodern

age,

weare

adapting

toamulti-cultured society.

Therefore,

it is

desired

toexamine

how

patterns

have been developed

by

different

andsuccessive cultures.

To

help

readers understand cross-cultural motifs and

patterns,

I

also

include

Semiotics

tosupportthis topic.

There

is

no

limitation

to the

study

of patterns.

I

have done

research on

culture,

history

of art and

craft,

graphic

theory, textiles, folk

art, religion,

cognitive

studies, psychology,

mathematics, crystallography

and even music.

As

result, I

believe

thatall ofthese

fields

are

interrelated in

analyzing

patterns.

No

aspect

is

more

important

thananother.

Each

discipline

supportstheother as

they

coexist.

The

goal ofthis thesis

is

tocreate

images

based

on

existing

theoriesrelated

I

am grateful

to

my

thesis committee,

Prof. R. Roger

Remington,

Prof.

Robert Keough

and

Dr. Richard D. Zakia

who

have

guidedme

through

stages of

accomplishing

this

thesis.

Special

recognitionto

my friends Gedeon Maheux

and

David Seah for

their

wholehearted support

in

knowledge

of computers and

encouraging

me allthe time withtheir

beautiful friendship.

My

appreciation also goesto

Andy

Lee

who

helped

me

by

reading

and com

menting

onthemanuscript ofthisthesis

report,

and

teaching

me other

knowledge

of computers with sincerity.

I

also givethanks to the

faculties

ofthe

Department

of

Graphic

Design,

Prof. Heinz

Klinkon,

Prof. James Ver

Hague,

Prof.

Nancy

A.Ciolek

and

Prof.

Mary

Ann

Begland

for

teaching

me graphic

design

and computer graphics.

My

gratitudeto

colleagues,

Yih-Chi

Wang,

Aaron

Lui,

Joseph A.

DiGioia,

Antoinette

Monnier

and

Jung-mei Tsen

for

theirwarm

friendship.

Special

thanks

to

Mr.

Eric Leader

at

Printing Prep

Inc.,

Buffalo

and

ETC

at the

RIT

Library

for

producing

the

final

implementation

and

publishing

this thesis

book

and

CD-ROM.

Above

all,

unlimited appreciation and

my

indebtness

to

my

parents,

Mr.

Choosing

a

Topic

During

the

Summer

of

1994,

1

startedtoconsider

my

thesis topic.

Anything

could

be

atopic

but I decided

tochoose patterns

because

it

containscontemporary,

multi-cultural and academic

issues in

which

I

have

a

strong

interest.

The

topicalso

had

to

be

ontheviewpoint of

fundamental

graphic

theory

andto

be

understood

by

both

graphic professionalsandnon-professionals.

I

rememberedthatwhile

taking

Theory

and

Methods

<amajor courseinthefirst quarter of1993)with

Prof. Roger

Remington,

I

wasattracted

to

William

Morris'

pattern

in

Art

S Craft Movement

and

decorative

arts

in

theperiod of

Art

Nouveau. Since I

had done

research onthe

Art Nouveau

period

extensively

atthe

time, I had

an

idea

whereto

begin my

research.

With

thosequestions

in mind,

I

found

it

necessary

to

understand graphic

theory

andapplications of pattern.

Since

1

had been

interested in computers,

I

thought

it

would

be

an

interesting

new

way

topresentthe

theory

of pattern with printedmedia and

interactive

multimedia applications.

I

believed

that

showing

thesubject patternwithnew

interactive

media would

be

a

interesting

way

to

satisfy

allrequirements

for

the thesisgoals.

[Research

8 analysis

]

Material Resources:

1

started

my

research atthe

RIT Wallace Library. I

printed a

list

ofrelated articles

from

theword pattern

using

the

Einstein

search engine.

Furthermore,

I

did

my

thesisresearch atthe

Rochester Public

Library

in

downtown Rochester

andthe

bookstores

aroundthe

Rochester

area.

I

tried to

familiarize

myself with wordsthat

describe

pattern and gatheredmore

information

on

books

on

how

patterns

have

been

studied

in

many

waysand

different

fields

ofstudy.

However,

I

found

thatmost ofthose

books

talked

just

withinart

history. I

wasnot satisfied

because I

couldn't

find

what

I

neededto

know.

Prof. Roger Remington

lent

me

Wucius Wong's Principle

of

Two

Dimensional

Design

and

Principle

of

Three Dimensional Design. Those

two

books

were

helpful for

me

to

understandthe

idea

of

different

structures

in

twoandthree

dimensional

composition.

He

also suggestedthe

book Basic Visual

Concept

and

Principles

(Charles

Wallschlaeger

S Cynthia

Busic-

Snyder). It

_{guidedmetoexpress}

my

pointof view on

the

graphictheory.

In

order

to

get

ideas

about comparison of

Eastern

and

Western

Culture,

1

brought

some

books

about

Korean

traditionaland

contemporary

art

from Korea. 1

wasabletoscansome

images

and quotes

into

thecomputer.

I

also gave a

list

ofthe

RESEARCH

During

the

research

stage,

I

read

The Visual Mind

by

L.

Loreto,

R. Farinato

and

M. Tonetti. The book

talked

about

Interactive Multi-Window Computer Graphics

in

chapter 27.

It

showed

illustrations

of production and comparison of

symmetry

patterns

in

a multi-window arrangement.

The

chapter usedthe

interactive

computer program

SYMPATI

software as reference.

SYMPATI

is

a

MS-DOS

application which works on

IBM

compatible

PC's

(see Appendix 7.a>

I

was

interested

to

find

theprogram and

learn how

it

works.

I

called a student who studies at

Illinois Institute

of

Technology

in

Chicago

tosee

if

he

could get

any

information

on

crystallography for

me,

especially SYMPATI.

However,

he

sent me a

demo

version of a program

LAZY,

a software program which can createthestructure of an

atom, instead

ofthe

SYMPATI. That

was a

better

pro gram

for

me

because

thesoftware was

for Macintosh

and

I

wasabletouse

it

on

my

Macintosh

computer.

Research

in

Cambridge &

Boston,

Massachusetts

During

the

Fall/Winter

Break,

I

went

down

to

Boston

to

find

more

information.

Since MIT

and

Harvard

University

are

located

in

the

Boston

area,

1

thought

they

must

have

a great

library

system with good resources.

On

top

of

that,

the

Boston

region

is

known

as one ofthe

biggest

publication centers.

When I

gotto

Cambridge

in

Massachusetts,

I

tried toget

into

the

Harvard

University

Library.

However,

the

only way

toget accessto their

library

wasto

have

a certificatethat

containsthe

list

ofthe

books

issued

by

the

RIT

library.

I

experiencedthesame problem at

MIT.

I

had better luck

withthe

Boston Public

Library

andthe

Cambridge Public

Library.

I

found

several good

books.

They

were

William Justema's The Pleasure

of

Pattern

(1968),

Pattern The Historical Panorama

(1976),

_and

M.

Richard Proctor's The

Principle

of

Pattern

(1969).

Nevertheless,

those

books

can not

be

checked out of

Massachusetts,

so

I

had

tospend a

lot

of

time

reading

the

books

and

making

photo copiesthere.

The

last

day

before coming back

to

Rochester,

I

stopped

by

the

MIT

Bookstore. The MIT Press

always publishes good

books,

so

I

expectedto

find

something

useful.

I

was

happy

that

I found

three

books

relatedto pattern.

They

were:

The Visual Mind

(Michele

Emmer,

The MIT

Press,

1993),

A Primer

of

Visual

Literacy

(Dondis,

Donis A.. The MIT

Press,

1973),

and

Strange Attract

on

Signs

of

Chaos

(Alice

Yang,

The New Museum

of

Contemporary

Art,

_1989).

Figure1

Mind_Mapping

1.aidsreadinganddevelopment/expression of creativity.

2.makesadvantageous use of visualmemory for increasedretention.

3. isanatural process.The brain hasa natural tendencyto cluster,orgroup ideas. 4.canbeteamed easily.

isadaptabletobothgroups andindividuals 5. is especiallyvaluablefor note-taking,

speechmakingandplaning, problem-solving, brain storming, planning session, and reviews exercises.

6. isanalternativetoformalnote-taking. Becauseof_flexibilityand adaptability, newmaterialcanbeeasilyadded. 7. isa whole-brain activity. Adapted

from;'

Use Both SidesofYour BrainbyTonyBuzan.

The Visual

Literacy

was another

important

source

for my

thesis

development.

It

helped

metounderstandthe

basic

elements of

design

and

how

topresentmean

ingful

synthesis of visual

information.

Although spending

just 4

short

days for

research, this

trip

to

Boston

was

very helpful

in

termsof

getting

goodmaterials.

After I

came

back

to

Rochester,

1

could propel myselftogo on

to

thenext stage.

Brain

Storming

and

Mind

Mapping

At

the

brain

storm stage of

my

thesis

development,

I

wrote

down anything

thatrelatedto theword pattern.

It

could

be any

simple

words,

dots, lines,

shapes,

and symbolsthatcame

from my

mind orthat

I

had

seen.

I

wasstill

brain storming

even attheend oftheresearch

because I believe

that

brainstorming

is

the

best way

for

idea

expansion.

Besides

brainstorming,

there

is

another

way

toexpand

ideas.

This

is

called mind mapping."">

Through

_the

development

process,

I

alwaysreminded myselfto get

feedback

through thismethod.

[THESIS

PROJECT

PLANNING

|

Through

the

Fall

quarter of

1994,

1

had been

writing

the thesis

planning

report.

The

thesis

planning

is

themeans of

managing

the

design

process,

containing

10

items

which are

Project

Title,

Designer's Name

and

Address,

Documentation

of

Needs,

Problem

Statement,

Mission

Statement, Methodology, Timeline,

Evaluation

Plan,

Bibliography

and

Glossary

of

Terms.

Although I

did

not

exactly follow

theplan

during

theentire

process,

I

was abletouse

it

as a guide

line

of what was expected.

With

theguide

line,

I

had freedom

toaddor change what

I originally

planned.

Basically,

I

stepped ahead

based

onthe

planning

report.

On

thestage of

doing

research on

methodologies,

experimented with other

fields

such as

anthropology, religion, culture,

etc..

It

also gaveme an

intuitive idea

to

have better

perspective,(seeAppendix2)

Based

onthe

research,

which

I

had done

during

the

phase ofthe thesis

planning,

I

was

seeking

more possibilities

in

relationtonewtheoriesabout pattern.

Prof. Roger Remington

emphasized and remindedme aboutthe

importance

of

methodology

several

times, constructing methodology

was one ofthemost

important

processes

for my

thesis.

In

fact,

from

this

methodology,

I

synthesized allmaterials and

information

on a

diagram

calledtheresearchmap. (see Appendix 3)

RESEARCH

had

tosupport

my

ideas

for

the

final design. For

this

I

spent

lot

oftime

setting up

theappropriatemethodology.

Within

the

limited

time, I

rushed myselftocoverother

unfamiliar

fields

ofstudy.

Topics

such as

crystallography,

mathematicaltheories and

theperception

theory

were studied.

I

tried tounderstandtheessential

ideas

of

the

theories

first,

andthen

I

madetheresearch

map

toconnectthese

methodology

topics.

My Study

became

more extensivethan

I

anticipated,

so

I

spenta

lot

of

time

deciding

which

category

to

focus

on

for my

thesis.

Collecting

Definitions

of

Pattern

First

of

all, it

was

necessary

to

have

the

definition

ofthesubject.

There

are

many

interpretations

oftheword

pattern,

so

I

wrote

down

themeanings as

I

went

along

withtheresearch.

The

richness of pattern

is

based

on

order,

and represents a

blend

ofthe

organic and geometric.

It

is

a sort of

design

thatconsistsof a number of similar com

positional

figures. Most

compositionof pattern

is

astructurethatgovernsposition

and arrangement of

figures

and

forms

to

bring

aboutvisualorder and

harmony.

Pattern

results

from

therepetition of ashape

size, color, texture,

value,

direction,

position,

and

orientation,

either

by

itself

or

in

combination.

[

The

termpattern

necessarily

implies

a

design

composed of one ormore

devices,

multiplied and arranged

in orderly

sequence.

A

single

device, however,

complicated or complete

in itself it

may

be,

is

not a

pattern,

but

a unit with which the

designer,

working according

tosome

definite

plan of

action,

may

composea

pattern.

]

Christie,Archibald H. Pattern Design₍₁₉₆₉₎

Once I

gatheredthe

definitions

of

pattern,

I

organized and

interpreted

them

into

my

own words.

Therefore,

it

helped

me

decide

onthemaintopicand

theme

for my

thesis.

Reading

Peter

Steven's

the

Handbook

of

Regular Patterns

(1980),

I began

to

have

more questions.

The book

is

full

of

illustrations

such as classes of

finite

one-dimensional

(line),

two-dimensional

(plane)

pattern classes.

It

is

helpful for

artists and

designers

tounderstandtheunderstructure of repeated pattern.

Although

the

book

has

good

information,

it

does

not

have

theexplanation of what

symmetry

group

theory

or

X-Ray

crystallography is

about.

I really

wantedto

know

whatthenotations such as_mt,pg, orc2mmmeant.

I

askeda student

majoring

in mathematics,

but he didn't

seemto

know. I

was

disappointed. I

asked a professor

in

the

Department

of

Imaging

Science

at

RIT. He

saidthat

he knows

the

theory but

I

did

not get

any helpful

knowledge

aboutthe

Figure2

FihnnarciSequence

Fabonacci sequenceisproduced_bystarting with1andadding theprevioustwonumbers toarriveat thenext,continuing to infinity. The firsttwelveFibonaccinumbers are1,1 2, 3, 5, 8, 13, 21, 34, 55, 89,and144. Thesenumbershaveinterestingproperties and are often evidentinnature and art.

I

looked for

some

books

about

X-Ray

crystallography

at

the RIT

library. It

wasnot

easy

to

learn

anew

field

of

study

depending

on

only books.

Fortunately,

I

found Elements

of

x-ray

Crystallography

written

by

Azaroff,

Leonid V.

(1968)

and

Symmetry

in

Science

and

Art

by

Shubnikov,

A.

V.,

and

V.A,

Kopstic

(1974).

These

two

books introduced

the

basic

theory

of crystallography.

Although

it

tooka

long

time

to

understand

the

mathematical

theory, I had

a great

time

studying

it.

I

willexplain

the

general

idea

of

crystallography

on page 13.

Mathematical Concept

Nature likes

to

repeat

designs. Recall

the

structure of

snowflakes, the

hexagonal

shape of

honeycombs,

and molecularstructure patterns.

With

these

examples

in mind, it

comes asno surprisethatmathematicians

have

attemptedto

classify

these

repeating

patterns.

There

are

many

typesof

patterns,

visual

pattern,

human behavior

pattern,

mathematical pattern... and so on.

Visual

pattern

is

associated with

point,

line,

shapes,

and geometry.

Human behavior

can act asa

pattern,

such as

the

sleep

pattern

of an average adult

is

atnight.

Mathematics

is

a

form

of pattern

derived from

number,

logic,

andspatial

forms. Mathematics

is

not

only

aboutnumbers and

only

useful

for

buying

orselling.

Western

cultures

usually

referstomathematics as

just

arithmetic

(as

thepublic

thinks),

orthe

study

ofnumber and space.

However,

mathematics

is

not

just

asthe

dictionary

defines,

it

has

evolved

into

the

"science

of patterns".

Numerical

systems are also part of pattern.

We

don't

recognizenumbers as

pattern

because

humans

are not

born

withthe

knowledge

ofmathematics

but

acquirethis througheducation.

The

infant

learn

numbers as an abstract

entity

which

cannot

be

seen,

heard, felt,

smelled,

or

tasted, but

remembered and utilized

for

the

rest of

his

or

her

life. Once

they

learn how

to

identify

numbers,

they

are abletouse the

knowledge

of

counting

torecognize mathematic patterns ratherthan

just

seeing

themas shapes.

Mathematical

patterns sometimes reflectvisualpatternsthat the

human

eye

finds particularly

aesthetic.

One

goodexample

is

thegolden-ratio.

The Greeks

found

the

goldenratio asthe

ideal

proportion

for

thesides of a rectanglethatare

most

pleasing

to

the

human

eye.

The

rectangular

face

ofthe

front

of

the

Parthenon

has

sides whose ratio

is

in

this

proportion,

and

it

may be

observed elsewhere

in

Greek

architecture.

The

goldenratio

is

also

found

in

nature-.

The

chambered shell of the

Nautilus

mollusk grownto

form

a

logarithmic

spiral,

amathematical curvethat

spirals

based

on

the

goldenratio.

The

goldenratio crops

up

in

variousparts of mathematics.

One

well-known

example

is in

connection withthe

Fibonacci

sequence."p"*

This

is

_{thesequence of} number at each stage

found

by

adding

together the

two

previous ones

(except

at

step

[image:16.580.77.534.71.747.2]

METHODOLOGY

Figure 3Interlaced

Islamic Pattern

<j>

yfc

SrRk'St

5k

L.

J>*i.

>JXL^fj^JJ^

What

is

Crystallography?

Crystallography

is the

study

of

the

external

form

ofthecrystals andtheir

internal

structures

(i.e.

the

arrangement of atoms

in

them).

Crystals

are solid matter

in

which atoms are

essentially

arranged

according

toan

underlying

crystal

lattice.

The symmetry

possible

in

a crystal

is

traditionally

called crystallographic symmetry.

Although

Crystallography

originated with

minerals, it includes

moregeneral

topics-undergoing development

into

a

broad

areaof pure applied research.

Crystallography

is

an

interdisciplinary

field

that

has

close contacts with

many branches

ofother

sciences such as

chemistry,

mathematics and

biology.

Islamic Pattern

The

vision of

Islamic

pattern

is

the

knowledge

ofthecultural orderof an

ordered universe.

The Islam

lands

werethecrossroads of

East

and

West,

the

nexus

of

the

greattrade

routes,

sothepeople were great navigators ofthe

desert

as

they

were

later

to

be

of

the

oceans oftheworld.

They

held

allnatural phenomena

in

greatrespect and noted

the

structure ofstar and plant

formations.

Their

land

of

desert developed

in

thepeople a cosmic sense of scale and

distance

in

relationto

topography

andthe

heavens

combined witha minute order and geometry.

Repeated

pattern and

interlaced

patterns are often characterized

in

Islamic

art.

Symmetric

tiling

of one

form

or another

have been

used as

decoration

on

buildings

in the

Islamic

world.

We

frequently

obtain or symbolize

meaning

by

theuse

of symmetry.

This

is

found

in the

religious

icons,

theexquisite mosaic and patterns

used

in

Islamic

art.

The Islamic

pattern

is

based

upon an

ingenious

application of

geometry

andthestructure as an

instrument to unify

the

diversity

encompassing

all

in

one reflects

Islamic

cosmology.

The

pattern

is

drawn

next

ending

to

infinity,

exploiting

ambiguity, illusion

and

flux.

The

largest

published collections of

interlace

patterns are

by

author

J. Bourgoin.

Therefore,

we will observemost ofthe

interlaced

patterns

in three

areas.

First,

interlaced

patterns consist either of

identical

strandsor oftwo shapes

of strands.

Second,

thesemeasures are relatedto the

symmetry

properties ofthe

pattern.

Third,

an explanation ofthe

frequent

appearanceof

interlace

patterns

consists of strands of a single shape or

just two

different

shape

in

Islamic

art.ns<"3

BrankoGnmbaumandG. C.Shephard,The Visual Mind₍₁₉₉₃₎

Tiling

Tiling

is

one ofthepracticalapplicationsof

symmetry.

The

repetitionof

some

basic

motifs

following

some rulestocover an_{entire surface makesa}

two-dimensionalspace group.

Tiling

can

be

used as

decorative

wallpapersor

textiles

[image:18.580.67.530.95.717.2]

Figure*

MauritzConelisEsther

Maurits Cornelis Escherwasborn in Leeuwarden in 1898. Hisgraphic artbursts

withcunning^planned visual surprises.

[

A

tiling

is

covering

of

the

plane

by

closedshapes without gaps or

overlaps;

twoother synonymoustermsaretessellationand parquetry.

Mathematically,

the

space

is

treatedas a

heavy

outline oftiles that

fit perfectly

together.

The

fitting

togetherofshapesto

fill

an area

is

a pleasurable

challenge-play

with colored polygon

shapes orthecompletion of a

jigsaw

puzzle are examples

in

whichtheshapes are

given,

and we expectthem to

fit

together

in

many different

ways.

]

doschamchnader,ne VisualMind(1993)

Islamic

and

Moorish

pattern

have

attracted people

to

their

intricate

pattern.

Dutch

graphic artist

Mauritz Conelis Escher

is the

most

famous

artist

displayed

tiling

asurface

by

a single motif or a

few

motifs of

interesting

shapes.

Mauritz Conelis Escher

At

the

first meeting

with

Dr. Richard Zakia

he

suggestedthat

I do

some

research on

Mauritz Conelis Escher (i898-i972).ns"4 1

found

two

books

about

M.

C.

Escher

at

Border's Bookstore

and

I

scanned over

his

bibliography

and chronological

artworks,

but I didn't pay

attentionto

his

works upontheregular

division

ofthe

planethatwere

derived from

mathematics and crystallography.

It

wasn't until after winter

break

when

I

found

the

Vision

of

Symmetry,

the

Symmetries

in

Culture,

andthe

Symmetry. 1

now

had

a

better

cluetounderstandthe

formation

of pattern

in

relationto

symmetry

theory.

I

resumed

researching M. C.

Escher. The

most

fancy

application ofthe

laws

of

crystallography

can

be

seen

in

the

work of

M. C.

Escher,

who arranged

people, animals,

and

insects in

two-dimensional

one-,

two-andmulti-colored symmetrical patterns.

One

ofthepractical applications of

symmetry

is

tiling-repetitionofsome

basic

motiftocover an entire available

surface,

making

atwo-dimensionalspace

group.

Tiling

can

be

used as

decorative

wallpapers or

textiles

orto

investigate

properties

ofgeometricalshape.

Crystallographers

usethe

symmetry group

of a pattern

in

theirclassification

scheme; it is

based

onthe

identification

of an

underlying

geometric

lattice

ofthe pattern relatedto

its

translationsymmetries andtheir

location,

in

relationto this

lattice,

of centers of

rotation,

axis of reflection and axis of glide-reflection.

The

crystallographers'

and

mathematicians'

quest

is

for

a

logical

analysis of a given

structure;

Escher's

quest was

to

discover

thevarious ways

in

whichtocreate original periodic patterns

in

theplane.(seeAppendix 7.c)

Tiling

&

Symmetry Group

The

mathematical model

for

internal

structure of crystals

is

one

in

which

atoms arealigned

in

a periodic

lattice,

so almost all mathematical

investigations

of

METHODOLOGY

always a minimal patch ofthe

tiling

that

fills

thewhole

tiling

by

translating

it

again

and again

in

two

different directions. One

can

first fill

out an

infinite

strip

withthe

patch,

then translate the

strip

to

fill

outtheplane.

Mathematicians

usegeometrictransformations oftheplanethatpreserve

shapeto

describe

theorder.

These

transformationsare

translation,

rotation,

reflection

and glide reflection.

Each

such geometrictransformation

is

called a

symmetry

ofthe

tiling;

thecollection of all symmetries of a

tiling

is

calledthe

symmetry group

of

tiling.

Symmetry

groups arenot

just

collections of geometrictransformations

but

also

have

algebraic structure.

Periodicity

of a

tiling severely limits

thepossibilities

for

symmetries otherthan

translations;

thereare

only

171

different

types of

symmetry

groups of periodictiling.

Symmetry

During

my

research, I

wassurprisedto

find

that

many

subjects such as

crystallography,

fractal,

Islamic

patterns,

music and

many

more all relatetosymmetry.

This

leads

metoquestion

myself;

Why

are

humans

attractedtosymmetry?

Why

did

it

make people

feel

comfortable?

Did it

happen

naturally?

Did God

create

it

thisway?

As

for

the

theology field,

it

coversthestudies on

theory

ofperception,cognitive

theory,

group

theory,

etc..

At

this point,

1

needto talkmore aboutthe

definition

of

symmetry

and

why I

had

to

study symmetry

and

how symmetry

theory

supportstheexplanation

of pattern.

Symmetry

is

a

very

simple

concept,

but

it

also means morethan

just

a

pleasing

visualresponse.

Symmetry

appealstoour visual

sense,

and

thereby

plays a

role

in

oursense of

beauty.

Symmetry

has

enjoyed a special place

in

both philosophy

andtheology.

Symmetry

is

amathematical

property

which generates repeated

patterns,

as well as a

feature

used

in

theperception and categorization of

form.

Symmetry

plays arole

in

a wide range of

decorative designs

and

textiles.

Symmetry

is

based

on

superposition.

Elements

which are

regularly

superpositioned

along

an axis arethus

regularly

repeated.

The resulting

pattern

is

symmetrical.

In both everyday

usage and

mathematics,

symmetry has

thesense of

repetition,

such as a wallpaper

repeating

one

design

over anumber oftimes.

The

perfect

symmetry

is

repetitive and

predictable,

but

sometimes people

consider

imperfect

symmetry

to

be

more

beautiful

thanexactmathematical symmetry.

Man

seemsto

like

visual

symmetry,

balance

or comfort so

that

is

why

man

usually

makes objects with symmetrical shape.

Symmetry

is

amathematicalas well as an

aesthetic

concept,

and

it

allows classification of

different

types

of regular pattern

and

distinguishes between

them.

The symmetry

_{concept provides perspective}

from

our world as an

integrated

whole.

The

collection of all such

symmetry patterns, based

_ontilings

in

_the

have been

classified

by

their symmetries;

These

symmetries are called either

the

wallpaper groups orthe two-dimensionalcrystallographic groups.

Symmetries

restrictthe

kinds

of pattern arrangements possible

forming

a sort of grammar.

All

patterns are produced

by

thesame rules.

Pattern

structures can

change,

but

the

rules cannot.

There

are

limited

possibilitiesto

build

patterns

based

on

group

theory.

I'm

not a mathematician and

I

don't know

thespecific

details beyond

the

basics,

so

I

am

going

to

introduce

the theoriesonthe

following

pages and appendixesto

help

interested

readers understand.

Humans

have

the

ability

to

discern

patterns.

Symmetry

relates

harmony

and proportion and

it

provides perspective

form.

Symmetry

is

divided

into

two

large

classes;

point

group

andspace group.

The symmetry

groups

have

algebraic structure and provide a

way

of

generating

periodictilingsas

I

talkedabout ontheprevious pages.

Author

Dorothy

K. Washburn

and

Donald W. Crowe

pointed outthe

differences

in

cross-cultural preferences

for

alltheplane pattern symmetries

associated with

culturally

meaningful

stimuli,

and

investigated

thesignificance of

different

geometries

in

different

cultural contexts.

Point Groups

are when at

least

one special point

in

theobject or pattern

differs from

alltheothers.

This

special point

has

an

important

distinguishing

feature:

it

remains unchanged no matter whattypeof

symmetry

operation

is

performed.

Such

symmetries

belong

to the

point-group

category.

For

an

example,

asystem of concentric rings resemblethepattern when a stone

is

thrown

into

still water.

The

pattern remainsunchanged

if it is

rotated around

its

center regardless of

its

position.

Those

are often called

finite

patterns andthereare

group

i and

m,

group2 and

2mm, group3

and

3m, group4

and

4mm, groups

and

5m,

group6 and

6mm,

group7

and

7m,

group8 and

8mm,

10mm, 12mm, 16mm, 24mm,

...,and

higher.

<seeAppendix 7.d )

Space

groupsare another

form

of symmetries.

However,

there

is

no point

in

theobject or patternthat

is

different from

alltheothers.

For example,

patterns of

hexagons like

a

beehive

are considered a space group.

In

general,

space groups can

be

one-dimensional,

two-dimensionalorthree

dimensional-according

towhetherthe

repetition extends

in

the

X,

Y,

or

Z

axis.

If

a

design

admitstranslations

in

only

one

direction

the

design

is

called a

band,

strip,

frieze,

orone-dimensionalpattern.

In

the

beehive

example,

the

hexagons

in

thepatternextend

in

two

directions-length

and width.

The corresponding symmetry

is

called atwo-dimensionalspace group.

If

a plane

figure

admitstranslations

in

two

ormore

directions,

it is

atwo-dimensionalpattern.

Two-dimensional

patterns are

commonly

found

on

wallpaper,

tile, textiles,

and othermedia where

broad

areas are covered

by

motif repetition.

A

pattern can

be

generated

simply

by

a

repeating

motif.

This symmetry

METHODOLOGY

distance

andthe

resulting

pattern

is

periodic created

by

infinite

repetition ofthe

same motif.

From any

one-dimensional pattern withperiodicity,

it is

easy

togenerate

a planar pattern

by

repetition,

extending

the

periodicity

in

two

directions.

Symmetry

Operation

Symmetry

operation and

its

combinations provided

7

possibilities

for creating

border decoration

with one color and

17

possibilities

for

twocolorone-dimensional

groups.

There

are another

17

possibilities with one color

for creating

planar

two-dimensional

space

group

and 46 possibilities

for

two

color,

two-dimensionalgroups.

For

three-dimensional periodicity,

thereare altogether 230 possibilities.

One-color finite

one-dimensional₇ two-dimensional₁₇

Reflection

Glide-reflection

Rotation

Two-colorfinite

one-dimensional₁₇ two-dimensional46

No

matter

how

complicated,

every

rigid motion oftheplane

is

one of

four

basic

rigid motions.

They

are

translation,

rotation, reflection,

and glide reflection.

The

term

"operation"

implies

action or movement.

Translation

themotif moves

up

or

down,

left

or right or

diagonally

while

keeping

thesame operation

in

translation,

the

resulting

pattern

is

periodic.

the

figure

across a vertical axis or across a

horizontal

axis

themotifturns

in

rotation

themotif

both

translateand reflects

(see

Appendix7.e and7.f

for

the

full

explanation ofthe

symmetry

operation)

Symmetry

in Culture

Pattern

arrangement reflects

culturally

meaningful patterns and

human

behavior. Since

the

discovery

and explanation of

human behavior

is

theessence of

anthropology, it

would seemthatan analytical

technique

which can

isolate

behavior

consistently

and

objectively

would

be

a great asset

for

the

study

of certain

kinds

of

cultural activities.

Practically

every

culture

in

theworld

is

known

to

decorate

at

least

some portions of

its

material possessions withrepeatedpatterns. Dorothyk.washbumand DonaldW.Crowe(1988)

Incidentally,

I

was

studying

Dorothy

K. Washburn

_and

Donald W. Crowe's

Symmetries

of

Culture

(1988).

They

talkedabout patterns

occurring

on

decorated

objects

from

culture.

They

also showed one-and

two-color,

one-and

two-dimensional,

andthecommon

finite design. I

borrowed

the

idea

pattern of

triangles

from

the

symmetry group

classification

for my

thesis.

While I

was

studying

withthis

book,

I

explored

many structurally

_{similar patterns which are}

from

different

Mgure5

BpnoltBManriplhrnt

In the 1960and1907sanIBM researcher,Benoit Mandelbrot, invented a newgeometry,whichhecalled

"fractal" geometry.Hecoinedthe term"fractal"

tosuggest "fractured"

and "fractional"

-a_geometrythatfocusesonbroken, wrinkled,and uneven shapes.

historical

periods, (see Appendix 7. gand7.h)

Chaos

Theory

and

Fractal

Fractal

is

a geometrical

figure

in

which an

identical

motif

keeps repeating

itself

on

infinite

scale.

Originally

the

geometry

ofnature

is

fractal

to theextentthat

when

looking

at shapes

in nature-clouds,

treesand coastline-small parts arethe same as

big

parts.

Traditional dynamics deals

with regular

patterns,

where aschaos

is

the

science ofthe

irregular.

It

aimstoextract new

kinds

of pattern

from

apparent

randomness,

patternthatwouldnot

be

recognized as such

in

atraditionalapproach.

Chaos

is

a

kind

of randomness of which origins are

entirely deterministic. It

also

involves

shapes which are almost

invariably

fractal.

Chaos

theory

has

resulted

from

asynthesis of

imaginative

mathematics and

readily

accessible computer power.

It

presents a universethat

is

deterministic,

obeying

fundamental

physical

laws,

but

with a predisposition

for

disorder,

complexity

and

unpredictability.

It

is

the

beautiful

graphics associated with chaotic systemsthatmake

it

appealing

topeople who seethem.

They

show

how

asimple

equation,

when

input

into

a

computer,

can produce

breathtaking

patterns of ever

increasing

complexity.

(seeQuickTimeMoviesonCD-ROM )

While playing

around with

HSC Software Kai's Power Tool

2.1,

a plug-in

for

the

Photoshop,

I

appreciatedthe

beautiful fractal

pattern.

To

me,

fractal

is

an

absolutely

revolutionary

approachthatappliesthe

theory

of pattern.

After

discovering

the

fact

that

fractal

is

relatedto

symmetry theory, I

used

fractal

as a part of

my

thesis.

The

fact

that

fractal

is

relatedto

symmetry

theory

wasnot

surprising any

more since

I

was

beginning

to

understandthatall researchtopicsare

interrelated.

Mandelbrot Set

The

fractal geometry

was

invented

by

Benoit B.

Mandelbrot,"*"^

and

has

become known

asthe

Mandelbrot

set.

The

fractal geometry

is

between

theexcessive

geometric order of

Euclid

andthegeometric order of general mathematics.

It

is

based

ona

form

of

symmetry

that

is infinite

self-similarity.

The

patternthat

Mandelbrot

and others

discovered

in

one region ofthecomplex plane was a

long

proboscidean

insect

shape ofstable points-themandelbrotset

itself,

usually

in

black-surrounded

by

a

flaming boundary

of

filigreed detail

that

includes

miniature,

slightly distorted

replicas ofthe

insect shape,

and

layer

upon

layer

of self-similar

forms.

METHODOLOGY

Figure6|Quasicrystal

Acrystallatticecanhave two, three-,four-, five-,and six-fold symmetiy.Thesealloys can notbecrystalsinthe usual sense,hence the new term'quasicrystal'

8

OP.

&

\

**

* &$

calltheprocessof

unfolding

the

detail

"zooming

in"

ontheset's

boundary

or

"magnifying"

it.

<seealsoQuickTime MoviesonCD-ROM)

Icons

of

Chaos

and

Strange Attractor

Points symmetrically

related

have

symmetrically

related

images.

This

condition

is

used

for

themathematical

form

ofthe

equation;

and once a symmetric equation

is

found,

we can use

it

to

do dynamics. This

symmetry-creation

is

called

chaotic

icons.

The

"icons"

for fluid flow

are calledattractor,and

they

are

in

an

imaginary

mathematical space of which points correspondtoall possible

flow-patterns.

People do

not

normally

thinkthatchaos

is

a means

for creating

pattern,

but

the

strange attractor

is

another

kind

of

fractal

patternmade

by

a

dynamical exhibiting

chaos.(seeAppendix 7.i)

Quasicrystal Pattern

Lattices

in

theplane orthree-dimensionalspace can

have

rotational

symmetries of orders

2,3,4,

and

6

but

cannot achieve

five-fold

rotational symmetry.

It

wasn't untilthe

early

1970's,

whena

mathematician, Roger

Penrose,

discovered

a

way

to tile theplanewith

figures

thatexhibit

local fivefold symmetry

as recreation

investigation.

The

significance ofthe

discovery

was addedto

in 1984

when

crystallographers

identified it

as Quasicrystals.F|"6

A

quasicrystal

is

amaterial

withouttheregular

lattice

structure of

ordinary

crystals,

but has

its

atoms arranged

in

a

highly

ordered

fashion

thatexhibits

local

symmetry.

Quasicrystals

are somewhere

between

amorphous

bodies

(like

glass)and perfect crystals

(like diamonds). The

study

of

Quasicrystals

is

still at

its

infancy,

but

it

demonstrated

the

possibility

of a mathematical

framework

thatcan serve as a

basis for understanding

these

newly

discovered

materials.

3D-Stereo

Random Dot Pattern

Not many

people

know

that the3D-Stereo random

dot

pattern

is

based

on the

symmetry

theory.

Bela Julez devised

thestereogram made

up

of

randomly

distributed

dots.

<seeAppendix7.

j

)

The

image

seen

by

theright eye

is identical

to that

seen

by

the

left

in

all

but

a central square

region,

which

has been

cut

out,

shifted a

little

tooneside and stuck

down

again ontothe

background. The

white

gap

left

behind

wasthen

filled

with arandom pattern of

dots. If

the two

images

are viewed

througha

stereoscope,

thesquarethatwas cut out appearsto

float

in

front

of

the

background.

Such

stereograms contain spatial

data

which are processed

automatically

by

theViSUalSystem.BrunoErnst.The Eye BeguOed-Optical illusions₍₁₉₈₆₎

Human Sense

of

Order

The

search

for

pattern

is

part of

human

nature.

Human

mind and culture

patterns.

As

in

human

behavior,

pattern

lies

atthe

heart

of cultural

systems,

allowing

both

participants and observers

to

anticipatethe

future. The

creation andperception

of pattern

has

a

logical

as well as aesthetic component.

Humans

areendowed

from

birth

withthe

capacity

to

judge

experiences

by

such prior categories as

similarly

or contrast.

We

distinguish

the

familiar from

the

unfamiliarandthesimple

from

thecomplex.

Moreover,

there

is

a

relationship between

our reactionstoremembered sights andtosimple shapes.

For

example, if

wesee

something

familiar,

it

becomes

theexpected andthus takenas

normal,

unless an unfamiliar context arouses attention.

Geometrical simplicity

can also

become

monotonous and

fail

toregister except where

it

contrasts with

less

ordered surroundings.

If

theaim of

history,

archaeology,

and

anthropology

to

describe

and

study

theproducts of

human behavior

which

consistently

reoccur andthus

form

non-random

patterns,

and

if

wetreat

these

patterns as manifestations of

ideas

held

in

common

by

makers and users ofthe

artifacts,

thenwe

must,

first

of

all,

give our attentionto

classificatory

aspectsofthosephenomena whichrelateto thosenon-random

ideas

and patterns of

behavior. We

offer

here

one

way

ofmore

rigorously

defining

the units of analysis

for

a whole class of phenomena-repeated

design-in

a

way

which

enables ustoaddress

important

problems

relating

to

group

formation,

maintenance,

and

interaction.

The

problem of

why

people

do

things

similarly

is pervasive, profound,

andnottrivial.

It

deserves

our

best

systematic efforts.

[

Order

and

complexity

are

antagonistic, in

thatordertends toreduce

complexity

while

complexity

tends toreduce order.

To

create order requires not

only

rearrangement

but

in

most cases alsotheelimination of what

does

not

fit

the

principles

determining

theorder.

On

theother

hand,

when one

increase

the

complexity

of an

object,

order will

be harder

toachieve.

Order

and

complexity,

however

cannot

exist withouteach other.

Complexity

without order produces

confusion;

order

without

complexity

produces

boredom. Although

order

is

neededtocope with

both

the

inner

and outer

world,

man cannot reduce

his

experiencetoa network of

neatly

predictableconnections without

losing

the

stimulating

riches and surprises of

life.

Being

complexily

designed,

man must

function complexity

if

he

is

to

be

fully himself;

and

to

thisendthe

setting

in

which

he

operates must

be

complex also.

It

has

long

been

recognizedthatthegreat worksof man combine

high

order with

high

complexity.

]

RudolfArnheim, PsychologyofArt(1972)

Gestalt

Theory

and the

Law

of

Pragnanz

Patterns

can

be

considered

in

termsof perception

theory.

Patterns

are

composed

from

a

basic

list

of visual

elements;

dot,

line,

shape,

direction,

tone,

color,

texture,

scale,

dimension,

orientation andmovement.

The

interaction

and effect of

human

perception on visual

meaning

is

from

theresearch and experimentation

in

Gestalt

psychology.

However,

Gestalt

thinking

has

moretoofferthan

just

the

METHODOLOGY

base

is

the

belief

thatan approachto

understanding

and

analyzing

all systems

requires

recognizing

that

the

system as a whole

is

made

up

of

interacting

parts,

which can

be

isolated

and viewed as

completely

independent

andthenresembled

in

thewhole.

Gestalt

theory

wasone principlethatcontributedtoorder and structure

in

a

pattern;

that

is,

symmetry

contributedto the

"goodness"

ofthepattern.

Gestalt

theory

of

perception,

the

Law

of

Pragnanz

defines

psychological organizationas

"good"

(regular,

symmetrical,

simple)as

prevailing

conditions allow.

It

would

be

defined emotionally least

provoking,

simplest,

least

complicated,all of which

describe

thestate arrived at

visually

through

bilateral

symmetry.

Axial

balance designs

are

not

only

easy

to

understand,

they

are

easy

to

do,

employing

the

least

complicated

formulation

Of COUnterpOiSe. Dondis,DonisA.. A PrimerofVisual_{Literacy (1973)}

Visual

Literacy

Visual

literacy

is

theanalysis of

points,

lines

and shapes

in space,

while art

is

often concerned with aesthetic appreciation or withtheuse ofspacetoevoke an

emotional response.

The

figures

and

forms

involve

theuse of

basic

visual

elements;

point,

line

and plane.

They

are part oftheattributes of

form

thatcreate

tone, texture,

orpattern.

The

mathematician

may

thinkabout and

define

words such as

point,

line,

plane,

and volume

in

abstractterms.

Point

used

repetitively

in

aregular configuration can

be

usedtocreate

various patterns.

Patterns

can

be

created

by

a simple shape or configuration called a

cell of

fundamental

region.

Line

can

be

used

for

representation of

shapes,

to

form

objects,

and structures.

Line

can

bring

meaning, symbolism,

and expressiontovisual

forms

andtheirmessage.

Like

points or

dots,

lines

can

be

usedtoproducetone.

Patterns

can

be

organized

from

a set of

lines

into

a configurationthatcan

be

repeated an

infinite

number oftimes.

Plane is

defined

in

Euclidian geometry

as

formless,

representing

an

infinite

expanse of

flat

surface areathatstretches

two-dimensionally

in

all

directions. Like

points and

lines,

planes can

be

usedtocreate

images

with

tone, texture,

and pattern.

Paul Klee's Form

Generation

theory

Prof. Robert Keough

suggestedthat

I

read

Paul Klee's Pedagogical

Sketchbook (Sibyl

Moholy-Nagy,

1953;when

I

talkedwith

him

about

starting

to

design

theapplication of graphicelementssuch as

dot, line,

plane,

_{and color, (see Appendix 7. k)}

Paul

Klee(i879-i94o),

artist and master

teacher

at

the

Bauhaus,

developed

his

theory

regarding

pictorial

dimension

of

form

generation

starting

with point asthe

comes

into

being-the

first dimension. If

the

line

shifts

to

form

a

plane,

weobtain a

two-dimensional

element.

In

the

movement

from

planes givesrise

to

a

body.

Summary

of

the

kinetic

energies which move

the

point

into

line,

the

line

into

a

plane,

and

the

plane

into

aspatialdimension."PauiKiee,TheThinHngEyefwei)

William

Morris

and

The

Arts

and

Craft

Movement,

19C.

Sometimes

the

Arts

and

Craft

Movement,

19C

is

criticized

for

its

lack

of

practical

tendency.

However,

anyone will appreciate

Sir William Morris

(1834-1881)

if

they

viewed

his

patterns.

Sir Morris

revolutionized

the

art of

pattern-making

and

altered

the

course of

Western

design. His

patterns

display

the

value of

form,

function

and

beauty.

Morris

himself

wrote;

'The

special

limitations

of

the

material should

be

a

pleasure

to you,

not a

hindrance;

a

designer,

therefore,

should always

thoroughly

understand

the

process ofthespecial manufacture

he

is

dealing

with or

the

result

still

be

a more

"tour de force".

'

wnuamMoms,textiles(im3)

Geometry

and

Pattern

The

human

visual-cognitivesystem

looks for

geometric shapes.

Geometry

is

the

mathematical

study

ofshape.

It

enablesmankindtorecognize a

triangle

as a

triangle,

a circle as a

circle,

but

notsize nor

color,

rather

its

shape.

Whenever

wesee

three

straight

lines

joined

attheirends

to

form

a closed

figure,

werecognize

it

as a

triangle.

Geometry

is the

conceptof

proportion,

which

is

related

to the

Pythagorean

axiom

that

everything

is

arranged

according

tonumber.

It

is

a

property

of

numbers,

harmony,

proportion and

has

been

used

in

aesthetics.

On

the

other

hand,

symmetry

is

alsoregarded as a

kind

of geometry.

The

name of geometrical proportion a:b=c:d

is

harmoniously

ordered or

rhythmically

repeated proportions

introduced

in

a

logical

recurrences

in

all

consciously

composed

plans.

Either Euclidean geometry

ornon-Euclidean

geometry

is

based

on

distance

and

angles,

but

the

transformations thatpreserve

distances

and angles are

precisely

the

rigid motions

in

Euclidean

geometry.

Moorish

or

Islamic

decorative

art,

Chinese

lattice,

and

Greek

and

Celtic

canon

is

famous for

the

geometric ornamentation andpatterns.

Those

arestructured

withnumerical and geometrical relationships

progressing

through

the

dimensions.

The

useofthegeometric principle of

symmetry for

the

description

and

understanding

of

forms

represents

the

unionof mathematics and

design. The only

limitation is

they

must consist of

regularly

repeated patterns.

The

geometric

METHODOLOGY

Hgure7

ien4JQescaaesil5aSJSffl_

Figures

Frenchmathematician and philosopher.He introduced the ideaof coordinate geometry. Incoordinategeometry, figurescanbe described_bymeansof algebraic equations.

Siprpinski Triangles

The Sierpinskigasketisthe simplest example with which we can makethepoint.This is obtained_byrepeatedlydeletingthemiddle

quarter of atriangle, removingsmallerand smallerpieces, forever. Itwasinvented

by

Waclaw Sierpinskiand named after.The Sierpinskigasket canbe thoughtofasbeing composedofthreeidenticalgaskets,eachhalf thesizeofthe original.

Golden

Section,

Golden

rectangle,

Fibonacci Numbers

and

Symmetry

In nature,

this

mathematical

relationship

appears

repeatedly

in

growth

pattern.

The

golden section

is

thata certain

length

is

divided

in

sucha

way

that

the

ratio of

the

longer

partto thewhole

is

thesame astheratio oftheshorterpartto

the

longer

part.

Mathematical

relationships

between

the

features

ofthegolden rectangle

and

the

spiral showtheconnection

between

thespiral and

the

golden section.

The

goldensection was said

by

Kepler

to

be

"one

ofthetwo treasuresof

geometry," and

was considered

by

Plato

asthe

key

to thephysics ofthecosmos.

This

mathematical

relationship

appears

repeatedly

in

growthpatterns

in

nature and

has fascinated

mathematiciansand artists

for

centuries.

There

is

also a connection

between

thespiral and

Fibonacci

series.

The

construction ofthegoldenrectangle

has

a connection

among

thegolden

section,

spirals andthe

Fibonacci

numbers.

They

arerelatedtoeach other sincethegolden

ratio

is

obtained.

The

golden ratio

is

a

symmetry because

whenpatterns are generated

by

simple rulesthe

definition

of

symmetry

includes

harmony

and proportion.

The

fundamental property

ofthe

logarithmic

spiral corresponds

precisely

to theprinciple thatgovernsthegrowth ofshells.

The

principle

is

thesimplest

possible;

thesize

increases,

but

the

shape remainsthesame.

The only

mathematical curve which

follows

thispattern of growth

is

the

logarithmic

spiral.<seeAppendix7.1)

The

essence of shell shape

is

captured

by

the

logarithmic

spiral,

characterized

mathematically

by

Decartesfis-7

in

_1638.

The

logarithmic

_{spiral was observed}

in

'many

man-made and organic

forms.

Algorithmic

Beauty

of

Seashells

Shells

are one ofthemost

interesting

creations of

God. The

shell

is

a

metaphor of geometrical

proportion,

symbolic,

_{and a pattern}

formation

_of

biology.

The

fundamental property

ofthe

logarithmic

spiralcorresponds

precisely

to the

principlethatgoverns

the

growth of shells.

The

mathematicalcurve

that

follows

this

pattern of growth

is

once againthe

logarithmic

spiral, (see Appendix7.m and7.n>

The

logarithmic

spiral shape alsorevealsvariouscolorsandpatternsonthe

surfaces effected

by

flowing

water.

Like

a

tree's

annual

ring,

mostshell patterns are

historical

records of what

happens

at

the

growing

edge.

Each kind

of shell

has

millions

of various patterns with

lines, dots,

ripples, regularity,

irregularity,

symmetry

and so on.

Several

shells

display

triangles

astheir

basic

pattern

element,

andthepatterns

aresimilarto the

famous

Sierpinsky

_{triangles"** with}

their

fractal

_geometry.

Multi-cultural Mathematics

The

connection

between geometry

and art

lies

in

theircommon

interest in

space.

In

some cases art can

be

thoughtof as a creative application ofgeometry.

The geometry

involved in

art and

design

can

be

much more accessible and enjoyable

tomost of

us,

andthe

development

of geometrical concepts and skills can

be

applied

to

help

people

enjoy

creative art.

The

multi-cultural connection

lies

in

the

naturally

multi-cultural context of

art.

By

looking

atthe

geometry

oftwo-dimensionalpatternsand

designs,

art ofthis

kind

can

be

found

all overtheworld

in

brickwork,

woodwork,

tiles,

weaving, carpets,

and other works of art.

The Arabic

geometrical

designs have

perhapsthemost

developed

and sophisticatedtradition

in

this

respect,

but many

ofthese

designs

are

found

in

ancient

India, China,

and

Africa

and,

of

course,

have been

integrated into

modern

Western

culture.

The

artistic and mathematical

beauty

ofthese

designs,

whether ornot

they

are part of

culture,

areexamples of multi-culturalmathematics.

The

pattern ofthe

number

bonds

is

reflected

in

thepattern of

lines

which

it generates,

thus

giving

strong

visual supporttoan

important

numerical relationship.

The

act of

drawing

may

itself

be

pleasurable,

and provide a contrastto therepetitive recall ofnumber

bonds.

An

example ofthe

link between

mathematics andtheart ofmulti-cultures

is Arabic

patterns and

designs

thatprovide an

ideal

context

for

work

in

geometry

and art.

The

analysis ofthe

design

by

synthesis

into

a

generating

algorithm or

algorithms can

be

exciting.

Mathematics

can

be

abstracted

from

certain patterns and

designs

in

an attemptto

describe

thepatternsmore or

less formally. As

I

described

in

theprevious

page,

the

group

theory

has been

usedtoprovethat

only

seven or

seventeen ofsuch patterns can exist.

The formal deduction

oftheresults on

symmetry

groups andtheirsubgroups would completethe

development

ofthe thirdstage of

geometric

thinking, recognizing

mathematics

in

the

design,

or

analyzing

and

describing

the

design

mathematically.

Semiotics

Semiotics,

a

term

derive

from

the

Greek,

is

the

theory

of

how different

signs are constituted and classified

according

to theiruseand

interpretation.

The

sharing

generates auniformresponse

to the

symbol and

tendency

to

define

its

integrity

in interpretation.

Semiotics

is

the

study

andapplication of signs.

Signs

are

anything

and

everything

that

convey

meaning.

For

example,

words are

signs,

pictures

are

signs,

colors aresigns.

Graphic designers

usesignstorepresent

information

and

ideas.

In my

thesis

I

attemptedto

convey

this

information;

i.

Patterns

can

be

created

using

mathematics and crystallography.

METHODOLOGY

3.

Patterns

are aesthetic.

4.

Patterns

are used

in

many

ways...

arts and

crafts,

graphic

design,

textiles,

etc.

5.

Patterns

are

culturally defined.

Motif

A

motif

itself is

not a

pattern,

but

the

basic

unitof apatternwith an

arbitrary

or abstract

shape,

or

derived

object.

Also it

canalso

be

aunitcell or

fundamental

region,

mostly consisting

of symmetry.

Although

different

cultures and

religions sharethesame

motifs,

thename and

meaning

attachedtoeach one varies.

I

showedtwomotifs as a main subjecttoexplain

how

motif reflects

different

cultures.

These

were aspiral and a cloud

having

swirling shape,

whichare

familiar

to

peopleand

have been

studied

in symmetry

theory.

The

arts of

living

in

the

West

have been

shaped

by

knowledge. The diversities

of

Eastern

wisdomand

Western

knowledge

seemtosharethesame

basic

pattern-forming

processesthatcreatethe

harmonies

ofnature and art.

In general,

decorative

motifs are signsthat

have

symbolic

meaning,

most of

which were

originally

conceived as symbols.

Computer Generated Pattern

The

consequences of computertechniquescannot yet

be

estimated,

but

we

learn

agreat

deal from

almost

any

sequence of order and

styling

of

its

effect.

The

ability

ofthecomputernot

only

toorganize

but

alsoto

introduce

an

exactly

calculated

dose

ofrandomnessresults

in giving

latitude

tochance.

Within

a

fixed framework

of

predeterminedmoves not

only

exhibit

features

of aesthetic

interest, they

also offer

unrivaled

insights into

theoperation of our sense of order

in

theperception of

Complex patterns.(seeAppendix 7.o andQuickTimeMoviesonCD-ROM)

Motion Pattern

Like

any

other

pattern,

motion pattern

is

worthy

of consideration.

It

is

used

for

virtualgames orscreen

design

and animation as well.

I

found

some

CD-ROMs,

such as

Head

Candy

and

GrooveThing

thatprovided good examples of motion patterns at several computerstores:

Lechmere,

Compu

U.S.A,

and

Computer City.

With

parts of animation

from

the

CD-ROMs

mixed with some other animation

from

online

service,

I

created an animation as an example of motion patter