Mathematics | Perspective
Time | Space
DESMA 9:
Art, Science and Technology
"“Clouds are not spheres, mountains are not cones,
coastlines are not circles, and bark is not smooth, nor
does lightning travel in a straight line.”
- Benoit Mandelbrot
TODAY 1/13/09:
•
•
•
•
•
ZERO
Perspective
Escher
Buckminster Fuller
Mandelbrot
ZERO
"zero" derives ultimately from the Arabic sifr which also gives us the word "cipher"
Around 700 BC, uses three hooks to denote an empty place in the positional notation.
On a tablet found at Kish, an ancient Mesopotamian city located east of Babylon in
what is today south-central Iraq. Not used as a number but a punctuation mark
Greek astronomers began to use the symbol O.
Some historians favor the explanation that it is omicron, the first letter of the
Greek word for nothing -- "ouden".
Zero appeared in India and spread to China and Islamic countries.
It was not until the 1600’s the zero was used in the West, after encountering a lot of
resistance!
ZERO
during the Middle Ages, zero was disparaged as a mark of infidel sorcery,
the sign of the Devil himself, the canceller of all meaning.
For the Mayans, Zero was the Death God among their lords of the underworld,
and men adopting the persona of Zero were ritualistically sacrificed in hopes of
staving off the day of zero, the time when time itself would stop.
Only much later was zero reinterpreted as a symbol of God's power to create
a lot out of naught.
Nothingness or Emptiness
“zero = infinity”, as alluded to in the Upanishads
and reflected in the Sanskrit word “poornam” which means both “zero” and “full”
Y2K bug:
zero causes problems
even recently!
Recently many people throughout the world celebrated the new millennium
on 1 January 2000. Of course they celebrated the passing of only 1999 years
since when the calendar was set up no year zero was specified.
Although one might forgive the original error, it is a little surprising that most
people seemed unable to understand why the third millennium and
the 21st century begin on 1 January 2001.
PERSPECTIVE
al-Haytham
Al Haytham, around 1000 A.D. gave the first correct explanation of vision,
showing that light is reflected from an object into the eye. He studied the complete
science of vision, called perspectiva in medieval times, and although he did not
apply his ideas to painting, the Renaissance artists later made important use of his
study.
Duccio:
first attempt
Annunciation
1316
13th century:
GIOTTO
The Visitation
Not a precise formulation –
Intuitive approach
Albrecht Dürer (German, 1471-1528), Melancholia 1, 1514
BRUNELLESCHI:
Credited with the first correct formulation of
linear perspective, about 1413
His perspective paintings were lost but a “Trinity” fresco by Massacio from
this same period still exists. He used Brunelleschi’s perspective principles.
VANISHING POINT
Developed by Brunellesci,
Massacio preserved
Alberti described
Alberti’s treatise “On Painting”, 1435
Written in Latin for scholars and Italian for the general audience
Perspective is necessary in order to understand painting. It is completely
mathematical, concerning the roots in nature from which arise
this graceful and noble art….
A painting is the intersection of a visual pyramid at a given distance,
with a fixed centre and a defined position of light, represented by art with lines
and colours on a given surface.
If you place the intersection one metre from the eye, the first object,
being four metres from the eye, will diminish by three-quarters of its height on the
intersection; and if it is eight metres from the eye it will diminish by
seven-eighths and if it is sixteen metres away it will diminish by fifteen-sixteenths,
and so on. As the distance doubles so the diminution will double.
Piero de la Francesca – leading artist of his time, also leading
Mathematician in 15tn century Rennaisance
Piero’s illustration of a dodecahedron
Leonardo da Vinci
Mathematics
& Art
Fused into a
Single concept
GOLDEN RATIO
"Vitruvian Man" by Leonardo da Vinci
Gallerie dell'Accademia, Venice, Italy
ARCHITECTURE
The Parthenon, Athens,
superimposed with a diagram demonstrating
the use of the Golden Mean in the design
Architect Charles E. Jeanneret, known as "Le Corbusier" (French, 1887-1965),
often used golden rectangles in his designs for buildings. One of these is the
United Nations building in New York. The proportions of its vertical and horizontal
dimensions correspond to the Golden Mean.
Golden ratio and Hollywood
Phi: Nature of Growth
DIMENSIONALITY:
MOVING AWAY FROM
FLATLAND
Maurits Escher
In a lecture in 1953, Escher said:
I have often felt closer to people who work scientifically
(though I certainly do not do so myself) than to my fellow artists.
Circle Limit III
In 1958 he met mathematician Coxeter and they became life-long friends. In 1995,
Coxeter published a paper which proved that Escher reached mathematical
perfection in one of his etchings.
[Escher] got it absolutely right to the millimetre, absolutely to the millimetre ....
Unfortunately he didn't live long enough to see my mathematical vindication.
In mathematical quarters, the regular division of the plane has
been considered theoretically. ...
[Mathematicians] have opened the gate leading to an extensive domain,
but they have not entered this domain themselves.
By their very nature they are more interested in the way in which
the gate is opened than in the garden lying behind it.
Buckminster Fuller
Octet Truss
Tensegrity
Geodesic dome
How they work
Synergetics
Artists influenced by Buckminster Fuller:
John Cage
Joseph Albers
Kenneth Snelson
Others impressed by him
Bucky class we Elaine de Kooning and
Joseph Albers, Black Mountain college, 1948
Piet Mondrian
1942
1926
Mandelbrot:
The Psychedelic world of
FRACTALS
Fibonacci, Fractals and Financial Markets - Socionomics.net
Contemporary Artists using Mathematics
http://www.sciencenews.org/articles/20061118/mathtrek.asp
http://www.sciencenews.org/articles/20060506/mathtrek.asp
Sol Lewitt
John Sims
Casey Reas
Mark Pomilio
MUSIC & MATHEMATICS
MATH ROCK – Gil Kuno presents on Wednesday
Yoshida Tatsuya, one of the originators of the RUINS and the
cited as the "indisputable master drummer of the Japanese underground"
On campus:
IPAM
In the city: