The Mayans, Incas, and Aztec are often grouped together in historical and geographical context. We will compare the mathematics of two of these civilizations, the Mayans and the Incas. It is highly unlikely that these two people groups ever came in contact with each other due to the distance between the two regions and base on the records of their expanses [6]. The Mayans inhabited the Yukatan peninsula, roughly 3488 miles from modern day Peru, the home of the Incas. The time periods of both civilizations only overlapped for about 300 years as well, a drop in the bucket of the almost 2000 year span of the Mayans. The height of the Mayan civilization, also known as the Classic Period, was from 200-900AD [3]. Both civilizations had a farming economy, but each had different motivations for the use of mathematics as well as different ways of calculating and recording.
The motivation for the Mayans delving into the field of mathematics stemmed from their focus on astronomy and use of a calendar. They worshipped many gods and viewed time as eternal and therefore of divine nature [8]. The culture had different classes of people, the highest of which were the astronomer priests and their noble families [6]. Women were sometimes included in ruling a city state as well as economics and religion. Though the Mayan civilization was made up of several independent city states with its own ruling family, the culture was mostly universal across all of them. The Mayan population is thought to have reached 50,000 people at its peak [2]. The city states built roads between them, traded, and were often combative towards each other, sometimes even forming alliances against others [6]. The Mayans are also known for their architecture and underground water reservoirs. The roads, religious centers, and basic solar calendar are thought to have been adapted from the Olmec society who inhabited the land before [6]. Some Mayan buildings towered at 200 feet tall, some with ornate friezes, or horizontal sculpted decoration. The tools mostly used in this region were stone, wood, and shells [6].
Most of what we know about the Mayan civilization comes from the remains left undestroyed by the Spanish missionary Diego de Landa or what was written in his book Relación de las cosas de Yucatán [3]. He found the Maya religious rituals such as animal and human sacrifices to be demonic and therefore had all books burned. However, three books survived the demolition: the Dresden Codex, the Madrid Codex, and the Paris Codex, named after where the artifacts are currently housed [3].
It is from these codices and Landa’s summarizing book that researchers were able to uncover the Mayan writing system, known as the Maya hieroglyphs, as well as their unique number system. The Mayans used about 800 symbols for numbers, sounds, words, and phrases which were read downward and rightward and written in folded books made of soft bark, called codices. One of the most important aspects of their number system was that it had a representation of zero [6], likely the first in history.
It is also from this ancient book that we find insight into the Mayan calendar, specifically based on three different dating systems: the Tzolkin, the Haab, and the Long Count [6].
The Tzolkin is the calendar of the divine and is made of thirteen groups of twenty, totaling 260 days. It lines up with nine cycles of the moon which is also the human gestational period [10]. Human birth is sacred and therefore tied to the religious count.
The Haab is the Mayan’s civil calendar. It estimates the solar year with nineteen different months, all twenty days except one which is five days, totaling 365 days. The five-day month is known as the “Wayeb,” during which several special ceremonies take place [10], and it is thought to be very unlucky. The main of the year outside of this significant month is known as the “Tun” [1]. This calendar is found at the remains of the Pyramid of Kukulkan at Chichen Itza [6].
The Long Count began on approximately August 11, 3114 B.C. [1] and, as its name implies, counts up from there. It was used to date buildings and historical events. The end of this cycle is on December 21, 2012, the winter solstice [10]. Though many thought the world was going to end on that date and it clearly did not, a scientist named Paolo Tagaloguin recently tweeted “Following the Julian Calendar, we are technically in 2012… The number of days lost in a year due to the shift into Gregorian Calendar is 11 days… For 268 years using the Gregorian Calendar (1752-2020) times 11 days = 2,948 days. 2,948 days / 365 days (per year) = 8 years” [9]. The tweet has since been deleted.
The Calendar Round is a combination of the Tzolkin and Haab calendars [10]. It describes that these divine and civil calendars align every 52 years [2], the human age at which Mayans believed they attained elder wisdom [10]. Every other Calendar Round lined up with the revolutions of Venus [2]. Some other dating terms used were “kin” for one day, “uinal” for a span of twenty days, “katun” for a span of 7200 days, and “baktun” for a span of 144000 days [1].
This fixation that the Mayans had on cycles of time shows why and how they developed mathematical skills. Most ancient civilizations formed a number system based on their needs regarding trade and farming. The Mayans, however, based their number system on observable solar happenings. In [10], they are regarded as “accomplished observers of the sky.” This is why their number system is not purely positional. It uses only three different symbols and begins with the units up to 19. The next place is the number of 20’s up to 19. However, in the third place the number of 360’s up to 19 are used instead of the number of 400’s. It then returns to the pattern of powers of 20’s. There may have been a system having a pure base of 20 used by merchants and civil transactions, but the one that uses 360’s is found in the Dresden Codex and is likely the one used by astronomers and priests [2].
Though there is no evidence of fractions, multiplication, or division, nor the use of advanced tools for calculations, the Mayans were able to approximate the solar year and the lunar month with astonishing accuracy [2]. Their solar year approximation of 365.242 found in [2] only has an error of 0.000054% from our modern calculation of
365.242198. Likewise, their lunar month approximations of 29.5302 and 29.5308 found in [2] lie almost symmetrically around our modern day calculation of 29.53059.
The motivation for the Incas use of mathematics differs quite a bit from that of the Mayans. The Incan Empire is also known to have a farming economy, but is uniquely credited with the invention of terrace farming and complex irrigation systems. They also played music, inventing the flute, drum, and Inca panpipe [5]. They, like the Mayans, worshipped several gods and offered human sacrifices. Unlike the Mayan civilization, however, they were grouped as one empire rather than multiple independent city states. There was one king and a structure of government officials underneath him that ruled the entire civilization. Beneath the royal class were other classes of people in the Inca society [5]. There were approximately 12 million people of different ethnic groups who spoke about 20 different languages [3].
Most remarkably, there are no records of any written language, only spoken language among the Incas [3]. It was possibly for this reason that a strong need for strong roads and communication carriers, known as roadrunners [5], emerged. Instead of passing down knowledge through texts such as the codices, information was transferred orally [7] and records were kept on a system of strings with knots, known as quipus [3].
Quipus were a positional number storage device which used regular spacing and knots to indicate different quantities. There are differing theories about what specifically was recorded on quipus. Some think they were used for traditions and history as a form of writing; some believe it was used as a translation tool, specifically after being conquered by the Spanish; and others think it was to record the magical and astronomical [4]. In [3] it is suggested that the Incas counted everything owned by peoples that they conquered.
Regardless of the quantities being represented, the quipus most definitely were the Incas way of recording permanent data. A standard quipu was comprised of several vertical cords (called H cords) attached to a horizontal cord (called a B cord). They are positional, having the largest quantities represented towards the top down to the smallest quantities represented towards the bottom of each string. Certain variety was obtained by using different colored cords as well as different types of knots [4]. For example, white strings may have signified peace and purity whereas red strings may have signified war [3].
Our knowledge of how the Incas stored and computed numbers come from the Spanish, who eventually conquered them. Both the quipu and the counting board explained above, also known as the yupana, were discussed in Filipe Guaman Poma de Ayala’s letter to the King of Spain. The Spanish priest Jose de Acosta gives high praise to the accomplishments of the Incas in his remark that “in what they undertake to do they are superior to us” [3]. This implies that there may in fact be more complexity than what we have evidence of regarding mathematics of the Incas.
The Incan civilization also seems to be somewhat secretive in their sharing of knowledge, even among themselves. They used herbs along with rituals to heal people and performed surgeries, such as trepanation with astonishing survival rates. However, because the healers were regarded as elite, there was a competitiveness that prevented sharing of techniques. Researchers today know about the brain surgeries of the Incas based on studying skull remains [7]. It is unclear whether the field of math would have experienced this same component of secrecy.
What we know about both the Mayan and Incan civilizations is undoubtedly limited by Landa’s burning of Mayan texts, the secretive nature of Incan knowledge, and war conflict in each. Though the Incan civilization was conquered by the Spanish, it is not certain what became of the Mayan civilization. Historians claim that the city states were deserted, but they are not sure why [6]. Nevertheless, centuries later we can marvel at the similarities and differences of their mathematics as well as the motives and uses behind each.
References
[1] dCode, “Mayan Numerals,” May 2020, https://www.dcode.fr/mayan-numbers.
[2] J. O’Connor & E. Robertson, “Mayan mathematics,” November 2000, https://mathshistory.st-andrews.ac.uk/HistTopics/Mayan_mathematics/.
[3] J. O’Connor & E. Robertson, “Mathematics of the Incas,” January 2001, https://mathshistory.st-andrews.ac.uk/HistTopics/Inca_mathematics/.
[4] L. Diana, The Peruvian quipu, Math. Teacher. 60 (1967) 623—628.
[5] L. Donn, “Inca Empire for Kids: Inventions & Achievements,” February 2007, https://www.mrdonn.org/faq.html.
[6] L. Donn, “Maya Empire for Kids: Inventions & Achievements,” February 2007, https://mayas.mrdonn.org/inventions.html.
[7] M. Crouse, “Inca Medicine: Religion, Culture, and Ethnobotany,” June 2014, https://www.central.edu/writing-anthology/2019/05/29/inca-medicine-religion- culture-and-ethnobotany/.
[8] M. Roque-Sol, “The Mayan Mathematics,” February 2019, https://www.math.tamu.edu/~roquesol/M629_6.html.
[9] P. Froelich, “Alternate reading of Mayan calendar suggests end of the world is next week,” June 2020, https://nypost.com/2020/06/13/reading-of-mayan-calendar- suggests-end-of-the-world-is-next-week/.
[10] Smithsonian Institution, “The Calendar System,” January 2020, https://maya.nmai.si.edu/calendar/calendar-system.
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