948 science: The Americas 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 originally 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 ancient Mayan system of mathematics, which they applied to fields 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 20 Such a system probably evolved because ancient people probably first counted with their fingers 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 mathematics are based on the number 10 and its multiples Thus in Mayan texts the numeral was represented by a thick dot, by two dots, and so on The numeral was represented by a straight horizontal line, by one dot above a line, by two dots above a line, and so on The 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 representing zero The Mayan system, though, had some irregularities that historians of mathematics believe related to the Mayan calendar—or rather, the two Mayan calendars One was a ritual 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 represented a god and the 20 days represented humans (because of their base 20 system of math) The other calendar was a solar calendar consisting of 360 days, with 18 periods each consisting of 20 days (and the extra five days regarded as unlucky) The two calendars coincided with each other after 18,980 days, equivalent to 52 years or 73 ritual years The Maya also observed that the planet Venus returned to its original spot after two 52-year cycles, and in fact they held a great celebration after 104 years—that is, after two 52-year solar cycles or one 104-year cycle of Venus But how does this explain the irregularity in the counting system? To answer that, it is necessary to recognize that the Maya also counted time as a linear sequence of days They 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 after the world was created Put differently, rather than counting determining the structure of the calendar, the structure of the calendar determined 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 Native American structures, from major temples to small ceremonial structures Geometry and architecture intertwined with astronomy as well, with many of these structures mirroring the movement of heavenly bodies, particularly the sun as it changes position in the sky throughout the year The geometric regularity of these buildings reflects geometric and astronomic understandings that date back at least 2,000 years Ancient American geometry in many respects reflected the geometry of the natural world It included not only simple squares and circles but also arcs, hexagons (six-sided figures), octagons (eight-sided figures), and dodecagons (12-sided figures, as in a clocks) Thus, geometric understanding among the ancient Americans was a process of discovering and unfolding the geometry found in nature, rather than an artificially constructed system of thought Ancient Americans began with the simple notion that two points could define a line that extended infinitely 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 “flower,” 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 (five-sided figures) hexagons, octagons, decagons (10-sided figures), dodecagons, and even polygons with 24 and 48 sides (The word polygon refers to any of these enclosed, flat shapes with angles; poly-, means “many.”) Once these shapes and how to derive them were understood, architects 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? Historians 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 subfield called archaeoastronomy—the study of the relationships between building practices and astronomical knowledge In connection with ancient American structures they have found that many are built with an understanding of the concept of azimuths, 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 When