ARTST 102 Algorithmic Visualization
Cynthia Nelson
Mathematics in Ancient Cultures
Numbers have never been my thing. I've always said that by going through college as an artist, I've stayed as far as humanely possible from anything math related. I can barely even balance my checkbook. It is with extreme irony that now, after I have "graduated" from UCSB I find myself working at a bank. Numbers, numbers, numbers. All day long. Perhaps this daily exposure to them has worn at the edges of my fear, making me less wary of their confusing and cryptic language.
But, as with all languages, it had to start somewhere to be able to grow
into what we know of it today. And always the "Cultural Anthropology" girl,
this is what I found the most fascinating.
I've discovered a variety of
number systems, some that go as far back as
3000 BCE. The majority of them have sprung up in the regions surrounding
the Mediterranean
and stretching to the Orient. From Egypt and Babylon to Ancient China,
numbers (and the systems to write them) have been and integral part of
the history
of the culture that surrounds them. The Mayan counting system is not quite
as ancient, but perhaps is more mysterious. The brilliance that sprang
from a rich and diverse culture suddenly ceased to exist and we have yet
to know
why.
There were seven main ancient numeric systems that have led modern day
mathematics to the point at which they stand today. Babylonian, Egyptian,
Chinese, Mayan,
Greek, Arabic, and Indian. I am going to explore three: Babylonian, Chinese,
and Mayan. These civilizations have
histories that cross threads little,
if at all. This will allow me to explore the rise of possibly significantly
original
ideas and concepts within the world of mathematics.
Babylon
The region had been the centre of the Sumerian civilization which flourished before 3500 BC. This was an advanced civilization; building cities and supporting the people with irrigation systems, a legal system, administration, and even a postal service. The Sumerians had developed an abstract form of writing based on cuneiform (wedge-shaped) symbols. Their symbols were written on wet clay tablets which were baked in the hot sun. There are thousands of these that have survived to this day. It was the use of a stylus on a clay medium that led to the use of cuneiform symbols since curved lines could not be drawn. The later Babylonians adopted the same style of cuneiform writing on clay tablets.
Cuneiform numbers could be written using a combination of two symbols:
a vertical wedge for '1' and a corner wedge for '10'. The Babylonians
had a
sexagesimal
system and used the concept of place value to write numbers larger
than 60. So they had 59 symbols for the numbers 1-59, and then the
symbols
were repeated
in different columns for larger numbers. For example, a '2' in the
second column from the right meant (2 x 60)=120, and a '2' in the
column third
from the right
meant (2 x 602)=7200.
Other advancements of the Ancient Babylonians that have survived
until today:
*The Babylonians divided the day into 24 hours, each hour into 60
minutes, each minute into 60 seconds. This form of counting has survived
for
4000 years.
*The base 60 number system of the Babylonians was successful enough
to have worked its way through time to appear in our present day
modern world. We
still have 60 minutes in an hour, 60 seconds in a minute, 360 degrees
in a circle
and 60 minutes in a degree.
*Perhaps the most amazing aspect of the Babylonian's calculating
skills was their construction of tables to aid calculation. They
created tables
of reciprocals
converted to sexagesimal notation.
*These tables help to aid them in finding square roots. From these
came the
earliest form of the Quadratic Equation:
x = sqrt[(b/2)2 + c] - (b/2) and x = sqrt[(b/2)2 + c] + (b/2).
Chinese
"
Chinese mathematics," was defined by Chinese in ancient times as the "art
of calculation" (suan chu). This art was both a practical
and spiritual one, and covered a wide range of subjects from
religion and astronomy to water
control and administration.
The first true evidence of mathematical activity in China can
be found in numeration symbols on tortoise shells and flat
cattle bones (commonly
called
oracle bones,
dated from the Shang dynasty (14th century B.C.).
These numerical inscriptions contain both tally and code
symbols
which are based on a decimal system, and they employed a
positional value
system. This
proves
that the Chinese were one of the first civilization to understand
and efficiently use a decimal numeration system.
In 1899 a major discovery was made at the archaeological
site at the village of Xiao dun in the An-yang district
of Henan
province. Thousands
of bones
and tortoise shells were discovered there which had been
inscribed with ancient Chinese characters. The site had
been the capital
of the kings
of the Late
Shang dynasty
from the 14th century BC. The last twelve of the
Shang
kings ruled
here until about 1045 BC and the bones and tortoise shells
discovered there
had been
used
as part of religious ceremonies. Questions were inscribed
on one side of a tortoise shell, the other side of the
shell was
then
subjected to the
heat of a fire,
and the cracks which appeared were interpreted as the answers
to the
questions coming from ancient ancestors.
Around 300 - 0 BCE the main chinese mathematical advancements
were calculating square and cube roots, measurement of
a circle, and
calculating the volume
of a pyramid.
Systems of linear equations also emerged.
In about the fourteenth century AD the abacus came into use in China. Arithmetical
rules for the abacus were analogous to those of the
counting board (even square roots and cube roots of numbers could be calculated)
but it
appears that the
abacus was used almost exclusively by merchants who
only
used the operations of addition and subtraction.
Mayan
The first findings or writings about the Mayan number system date back to the fourth century A.D. Evidence shows that the Mayan culture of Yucatan and Central America were extremely advanced not only in mathematics, but were believed to be geniuses when it came to time and calendars, astronomy, architecture, and commerce. It is believed that the Mayan culture was obsessed by time and numbers which studies have concluded based on drawings found on historical monuments and stela.
This has to be the ancient numeric system that I am most familiar with, having
taken a number of anthropology classes on this culture. The Mayans came up
with a vigesimal system that is based on the number 20. (ten fingers, ten
toes). The Mayans used a system of dots and bars for counting. A dot (pebble)
stood
for one and a bar (stick or rod) stood for five. Depending on what level
in the column the dots and lines were in would determine how many times
it would
need to be multiplied by twenty to give the right number. The Mayans wrote
their numbers vertically instead of horizontally with the lowest denominations
at the bottom, increasing as we move to the
top.
Mathematics factored greatly in everyday mayan life. The Maya kept time with
a combination of several cycles that meshed together to mark the movement
of the sun, moon and Venus. Their ritual calendar, known as the Tzolkin,
was composed
of 260 days. It pairs the numbers from 1 through 13 with a sequence of
20 day-names. It works something like our days of the week pairing with
the
numbers of the
month. It will take 260 days before the cycle gets back to the begining
again (13 x 20).
The Tzolkin calendar was meshed with a 365-day solar cycle called the "Haab".
The calendar consisted of 18 months with 20 days (numbered 0-19) and a short "month" of
only 5 days that was called the Wayeb and was considered to be a dangerous
time. It took 52 years for the
Tzolkin and Haab calendars to move through a
complete cycle.
Kin = 1 day
Uinal = 20 days (like a month)
Tun = 360 days (year)
K'atun = 7,200 days
Baktun = 144,000
http://cs-exhibitions.uni-klu.ac.at/index.php?id=323
http://cs-exhibitions.uni-klu.ac.at/index.php?id=323
http://cs-exhibitions.uni-klu.ac.at/index.php?id=326
http://cs-exhibitions.uni-klu.ac.at/index.php?id=326
http://cs-exhibitions.uni-klu.ac.at/index.php?id=326