of these texts survived and are housed in museums in Paris,
France; Dresden, Germany; and Madrid, Spain. While these
texts, called codexes (or codices), date to the centuries just
preceding European contact, they were based on texts origi-
nally written in earlier periods, which in turn were based
on knowledge that had been acquired for hundreds of years
before that.
Historians of science are especially intrigued by the an-
cient Mayan system of mathematics, which they applied to
fi elds like astronomy and architecture. Much of this system
has been preserved in the codices. Mayan mathematics was a
vigesimal system, meaning that it was based on the number- Such a system probably evolved because ancient people
probably fi rst counted with their fi ngers and toes, and when
they reached the number 20 they had to start over with a new
set of 20 in much the same way that modern systems of math-
ematics are based on the number 10 and its multiples.
Th us in Mayan texts the numeral 1 was represented by a
thick dot, 2 by two dots, and so on. Th e numeral 5 was repre-
sented by a straight horizontal line, 6 by one dot above a line,
7 by two dots above a line, and so on. Th e numeral 10 was
then represented by two horizontal lines, 11 by a dot above
two lines, and so on up to 15, represented by three lines, and
so on up to 20, represented by a single dot above a shell rep-
resenting zero.
Th e Mayan system, though, had some irregularities that
historians of mathematics believe related to the Mayan cal-
endar—or rather, the two Mayan calendars. One was a rit-
ual calendar that consisted of 13 periods, each with 20 days.
Historians are unsure why this calendar was structured in
this way, but they speculate that each of the 13 periods repre-
sented a god and the 20 days represented humans (because of
their base 20 system of math). Th e other calendar was a solar
calendar consisting of 360 days, with 18 periods each consist-
ing of 20 days (and the extra fi ve days regarded as unlucky).
Th e two calendars coincided with each other aft er 18,980
days, equivalent to 52 years or 73 ritual years. Th e Maya also
observed that the planet Venus returned to its original spot
aft er two 52-year cycles, and in fact they held a great celebra-
tion aft er 104 years—that is, aft er two 52-year solar cycles or
one 104-year cycle of Venus.
But how does this explain the irregularity in the count-
ing system? To answer that, it is necessary to recognize that
the Maya also counted time as a linear sequence of days. Th ey
calculated, for example, that the world was created on a date
that coincides with August 12, 3113 b.c.e., and they dated
many of their historical monuments in terms of the number
of days that had passed since this creation. In the city of Tikal,
for example, is a historical marker dating construction of the
structure as occurring 1,253,912 days aft er the world was cre-
ated. Put diff erently, rather than counting determining the
structure of the calendar, the structure of the calendar deter-
mined counting, enabling the Maya to incorporate into their
mathematical system their conception of the gods, creation,
the movement of heavenly bodies, and the like.
GEOMETRY, ARCHITECTURE, AND ASTRONOMY
Ancient Americans had a sophisticated grasp of geometry,
and geometric patterns can be found in a wide range of Na-
tive American structures, from major temples to small cer-
emonial structures. Geometry and architecture intertwined
with astronomy as well, with many of these structures mir-
roring the movement of heavenly bodies, particularly the sun
as it changes position in the sky throughout the year. Th e geo-
metric regularity of these buildings refl ects geometric and as-
tronomic understandings that date back at least 2,000 years.
Ancient American geometry in many respects refl ected
the geometry of the natural world. It included not only simple
squares and circles but also arcs, hexagons (six-sided fi gures),
octagons (eight-sided fi gures), and dodecagons (12-sided fi g-
ures, as in a clocks). Th us, geometric understanding among
the ancient Americans was a process of discovering and un-
folding the geometry found in nature, rather than an arti-
fi cially constructed system of thought. Ancient Americans
began with the simple notion that two points could defi ne a
line that extended infi nitely in either direction. Additionally,
either or both of those two points could become the center of
a circle, and the line connecting the two points could serve as
the radius of a circle. If circles were drawn using each of the
two points as the center, the two circles would intersect at two
points. In turn, each of those points could serve as the center
of yet another circle. If the process is continued, the result is
a six-petaled “fl ower,” known among some ancient American
cultures as the Flower of Life.
More important, this system of intersecting points can be
used to make many other geometric shapes, including perfect
triangles, pentagons (fi ve-sided fi gures) hexagons, octagons,
decagons (10-sided fi gures), dodecagons, and even polygons
with 24 and 48 sides. (Th e word polygon refers to any of these
enclosed, fl at shapes with angles; poly-, means “many.”) Once
these shapes and how to derive them were understood, ar-
chitects could apply them to the construction of buildings,
including temples, homes, and ceremonial structures, with
simple tools like string. Each of the points in an octagon, for
example, would become the position for a vertical post, which
then evolved into more complex architectural designs.
What role, though, did astronomy play in all this? His-
torians and archaeologists have long noted that many of the
structures of ancient America were oriented along north–
south and east–west axes. In recent decades, however, many
scholars have turned their attention to a subfi eld called ar-
chaeoastronomy—the study of the relationships between
building practices and astronomical knowledge. In connec-
tion with ancient American structures they have found that
many are built with an understanding of the concept of azi-
muths, particularly the azimuth of the sun at the points of the
winter and summer solstice.
An azimuth is an angle measured from a reference point.
Perhaps the best way to understand azimuths is to think about
satellites in the sky that beam down television signals. When948 science: The Americas0895-1194_Soc&Culturev4(s-z).i948 948 10/10/07 2:30:32 PM