Table Of ContentKristóf Fenyvesi
Tuuli Lähdesmäki
Editors
Aesthetics of
Interdisciplinarity:
Art and
Mathematics
Aesthetics of Interdisciplinarity:
Art and Mathematics
Kristo´f Fenyvesi (cid:129) Tuuli La¨hdesma¨ki
Editors
Aesthetics of
Interdisciplinarity:
Art and Mathematics
Editors
Kristo´fFenyvesi TuuliLa¨hdesma¨ki
DepartmentofMusic, DepartmentofMusic,
ArtandCultureStudies ArtandCultureStudies
UniversityofJyva¨skyla¨ UniversityofJyva¨skyla¨
Jyva¨skyla¨,Finland Jyva¨skyla¨,Finland
ISBN978-3-319-57257-4 ISBN978-3-319-57259-8 (eBook)
DOI10.1007/978-3-319-57259-8
LibraryofCongressControlNumber:2017957991
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In memory of Reza Sarhangi (1952–2016)
Foreword
Itishardtobelievethat25yearshavegonebysinceIwrotethebook,Connections:
TheGeometricBridgebetweenArtandScience(Kappraff1990,2ndedition2000).
Atthattime,onlyafewindividualswerestudyinganewdisciplinethathasbecome
knownasDesignScience.WhatisDesignScience?Itisasubjectthathasadvanced
from the twin perspectives of the designer and the scientist sometimes in concert
with each other and sometimes on their own and may be considered a bridge
between art and science. Design Science owes its beginnings to the architect,
designer,andinventorBuckminsterFuller(1975).
The chemical physicist Arthur Loeb is one of the individuals most responsible
forrecognizingDesignScienceasanindependentdiscipline.Heconsidersittobe
thegrammarofspaceanddescribesitasfollows:
Justasthegrammarofmusicconsistsofharmony,counterpoint,andform(sonata,rondo,
etc.) which describes the structure of a composition and poetry has its rondeau, ballad,
virelaiandsonnet,sospatialstructureswhethercrystalline,architectural,orchoreographic,
havetheirgrammar,whichconsistsofsuchparametersassymmetry,proportion,connec-
tivity,valency,stability.Spaceisnotapassivevacuum;ithaspropertieswhichconstrainas
wellasenhancethestructureswhichinhabitit.(Loeb1993,1)
Atfirst,Prof.LoebwasalmostaloneinteachingtheprinciplesofDesignScience
to generations of students at Harvard over a 25-year span. At the same time and
duration,MaryBladealsodevelopedsimilarideasattheCooperUnionandHaresh
LalvanibeganaDesignScienceprogramatPratt.
Prior to the ideas of Design Science coming together, a person who wished to
participateinthisacademicadventurewouldhavetoconsultalibraryofbookson
variousaspectsofthisdiscipline.Therewerefewattemptstounifythisknowledge
andfewoutletstocommunicatetheresultsofanyendeavortoanoutsideaudience.
There was even the question as to who was the audience. Due to the spirit of
community of the early pioneers, this problem was soon remedied. Two confer-
encesaroseintheearly1980sinitiatedbyMarjorieSenechalonShapingSpaceand
Symmetry (1988). The vastness of the scope of Design Science was illustrated in
Istvan Hargittai’s two-volume set Symmetry: Unifying Human Understanding
vii
viii Foreword
(1986) which demonstrated that art and science were indeed close companions.
Other early pioneers were Nat Friedman with his conferences in Albany and his
ISAMA conferences. This was followed in 1998 by Reza Sarhangi’s Bridges
Conferences (just having completed its 18th conference), Kim William’s Nexus
conferences, the Symmetry Festival Conferences of Gy€orgy Darvas, the series of
Symmetry conferences led by De´nes Nagy, and the Ars GEometrica summer
workshops led by Da´niel Erde´ly and Kristo´f Fenyvesi in Pe´cs, Hungary. Fast-
forwarding to the present, we see Finland beginning to place Design Science at
thecenterofitseducationalsystem.
The air was electric with ideas which led to new journals such as Structural
TopologycreatedbyJa´nosBaracsandHenryCrapo,theProceedingsoftheBridges
Conferences edited by Sarhangi,the electronic journal Visual Mathematics edited
by Slavik Jablan and Ljiljana Radovic (Vismath), the Nexus Network Journal of
Kim Williams (Nexusjournal.com), the Journal of Mathematics and the Arts
inspired by the Bridges Conference, the journal Symmetry and Culture, inspired
by the Symmetry Festival directed by Gy€orgy Darvas, and the International
JournalofSpaceStructureseditedbyTiborTarnai.
The beauty of these publications and events was that all contributions were
welcomeanditturnedoutthatartists,sculptors,andmusicians,oftenwithlittleor
noformalbackgroundinmathematics,mingledattheconferenceswithmathema-
ticians and scientists and contributed significant geometric ideas. This led the
renownedgeometerBrankoGrunbaumtolament:
Itisaratherunfortunatefactthatmuchofthecreativeintroductionofnewgeometricideas
isdonebynon-mathematicians,whoencountergeometricproblemsinthecourseoftheir
professionalactivities.Notfindingthesolutioninthemathematicalliterature,andoftennot
finding even a sympathetic ear among mathematicians, they proceed to develop their
solutionsasbestastheycanandpublishtheirresultsinthejournalsoftheirdisciplines.
(Grunbaum1983,166)
Grunbaum felt that mathematics had abandoned the concrete problems of
Euclidean geometry for more abstract and distant areas of mathematics remote
fromtheinterestsofnonmathematicians.
Also up to this time, there occurred a kind of Tower of Babel of academic
disciplinesinwhichmathematicians,chemists,crystallographers,artists,architects
engineers,andcraftspeopleeachhadtheirownlanguage.Infact,oftenpeoplefrom
differentdisciplinesweresayingthesamethingbutintheirownlanguages.These
interactions enabled emerging ideas and energies the opportunity to merge with
eachotherandcelebratetheircommonality.Mathematicsarosefromthesediverse
domainsasthegrandunifierduetoitsabstractionwithmathematicsrediscovering
itsrootsinthewellspringofgeometry.
The Design Science movement also acknowledged the contribution of ancient
culturesandethnicitiesthroughtheanalysisofancientartandarchitectureandthe
studyofethnomathematics.AsSlavikJablandiscussedinhisarticle,“DoYouLike
Paleolithic op-art?” (Kappraff et al. forthcoming), he traced these designs as far
backas23000BC.Overtheyears,aconstantthemewasIslamicpatternscelebrating
thegoldenageofIslam.PaulusGerdesdiscoveredthatsanddrawingsintheSona
Foreword ix
traditionoftheTchokwepeopleofAngolaandZaireresultedinmirrorcurvesand
Lunda patterns (Gerdes 1999a, b). At the same time as these folk arts resulted in
beautifulpatterns,theyalsohadaninnerstructurebasedonfamiliarmathematical
principles such asmodular arithmetic, abstractalgebra, fractals, and the theoryof
knots.
ItwasalsonotedthatwhilethethemesofDesignSciencegaverisetoscholarly
research,theideascouldalsobeappreciatednotonlybyartistsandcraftspeoplebut
alsobyyoungchildrensothatworkshopsforchildrenandtheuninitiatedbecamea
regular part of the conferences. And this has played a role for young children to
developaninterestinSTEAMprograms.Theseideasalsofilteredintoplacessuch
as the Museum of Mathematics (MoMath, New York). I have been teaching a
coursetoStudentsfromtheCollegeofArchitectureandnowthecollegeofDesign
inMathematicsofDesignatNewJerseyInstituteofTechnologysince1978.Some
of the students’ work can be found in Connections, and a textbook of teaching
materialswillsoonbeavailable(Kappraffetal.forthcoming).Anotherfineeduca-
tional resource came about due to the collaboration of Annalisa Crannel, a math-
ematician,andMarcFrantz,anartistwholedaseriesofViewpointsworkshops,an
NSFprogram,leadingtothebookViewpointsonaStudyoftheIterativeFunction
System of Fractals and Perspective Geometry in an Artistic Context (Frantz and
Crannel2011).MichaelFrameandBenoitMandelbrotalsocontributedabookon
fractalsintheclassroom(FrameandMandelbrot2002).
Always in the background of Design Science were what I would call the
“elders,”luminariessuchasBuckminsterFuller,ArthurLoeb,M.C.Escher,HSM
Coxeter, Branko Grunbaum, and Magnus Wenninger whose work has had such a
greatimpactonthefield.ThisbringsmetothecontentofAestheticsofInterdisci-
plinarity: Art and Mathematics, by Kristo´f Fenyvesi and Tuuli La¨hdesma¨ki. At a
manifestlevel,thisbookpresentsasetofcoherentessaysonthethemeofartand
design.Butatadeeperlevel,thebookgrappleswiththequestionofwhatmakesa
design have an instantaneously felt sense of “rightness.” A design, by nature, is
abstract,yetweknowagreatdesignwhenweseeit,andthisisnotarandomvalue
judgment.Thereareindeedobjectivecriteria.
Thinkofwhatgoesintoadesign.Thedesignmustfirstofallexpresstheskillof
itsmaker.Weoftenrefertothisasadesignbeing“elegant,”anditisinterestingthat
mathematiciansusethesametermtoexpressitsaesthetics.AlthoughIspeakofart,
architecture, and design, the same criteria can be shown to hold for mathematics,
particularly geometry, although patterns of number have a charm of their own.
Although skill is essential, the skill of an artist or designer or, for that matter, a
mathematician is not enough. The work must have discernable content. In other
words,itmustbetheproductofasystem.Forexample,thegreatarchitectureofLe
Corbusierwasbased onhis systemoftheModulorderived fromthegoldenmean
(Kappraff2000,Kappraffetal.forthcoming).ThegloriousdesignsoftheAlhambra
were based on the 17 wallpaper symmetries of the plane and the geometry of the
goldenageofIslam.MyworkontheproportionsoftheParthenon,alongwiththe
workofAnneBulckensandErnestMcClain(KappraffandMcClain2005),showed
that the proportions were based on the tones of the pentatonic scale. Although
x Foreword
mathematics can survive as a simple manipulation of symbols, it will be vacuous
unless it is based on a system, the richer the better. I can say that all great works
havesomesystematitsroot.
Mathematicshasplacedgreatcurrencyintheconceptofduality.Oftenwhena
design or a mathematical theory is derived, a second result, dual to the first, is
derivedatthesametime.Thisconceptpervadesthetheoryofgraphs,mathematical
logic,andprojectivegeometry,andM.C.Escherhasfounditinhisart.
Itisgenerallyrecognizedthatagreatdesignhasakindofsimplicityorlackof
clutter,aneconomyofform.JablanhasshownthateventhedesignsoftheMezin
culture in Ukraine from 23000 BC, and other early civilizations from Eastern
Europe, were based on a single tile with stripes on it derived from basketry and
weaving or clothing design. A great deal of diversity can be derived from even a
singlemodule.
Inanotherwisechaoticworld,theelementofsymmetrypresentsthemindwith
repeatingthemes.Ingeneral,themindrecoilsfromendlessnoveltyandprefersto
see something that it has seen before. In mathematics, it can be said that without
symmetry there are no laws or theorems or, for that matter, mathematics, only
endless,featurelessserendipity.Infact,thetheoriesofscienceandthetheoremsof
mathematicscanbethoughttobebasedonsymmetryprinciplesandcanbelikened
to narrow channels lying in the otherwise featureless terrain of art, science, and
mathematics. Some would say that a great design should contain the element of
surpriseoftenachievedbyanadmixtureofbothsymmetryandsymmetrybreaking
justassmoothmathematicslargelyprevailsinamathematicaldomainonlytocome
upon singularities, which break the symmetry and signify some unusual event or
propertyofthesystem.Thisalsobreaksthemonotonyofthesamenessintroduced
bysymmetry.Insteadofsurprise,onecantalkintermsofstabilityandinstabilityor
balance and imbalance. For example, the golden mean has found its way into the
organizationofgreatartbecausetheratio1:ϕ,whereϕ¼1.618...isthesymbolof
thegoldenmean,isawayofdefininga“middle”thatisnotexactlyinthemiddle
andsointroducestheelementoftensionintotheworkofart.Thegoldenmeanalso
introducestheelementofself-similaritywhereveritisfound.
Itisafurtherchallengeandsourceofdelighttofindculturalconnectionswithina
designsuchasremnantsofIslamictilingsorthegeometryderivedfromthe“flower
of life,” a component of the subclass of design referred to as sacred geometry,
bestowingspiritualcontenttoadesign.ThefolkartofGerdesmentionedaboveis
anotherexampleofdesigninaculturalcontext.Ethnomathematicshasgrownupto
followtheculturalaspectsofthissubject.Perhapsthemostadvancedapplicationof
theuseofculturalconnectionsistheworkofCroweandWashburnintheirbook,
Symmetries of Culture (1988), which correlated folk design with the symmetry
group employed to create the design with each design providing clues as to its
fabricator.
The artist or designer should also give as much consideration to the space left
over inadesign astothebuiltspace itself,i.e.,both figure andground.Finally,a
designcomes alivewhenthemagicalelementofcolorisintroduced.Ialways tell
mystudentstofindsomeonewhoknowsaboutcolortheoryandstudywiththem.
Foreword xi
In summary, the impact and interpretation of a work of art or design can be
related to the following concepts which also have importance for mathematics:
elegance,content,duality,simplicity,symmetry,symmetrybreaking,managingof
stabilityandinstability,culturalcontext,figureandground,andcolor.
In large part, this book will explore this terrain through a set of well-chosen
essays.
Newark,NJ JayKappraff
10March2017
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