총 36페이지

Digest&Focus 09
traditional West African balafon breaks
the octave into 12
Lithograph.
from the Pythagorean scale, instead
logarithmically equal
usingthepulseofthehumanheartforits
parts and uses the 12th
1948,
tonal structure. Examples such as these
root of two as the ratio
Escher,
demonstrate the intimate relationship
of frequencies between
between art and science, between the
two adjacent semitones.
C.
M.
mathematical world and the world of
The 12 intervals are the
by
music. Even the simplest pieces contain
sameasthegeometrical
Hands
thisseedofmath.
sequence that has the
Drawing
12th root of two as its
commonratio.
Focus
In her lecture, Professor Kim
Numerous works such
emphasized the ties between art and
as the Parthenon and
science, stressing that both are born
the Venus de Milo also
out of human creativity. “We find
show the relationship
harmony in mathematical structures,
between math and
just as we can find the underlying
art, making use of the
1950.
scientificfoundationsofbeautifulworks.
goldenratio,theratioof
Pollock,
Pythagoras actually did just that a long
thesumofthequantities
timeago.”
to the larger one equals
Jackson
Numerical progression is an example
the ratio of the larger
by
of the mathematics hidden in elements
one to the smaller.
1
Number
of music. When Pythagoras studied
The irrational number,
the sounds made by the blacksmiths,
approximately 1.618,
he found that the lengths of the anvils,
is usually called the
separate by regular intervals, were
golden ratio. It can also
in proportion to the sounds that each
beproducedbydividing
blacksmith made. If the length of the
the bigger number by
first anvil is 1, and the length of the
the smaller one from
second is half the first, while another
twoadjacentnumbersin
is two-thirds the first, each one would
a Fibonacci sequence,
Underlyingbothillustrationsaremathematicalstructures.
produceasoundthefrequencyofwhich
a list of numbers where
is inversely proportional to its length.
eachsuccessivenumberisthesumofthe
featurearepetitionofsimilarparts.“The
That is, the first anvil would produce a
previoustwo.
Writer,” by the American poet Richard
frequency of 1, the second a frequency
Another example is a fractal, a rough
Wilbur, uses expressions that iterate
of 2/1 and the third a frequency of
or fragmented geometric shape that can
similar meanings and connect varied
3/2. As is clear, the sounds represent a
be split into parts, each of which is a
partstothewhole.Fractalsaretheliving
numerical sequence more commonly
smallercopyofthewhole.Thetermwas
example of how nature brings together
knownastheharmonicprogression.
coined by Benoit Mandelbrot in 1975
scientific, mathematic, and artistic ways
Another example of the role of
and derives from the Latin “fractus,”
ofthinking.
numbers in art is equal temperament, a
meaning “broken” or “fractured.” The
“Students of any field should keep in
system of tuning in which every pair of
symmetry in a mathematical fractal led
mind that all areas of study connect
adjacentnoteshasanidenticalfrequency
todiscoveriesoffractalshapesinnature,
with each other,” Professor Kim said
ratio. In equal temperament tunings,
such as snowflakes and lightning. In
in response to a question form the
an interval -- usually the octave -- is
art, musicians, painters and writers all
audience, “as they begin to discover the
divided into a series of equal frequency
imbuetheirworkwithfractals.Paintings
phenomena connecting human beings,
ratios. For modern Western music, the
such as “Drawing Hands” by M. C.
nature,andtheuniverse.”
most common tuning system is the 12-
Escher and “Number 1” by Jackson
tone equal temperament, which divides
Pollock,afamousabstractexpressionist,
firestorm@hufs.ac.kr
SEPTEMBER 2009

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