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@jorge333morales
It is possible to write the number 1 as a sum of 48 different fractions, where every numerator is 1 and every denominator is a product of exactly two primes.
It is conjectured that such a decomposition holds for any positive integer— known to be true in a slightly less elegant situation, where every denominator is a product of exactly three primes, as shown in Egyptian fractions with each denominator having three distinct prime divisors, by Butler, Erdös and Graham.
Aztec Deities
Part 3 of this series. *All based in Aztec Mexico
Acolmiztu - God of the Underworld
Acuecucyoticihuati - Goddess of Rivers and Streams
Atlatonan - Goddess of the Earth
Ayauhteotl - Goddess of Infamy
Centeotl - God of Maize
Centzon-Totochtin - Drunken God
Centzonuitznaua - Gods of the Stars
Chalchihuitlicue - Goddess of Water
Chicomeccatl - Goddess of Corn and Maize
Chalchiuhtecolotl - God of the Night
Chalchiuhtotolin - God of Disease
Chalmecatecuchtli - God of the Underworld
Chantico - Goddess of the Hearth
Chicomexochtli - Gods of Painting
Chiconahuiehecatl - God of Creation
Chihucoatl - Goddess of the Moon and War
Cinteotl - God of Maize and Corn
Cipactli - Creator God
Cipactonal - God of Astrology and Calendars
Coatlicue - Goddess of Earth and Fire
Cochimetl - God of Merchants
Cocijo - God of Rain
Coyolxauhqui -Goddess of the Moon
Cuaxolotl - Goddess of the Stars
Ehecatl - God of the Winds
Huehuecoyotl - God of Mischief
Huehueteotl - God of Fire
Huitzilopochtli - God of War
Huixtocihuatl - Goddess of Fertility
Ilamatecuhtli - Old Mother Goddess of Midwifery
Itztli - God of Sacrifice
Itzpapalotl - Goddess of the Heavens
Itztlacoliuhqui - God of Justice
Ixtlilton - God of Healing
Macuil-Cozcacuahtli - God of Pleasure
Macuil-Cuetzpalin - God of Pleasure
Macuil-Malinalli - God of Pleasure
Macuil-Tochtli - God of Alcohol
Macuil-Tonaleque - Gluttony Gods
Macuilxochitl - God of Games and Sports
Malinalxochi - Goddess of Insects
Mayahuel - Goddess of Alcohol
Metzli - God of the Moon
Mextli - God of War and Sacrifice
Mictecacihuatl - Goddess of Death
Mictlantecuhtli - God of the Underworld
Mixcoatl - God of the Hunt
Nanautzin - God of Sickness
Omecihuatl - Goddess of Creation
Ometecuhtli - Supreme God of Creation
Ometotchtli - God of Pulque and Drunkness
Opochtli - God of Fishing and Hunting
Oxomoco - Goddess of Astrology and Calendars
Patecatl - God of Healing and Medicine
Piquete-Ziña - God of Bats and the Night
Quaxolotl - Goddess of Duality and Schizophrenia
Quetzalcoatl - God of Creation
Tecuciztecal - God of the Moon
Temazcalteci - Goddess of Purification
Teoyaomiqui - God of Flowers
Tepeyollotl - God of Caves and Earthquakes
Tepoztecatl - God of Alcohol
Toci - Creator Goddess
Tezcatzontecatl - God of Beer
Tlazoltéotl - Goddess of Purity and Confession
Tezcatlipoca - God of Death and the Night
Tlaloc - God of Rain
Tlahuizcalpantecutli - God of Dawn
Tlaoque-Nahuaque - Supreme God
Tonacatecuhtli - God of Food
Tonatiuh - God of the Sun
Tzitzmitl - Grandmother Goddess of Alcohol
Ueuecoyotl - God of Happiness
Xipe-Totec - God of Agriculture and Spring
Xiuhcoatl - God of Fire and Drought
Xiuhtecuhtli - Ancient God of Fire
Xmulzencab - God of Bees
Xochipili - God of Love
Xochiquetzal - Goddess of Fertility
Xólotl - God of Lightning and the Underworld
Yacatecuhtli - God of Merchants and Travel
Sources : Wikipedia , Godfinder and God search
Buena publicación
Ryan Coogler's Marvel movie sports a 100 percent score on Rotten Tomatoes.
That’s right!!
Introduction to Mesoamerica reading material
This list is drawn from old school syllabi, what I’ve read, and what I’ve seen others recommend. I will try to update this periodically with more books, journal articles, book chapters, and even documentaries. I’ll try and provide an open access (or limited access like a free JSTOR account) for the journal articles and book chapters.
Books
General
* Mann, Charles C. 1491: New revelations of the Americas before Columbus. Alfred a Knopf Incorporated, 2005.
* Coe, Michael D., and Rex Koontz. Mexico: from the Olmecs to the Aztecs. Vol. 29. Thames & Hudson, 2008.
* Evans, Susan Toby. Ancient Mexico and Central America: archaeology and culture history. Thames & hudson, 2013.
* Coe, Sophie D. America’s first cuisines. University of Texas Press, 1994.
* Matthew, Laura E., and Michel R. Oudijk. Indian conquistadors: Indigenous allies in the conquest of Mesoamerica. University of Oklahoma Press, 2007.
* Restall, Matthew. Seven myths of the Spanish conquest. Oxford University Press, 2004.
* Miller, Mary Ellen, and Karl Taube. An illustrated dictionary of the gods and symbols of ancient Mexico and the Maya. London: Thames and Hudson, 1997.
* Tiesler, Vera, and Andrea Cucina, eds. New perspectives on human sacrifice and ritual body treatments in ancient Maya society. Springer Science & Business Media, 2007.
Aztec
* Smith, Michael E. The Aztecs. John Wiley & Sons, 2013.
* Hassig, Ross. Aztec warfare: Imperial expansion and political control. Vol. 188. University of Oklahoma Press, 1995.
* Soustelle, Jacques. Daily life of the Aztecs. Courier Corporation, 2002.
* Lêaon-Portilla, Miguel. Aztec Thought and Culture. University of Oklahoma Press, 1963.
* Anderson, Arthur JO, and Charles E. Dibble. Florentine Codex. School of American Research and University of Utah, Sante Fe, New Mexico, II(1950).
* Portilla, Miguel León. The broken spears: The Aztec account of the conquest of Mexico. Beacon Press, 2006.
Maya
* Houston, Stephen D., and Takeshi Inomata. The Classic Maya. Cambridge University Press, 2009.
* Freidel, David, and Linda Schele. A forest of kings: The untold story of the ancient Maya. Harper Collins, 1992.
* Freidel, David A., Linda Schele, and Joy Parker. Maya Cosmos Three Thousand Years on the Shaman’s Path. (1993).
* Martin, Simon, and Nikolai Grube. Chronicle of the Maya kings and queens: Deciphering the dynasties of the ancient Maya. Thames & Hudson, 2008.
* Coe, Michael D. “Breaking the Maya Code, rev. ed.” London and NewYork(1999).
* American Anthropological Association. Ancient Maya Commoners. Eds. Jon C. Lohse, and Fred Valdez Jr. University of Texas Press, 2004.
* Demarest, Arthur. Ancient Maya: the rise and fall of a rainforest civilization. Vol. 3. Cambridge University Press, 2004.
* Sharer, Robert J., and Loa P. Traxler. The ancient maya. Stanford University Press, 2006.
* Iannone, Gyles, and Samuel V. Connell. Perspectives on Ancient Maya Rural Complexity. The Cotsen Institute of Archaeology Press, 2003.
* Scarborough, Vernon L., Fred Valdez, and Nicholas P. Dunning, eds. Heterarchy, Political Economy, and the Ancient Maya: The Three Rivers Region of the East-central Yucatˆn Peninsula. University of Arizona Press, 2003.
* Houston, Stephen, David Stuart, and Karl Taube. The memory of bones: Body, being, and experience among the Classic Maya. University of Texas Press, 2013.
* Jones, Grant D. The conquest of the last Maya kingdom. Stanford University Press, 1998.
Olmec
* Pool, Christopher. Olmec archaeology and early Mesoamerica. Cambridge University Press, 2007.
Teotihuacan
* Moctezuma, Eduardo Matos. Teotihuacan. Fondo de Cultura Económica, 2009.
* Sugiyama, Saburo. Human sacrifice, militarism, and rulership: materialization of state ideology at the Feathered Serpent Pyramid, Teotihuacan. Cambridge University Press, 2005.
* Manzanilla, Linda. Teotihuacan, ciudad excepcional de Mesoamérica. El Colegio Nacional, 2017.
* Headrick, Annabeth. The Teotihuacan trinity: the sociopolitical structure of an ancient Mesoamerican city, 2007.
West Mexico
* Pollard, Helen Perlstein. Tariacuri’s Legacy: The Prehispanic Tarascan State. University of Oklahoma Press, 1993.
* Warren, Joseph Benedict. The conquest of Michoacan: the Spanish domination of the Tarascan kingdom in western Mexico, 1521-1530. University of Oklahoma Press, 1985.
* Von Winning, Hasso, and Olga Hammer. Anecdotal sculpture of ancient West Mexico. Ethnic Arts Council of Los Angeles, 1972.
* Von Winning, Hasso. The shaft tomb figures of West Mexico. No. 24. Southwest Museum, 1974.
* Hosler, Dorothy. The sounds and colors of power: The sacred metallurgical technology of ancient west Mexico. MIT Press, 1994.
* Townsend, Richard F. Ancient West Mexico: Art and archaeology of the unknown past. Thames and Hudson, 1998.
* Beekman, Christopher S. and Robert B. Pickering. Shaft Tombs and Figurines in West Mexican Society: A Reassessment. Gilcrease Museum, 2016.
* Altman, Ida. The War for Mexico’s West: Indians and Spaniards in New Galicia, 1524-1550. University of New Mexico Press, 2010.
Oaxaca
* Flannery, Kent V. The cloud people: Divergent evolution of the Zapotec and Mixtec civilizations. Percheron Pr, 2003.
* Byland, Bruce, and John MD Pohl. In the Realm of Eight Deer. (1994).
* Joyce, Arthur A. Mixtecs, Zapotecs, and Chatinos. Malden, MA: Wiley-Blackwell (2010).
* Winter, Marcus. Oaxaca: the archaeological record. Editorial Minutiae Mexicana, 1989.
* Joyce, Arthur A., ed. Polity and ecology in Formative period coastal Oaxaca. University Press of Colorado, 2013.
* Spores, Ronald, and Andrew K. Balkansky. The Mixtecs of Oaxaca: Ancient Times to the Present. Vol. 267. University of Oklahoma Press, 2013.
* Terraciano, Kevin. The Mixtecs of colonial Oaxaca: Ñudzahui history, sixteenth through eighteenth centuries. Stanford University Press, 2004.
Film
* The Dawn of the Maya
* Edgewalker: A Conversation with Linda Schele
* Glyphers: Deciphering Mayan Society
* The Popol Vuh - English / Español
I noticed that there’s only quite a few of these floating around and I thought that it would be a good idea to spread some love to the math community, so here you go: a compilation of math resources that I found really helpfulヽ(＊>∇<)ﾉ by the way, HAPPY CHINESE NEW YEAR ILY <3 祝大家新年快樂、 身體健康！
HOW TO + STUDYING
learn algebra
learn math
how to ace math by @hastag-med-school
math masterpost by @pythgaoras
getting the most out of math class
how to cram for a math test
the art of doing well in technical courses
advice from a mathematics student by @mashaczstudies
NOTE TAKING
note taking tips for math
best way to take math notes (advice)
digital math notes (advice)
taking notes in math lectures (advice)
organizing math notes
GREAT WEBSITES
sparknotes math study guides
pbs learning media
purplemath
algebra for dummies
funny algebra puns
online calculators w/ solution
khan academy
mathxplain
YOUTUBE / PODCASTS
patrickJMT
math class with terry v
algebra song
relatively prime
math dude podcast
math mutation
math for primates
math factor
taking maths further
SOME HELPFUL PDF FILES
high school playbook
abstract algebra guide
algebra real world
MY OTHER MASTERPOSTS
back to school tips, part 1
back to school tips, part 2
note taking alternatives
self-care and motivation
essay writing
Scans of the inside covers of Strang’s Calculus, which you can legally-download for free here from the MIT website. This is my all-time favorite math or physics textbook. Scanned it so I could cut and paste it into my new sketchbook, wanna try and make a ~cool artistic~ reference poster out of it, ‘cuz I’ve been real into that idea since I took notes about rings for the algebra midterm on a big piece of watercolor paper.
How to solve square roots on paper, in this case the square root of 651,478 to one digit after the decimal.
Easy
I am thinking about Women in computing
“Look like a girl, act like a lady, think like a man, and work like a dog." — Betty Snyder Betty Snyder was a one of the six original programmers of the Electronic Numerical Integrator and Computer (ENIAC), the first general-purpose electronic digital computer. She, also made a significant contribution to the development of the COBOL and FORTRAN programming languages and created instruction code for the Universal Automatic Computer I (UNIVAC I), the earliest commercially sold computer.
Check-in to Women in computing on GetGlue.com
The Pythagorean theorem can be extended to any number of dimensions. In 2D space, the Pythagorean theorem gives us the length of the diagonal of a rectangle. It turns out, a simple modification (adding another square term) to the formula gives the diagonal of a rectangular prism. In the same way, the formula can be further extended to apply to 4 dimensional situations, and give the diagonal of a hypercube (4D cube) or tesseract (yes that’s where Marvel got the name for that glowing blue cube in the first Captain America movie). Keeping with the pattern, the Pythagorean formula can be generalized to any arbitrary whole number dimension to apply to finding diagonals of n-dimensional rectangle analogs.
Okay, but is this 4th dimension, hypercube stuff even real? IDK, is any math real? It doesn’t matter. But I can tell you this type of mathematics is useful. For example, Einstein’s theory of general relativity relies on the mathematics of 4-dimensional space. Without knowing this, the GPS in your phone would not work (GPS relies on precise timing. So precise that gravitational effects of the Earth become necessary to consider).
See also Weisstein, Eric W. “Tesseract.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Tesseract.html
“An Unorthodox ‘Test’” by Abe Shenitzer
Students are often taught, and tested for, technical skills. We miss the mark in not teaching the intellectual context and critical analysis of the material we present. What I mean is illustrated by the “test questions” below. These “test questions” reflect the kind of intellectual bias I consider vital in the teaching of mathematics.
Beautiful
Computing’s Search for Quantum Questions
Learn about the viability of quantum annealing to solve NP-hard problems.
Unlike a universal quantum computer, which is designed to tackle a range of problems, a quantum annealer is designed to tackle a specific type of problem called an NP-hard optimization problem. These can arise in many situations, but here’s an example: Say you went to a city and wanted to find the largest cluster of people who all know each other. (This is called the “maximum clique problem” in computer science.) You could try to tackle the problem with a brute-force approach, identifying every group of friends and comparing sizes until you settled on the largest. Such a strategy would work for a small village, but it becomes untenable if you want to identify and compare every group of friends in, say, Tokyo. […] To solve this kind of problem, the D-Wave 2X uses a process called annealing, which gets its name from a 7,000-year-old metallurgical practice. Early metalworkers figured out that if they made a material hot enough and then let it cool down slowly, it would make the final product stronger. The process works because the heat randomly jiggles around all the atoms in the metal, and then as it cools, the atoms slowly arrange themselves into their lowest-energy (and therefore most stable) configuration. Bartenders may also recognize the process: The secret to creating a flawless and clear ice cube is to let the water freeze very, very slowly.
Quanta Magazine - Image by Daan Botlek
Woow
This article is awesome. Everybody go read it. I bought watercolor paper yesterday and damn, why didn’t I do that earlier, ‘cuz I’ve been accidentally drawin’ holes in my notebook pages a lot recently. New leadholder, too. E&M homework down, thank god; DE 2 hw, DE 2 exam on Monday, unreleased E&M take-home final due Tuesday, and makin’ more/better lil animated ditties about matrix algebra left to go. And lord help me if this all don’t get as done as possible by the time I get dealt that E&M take-home final. Because the second I get it… well, I have a lot of Taco Bell in my future, put it that way. Off to K Brew for A+ 10/10 coffee and groove establishment for the day. DE2 hw, your ass is grass.
I found this page in that watercolor pad yesterday, even though I’d been using it for a few days for my sketches for my paper revisions. Was totally not expecting that, cool surprise, esp since I found it only a few days before its first birthday. Also “w/e matrices are hella arbitrary” made me laugh just now. Think I’mma hang this up in my studio, I like it. (And I have no idea what the hell my train of thought was here.)
Scans of the inside covers of Strang’s Calculus, which you can legally-download for free here from the MIT website. This is my all-time favorite math or physics textbook. Scanned it so I could cut and paste it into my new sketchbook, wanna try and make a ~cool artistic~ reference poster out of it, ‘cuz I’ve been real into that idea since I took notes about rings for the algebra midterm on a big piece of watercolor paper.
In the fall of 2003, a heavy rainstorm swept through the ruins of Teotihuacán, the pyramid-studded, pre-Aztec metropolis 30 miles northeast of present-day Mexico City. Dig sites sloshed over with water; a torrent of mud and debris coursed past rows of souvenir stands at the main entrance. The grounds of the city’s central courtyard buckled and broke. One morning, Sergio Gómez, an archaeologist with Mexico’s National Institute of Anthropology and History, arrived at work to find a nearly three-foot-wide sinkhole had opened at the foot of a large pyramid known as the Temple of the Plumed Serpent, in Teotihuacán’s southeast quadrant. “My first thought was, ‘What exactly am I looking at?’” Gómez told me recently. “The second was, ‘How exactly are we going to fix this?’”
Gómez is wiry and small, with pronounced cheekbones, nicotine-stained fingers and a helmet of dense black hair that adds a couple of inches to his height. He has spent the past three decades—almost all of his professional career—working in and around Teotihuacán, which once, long ago, served as a cosmopolitan center of the Mesoamerican world. He is fond of saying that there are few living humans who know the place as intimately as he does.
And as far as he was concerned, there wasn’t anything beneath the Temple of the Plumed Serpent beyond dirt, fossils and rock. Gómez fetched a flashlight from his truck and aimed it into the sinkhole. Nothing: only darkness. So he tied a line of heavy rope around his waist and, with several colleagues holding onto the other end, he descended into the murk.
Gómez came to rest in the middle of what appeared to be a man-made tunnel. “I could make out some of the ceiling,” he told me, “but the tunnel itself was blocked in both directions by these immense stones.”
In designing Teotihuacán (pronounced tay-oh-tee-wah-KAHN), the city’s architects had arranged the major monuments on a north-south axis, with the so-called “Avenue of the Dead” linking the largest structure, the Temple of the Sun, with the Ciudadela, the southeasterly courtyard that housed the Temple of the Plumed Serpent. Gómez knew that archaeologists had previously discovered a narrow tunnel underneath the Temple of the Sun. He theorized that he was now looking at a kind of mirror tunnel, leading to a subterranean chamber beneath the Temple of the Plumed Serpent. If he was correct, it would be a find of stunning proportions—the type of achievement that can make a career.
“The problem was,” he told me, “you can’t just dive in and start tearing up earth. You have to have a clear hypothesis, and you have to get approval.”
Gómez set about making his plans. He erected a tent over the sinkhole, to keep it away from the prying eyes of the hundreds of thousands of tourists who visit Teotihuacán each year, and with the help of the National Institute of Anthropology and History arranged for the delivery of a lawnmower-size, high-resolution, ground-penetrating radar device. Beginning in the early months of 2004, he and a handpicked team of some 20 archaeologists and workers scanned the earth under the Ciudadela, returning every afternoon to upload the results to Gómez’s computers. By 2005, the digital map was complete.
As Gómez had suspected, the tunnel ran approximately 330 feet from the Ciudadela to the center of the Temple of the Plumed Serpent. The hole that had appeared during the 2003 storms was not the actual entrance; that lay a few yards back, and it had apparently been intentionally sealed with large boulders nearly 2,000 years ago. Whatever was inside that tunnel, Gómez thought to himself, was meant to stay hidden forever.
Gómez believes the tunnel is “one of the most important discoveries in the history of Mexico.”
Covering 40 acres, the Ciudadela (”the Citadel”) was able to hold tens of thousands of the city’s residents during public ceremonies.
Teotihuacan’s ceremonial city center was built around the Avenue of the Dead, which runs for more than two miles. As many as 200,000 people lived in the surrounding areas, in some 2,000 structures not unlike apartment complexes.
Nearly 100,000 tons of earth have been removed from the tunnel, which Gómez hopes to finish excavating this summer.
Teotihuacán has long stood as the greatest of Mesoamerican mysteries: the site of a colossal and influential culture about which frustratingly little is understood, from the conditions of its rise to the circumstances of its collapse to its actual name. Teotihuacán translates as “the place where men become gods” in Nahuatl, the language of the Aztecs, who likely found the ruins of the deserted city sometime in the 1300s, centuries after its abandonment, and concluded that a powerful ur-culture—an ancestor of theirs—must have once resided in its vast temples.
The city lies in a basin at the southernmost edge of the Mexican Plateau, an undulating landmass that forms the spine of modern-day Mexico. Inside the basin the climate is mild, the land riven by streams and rivers—ideal conditions for farming and raising livestock.
Teotihuacán itself was likely settled as early as 400 B.C., but it was only around A.D. 100, an era of robust population growth and increased urbanization in Mesoamerica, that the metropolis as we know it, with its wide boulevards and monumental pyramids, was built. Some historians have theorized that its founders were refugees driven north by the eruption of a volcano. Others have speculated that they were Totonacs, a tribe from the east.
Whatever the case, the Teotihuacanos, as they are now known, proved themselves to be skilled urban planners. They built stone-sided canals to reroute the San Juan River directly under the Avenue of the Dead, and set about constructing the pyramids that would form the city’s core: the Temple of the Plumed Serpent, the even larger 147-foot-tall Temple of the Moon and the bulky, sky-obscuring 213-foot-tall Temple of the Sun.
Clemency Coggins, a professor emerita of archaeology and art history at Boston University, has suggested that the city was designed as a physical manifestation of its founders’ creation myth. “Not only was Teotihuacán laid out in a measured rectangular grid, but the pattern was oriented to the movement of the sun, which was born there,” Coggins has written. She is far from the only historian to see the city as large-scale metaphor. Michael Coe, an archaeologist at Yale, argued in the 1980s that individual structures might be representations of the emergence of humankind out of a vast and tumultuous sea. (As is in Genesis, Mesoamericans of the time are thought to have envisioned the world as being born from complete darkness, in this case aqueous.) Consider the Temple of the Plumed Serpent, Coe suggested—the same temple that hid Sergio Gómez’s tunnel. The structure’s facade was splashed with what Coggins called “marine motifs”: shells and what appear to be waves. Coe wrote that the temple represents the “initial creation of the universe from a watery void.”
Hot-air balloons above Teotihuacan just after dawn. In the foreground is the Pyramid of the Moon, with the Pyramid of the Sun in the distance.
The view atop the Temple of the Moon
The heads of feathered serpents and the god Tlaloc peer from the Temple of the Plumed Serpent. They are thought to have ideological significance.
Elaborately decorated conch shells are found throughout the city.
Recent evidence suggests that the religion practiced in these pyramids bore a resemblance to the religion practiced in the contemporaneous Mayan cities of Tikal and El Mirador, hundreds of miles to the southeast: the worshiping of the sun and moon and stars; the veneration of a Quetzalcoatl-like plumed serpent; the frequent occurrence, in painting and sculpture, of a jaguar that doubles as deity and protector of men.
Yet peaceful ritual was apparently not always enough to sustain the Teotihuacanos’ connection to their gods. In 2004, Saburo Sugiyama, an anthropologist from the University of Japan and Arizona State University, who has spent decades studying Teotihuacán, and Rubén Cabrera, of Mexico’s National Institute of Anthropology and History, located a vault under the Temple of the Moon that held the remains of an array of wild animals, including jungle cats and eagles, along with 12 human corpses, ten missing their heads. “It is hard to believe that the ritual consisted of clean symbolic performances,” Sugiyama said at the time. “It is most likely that the ceremony created a horrible scene of bloodshed with sacrificed people and animals.”
Between A.D. 150 and 300, Teotihuacán grew rapidly. Locals harvested beans, avocados, peppers and squash on fields raised in the middle of shallow lakes and swampland—a technique known as chinampa—and kept chickens and turkeys. Several heavily trafficked trade routes were established, linking Teotihuacán to obsidian quarries in Pachuca and cacao groves near the Gulf of Mexico. Cotton came in from the Pacific Coast, ceramics from Veracruz.
By A.D. 400, Teotihuacán had become the most powerful and influential city in the region. Residential neighborhoods sprang up in concentric circles around the city center, eventually comprising thousands of individual family dwellings, not dissimilar to single-story apartments, that together may have housed 200,000 people.
Recent fieldwork by scholars like David Carballo, of Boston University, has revealed the sheer diversity of the citizenry of Teotihuacán: Judging by artifacts and paintings found inside surviving structures, residents came to Teotihuacán from as far afield as Chiapas and the Yucatán. There were likely Mayan neighborhoods, and Zapotec ones. As the scholar Miguel Angel Torres, an official at Mexico’s National Institute for Anthropology and History, told me recently, Teotihuacán was probably one of the first major melting pots in the Western Hemisphere. “I believe that the city grew a little like modern Manhattan,” Torres says. “You walk around through these different neighborhoods: Spanish Harlem, Chinatown, Koreatown. But together, the city functions as one, in harmony.”
The harmony did not last. There is a hint, in the demolition of some of the sculptures that adorn the temples and monuments, of periodic regime change in the ruling class of Teotihuacán; and, in the depiction of shield- and spear-toting warriors, of clashes with other local city-states. Perhaps, as several archaeologists suggested to me, civil war swept through Teotihuacán, culminating in a fire that seems to have damaged vast sections of the interior of the city around A.D. 550. Perhaps the fire was caused by a visiting army. Perhaps a large-scale migration occurred.
In A.D. 750, nearly 700 years after it was established, the city of Teotihuacán was abandoned, its monuments still filled with treasures and artifacts and bones, its buildings left to be eaten by the surrounding brush. The former residents of Teotihuacán, if they were not killed, were presumably absorbed into the populations of neighboring cultures, or returned along the established trade routes to the lands where their ancestral kin still lived throughout the Mesoamerican world.
They took their secrets with them. Today, even after more than a century of excavation at the site, there is an extraordinary amount we do not know about the Teotihuacanos. They did have some kind of quasi-hieroglyphic written language, but we haven’t cracked it; we don’t know what tongue was spoken inside the city, or even what the natives called the place. We have a conception of the religion they practiced, but we don’t know much about the priestly class, or the relative piety of the city’s citizenry, or the makeup of the courts or the military. We don’t know exactly what led to the city’s founding, or who ruled over it during its half-millennium of dominance, or what exactly caused its fall. As Matthew Robb, the curator of Mesoamerican art at San Francisco’s de Young Museum, told me, “This city wasn’t designed to answer our questions.”
In archaeology and anthropology circles—to say nothing of the popular press—Sergio Gómez’s discovery was greeted as a major turning point in Teotihuacán studies. The tunnel under the Temple of the Sun had been largely emptied by looters before archaeologists could get to it in the 1990s. But Gómez’s tunnel had been sealed off for some 1,800 years: Its treasures would be pristine.
In 2009, the government granted Gómez permission to dig, and he broke ground at the entrance of the tunnel, where he installed a staircase and ladders that would allow easy access to the subterranean site. He moved at a painstaking pace: inches at a time, a few feet every month. Excavating was done manually, with spades. Nearly 1,000 tons of earth were removed from the tunnel; after each new segment was cleared, Gómez brought in a 3-D scanner to document his progress.
The haul was tremendous. There were seashells, cat bones, pottery. There were fragments of human skin. There were elaborate necklaces. There were rings and wood and figurines. Everything was deposited deliberately and pointedly, as if in offering. The picture was coming into focus for Gómez: This was not a place where ordinary residents could tread.
A university in Mexico City donated a pair of robots, Tlaloque and Tláloc II, playfully named for Aztec rain deities whose images appear in early iterations throughout Teotihuacán, to inspect deeper inside the tunnel, including the final stretch, which descended, on a ramp, an extra ten feet into the earth. Like mechanical moles, the robots chewed through the soil, their camera lights aglow, and returned with hard drives full of spectacular footage: The tunnel seemed to end in a spacious cross-shaped chamber, piled high with more jewelry and several statues.
It was here, Gómez hoped, that he’d make his biggest find yet.
The three-foot long remote-controlled Tlaloc II robot is equipped with an infrared scanner and video camera.
Workers examine earth from the Adosada Platform, a smaller structure abutting the Temple of the Plumed Serpent.
A worker removes dirt from a tunnel discovered under the Pyramid of the Plumed Serpent. So far, 70,000 objects of interest have been found there.
Gabriel Garcia Sarabia pieces together an ancient vase from fragments found in the tunnel.
A conservator restores a vase depicting a Tlaloc-like deity
A “flying dog” saucer was found intact
Archaeologist Eduardo Ramos walks behind the Pyramid of the Plumed Serpent. He believes the structure was torn down and rebuilt many times.
I met Gómez late last year, on a smoldering afternoon. He was smoking a cigarette and drinking coffee out of a foam cup. Tides of tourists swept to and fro over the grass of the Ciudadela—I heard scraps of Italian, Russian, French. An Asian couple stopped to peer in at Gómez and his team as if they were tigers at a zoo. Gómez looked back stonily, the cigarette hanging off his bottom lip.
Gómez told me about the work his team was doing to study the 75,000 or so artifacts they had already found, each of which needed to be carefully cataloged, analyzed and, when possible, restored. “I would estimate that we’re only about 10 percent through the process,” he said.
The restoration operation is set up in a cluster of buildings not far from the Ciudadela. In one room, a young man was sketching artifacts and noting where in the tunnel the objects had been found. Next door, a handful of conservators sat at a banquet-style table, bent over an array of pottery. The air smelled sharply of acetone and alcohol, a mixture used to remove contaminants from the artifacts.
“It might take you months just to finish a single large piece,” Vania García, a technician from Mexico City, told me. She was using a syringe primed with acetone to clean a particularly tiny crack. “But some of the other objects are remarkably well preserved: They were buried carefully.” She recalled that not long ago, she found a powdery yellow substance at the bottom of a jar. It was corn, it turned out—1,800-year-old corn.
Passing through a lab where wood recovered from the tunnel was being carefully treated in chemical baths, we stepped into the storeroom. “This is where we keep the fully restored artifacts,” Gómez said. There was a statue of a coiled jaguar, poised to pounce, and a collection of flawless obsidian knives. The material for the weapons had probably been brought in from the Pachuca region of Mexico and carved in Teotihuacán by master craftspeople. Gómez held out a knife for me to hold; it was marvelously light. “What a society, no?” he exclaimed. “That could create something as beautiful and powerful as that.”
In the canvas tent erected over the entrance to the tunnel, Gómez’s team had installed a ladder that led down into the earth—a wobbly thing fastened to the top platform with frayed twine. I descended carefully, foot over foot, the brim of my hard hat slipping over my eyes. In the tunnel it was damp and cold, like a grave. To get anywhere, you had to walk on your haunches, turning to the side when the passage narrowed. As protection against cave-ins, Gómez’s workmen had installed several dozen feet of scaffolding—the earth here is unstable, and earthquakes are common. So far, there had been two partial collapses; no one had been hurt. Still, it was hard not to feel a shiver of taphophobia.
Through the middle of Teotihuacán studies runs a division like a fault line, separating those who believe that the city was ruled by an all-powerful and violent king and those who argue that it was governed by a council of elite families or otherwise bound groups, vying over time for relative influence, arising from the cosmopolitan nature of the city itself. The first camp, which includes experts like Saburo Sugiyama, has precedent on its side—the Maya, for instance, are famous for their warlike kings—but unlike Mayan cities, where rulers had their visages festooned on buildings and where they were buried in opulent tombs, Teotihuacán has offered up no such decorations, nor tombs.
Initially, much of the buzz surrounding the tunnel beneath the Temple of the Plumed Serpent centered on the possibility that Gómez and his colleagues might finally locate one such tomb, and thereby solve one of the city’s most fundamental enduring mysteries. Gómez himself has entertained the idea. But as we clambered through the tunnel, he laid out a hypothesis that seemed to stem more directly from the mythological readings of the city laid out by scholars like Clemency Coggins and Michael Coe.
Fifty feet in, we stopped at a small inlet carved into the wall. Not long before, Gómez and his colleagues had discovered traces of mercury in the tunnel, which Gómez believed served as symbolic representations of water, as well as the mineral pyrite, which was embedded in the rock by hand. In semi-darkness, Gómez explained, the shards of pyrite emit a throbbing, metallic glow. To demonstrate, he unscrewed the nearest light bulb. The pyrite came to life, like a distant galaxy. It was possible, in that moment, to imagine what the tunnel’s designers might have felt more than a thousand years ago: 40 feet underground, they’d replicated the experience of standing amid the stars.
If, Gómez suggested, it was true that the layout of the city proper was meant to stand in for the universe and its creation, might the tunnel, beneath the temple devoted to an all-encompassing aqueous past, represent a world outside of time, an underworld or a world before, not the world of the living but of the dead? Up above, there was the Temple of the Sun and the eternal day. Down below, the stars—not of this earth—and the deepest night.
I followed Gómez down a short ramp and into the cross-shaped chamber directly under the heart of the Temple of the Plumed Serpent. Four archaeologists were kneeling in the dirt, brushes and thin-bladed trowels in hand. A nearby boombox blared Lady Gaga.
Gómez told me he had not been prepared for the sheer diversity of the objects he encountered in the farthermost reaches of the tunnel: necklaces, with the string intact. Boxes of beetle wings. Jaguar bones. Balls of amber. And perhaps most intriguingly, a pair of finely carved black stone statues, each facing the wall opposite to the entryway of the chamber.
Writing in the late 1990s, Coggins speculated that religious tradition at Teotihuacán would have been “perpetuated in the linked repetition of ritual,” likely on the part of a priesthood. That ritual, Coggins went on, “would have concerned the Creation, Teotihuacán’s role in it, and probably also the birth/emergence of the Teotihuacán people from a cave”—a deep and dark hole in the earth.
Gómez gestured at the area where the twin figures once stood. “You can imagine a scenario where priests come down here to pay tribute to them,” he explained—to the Creators of the universe, and of the city, one and the same.
Gómez has one more crucial task to undertake: the excavation of three distinct, buried sub-chambers located below the resting place of the figurines, the final sections of the tunnel complex as yet unexplored. Some scholars speculate that the elaborate ritual offerings on display here, and the presence of pyrite and mercury, which held known associations with the supernatural among ancient Mesoamericans, provide further evidence that the buried sub-chambers represent the entryway to a particular type of underworld: the place where the city’s ruler departed the world of the living. Others argue that even the discovery of long-sought human remains buried in spectacular fashion would hardly close the book on the mystery of Teotihuacán’s rulers: Whoever is buried here could be just one ruler among many, perhaps even some other kind of holy person.
For Gómez, the sub-chambers, whether they are filled with more ritual relics, or remains, or something entirely unexpected, might be best understood as a symbolic “tomb”: a final resting place for the city’s founders, of gods and men.
A few months after leaving Mexico, I checked in with Gómez. He was only marginally closer to uncovering the chambers beneath the end of the tunnel. His archaeologists were literally often working with toothbrushes, so as not to damage whatever lay beneath.
Regardless of what he found at the end of the tunnel, once his excavation was complete, he promised me, he’d be satisfied. “The number of artifacts we’ve uncovered,” he said, pausing. “You could spend a whole career evaluating the contents.”
Logarithms
A very important class of functions in many areas of math are the logarithms.
There are many ways to define them, because they are so fundamental to math.
First, a note on terminology: sometimes log(x) means “a generic function in the family,” and sometimes it means “a specific function (usually the ones called “log base 10″ or “log base e/ the natural log”). I am using it to mean anything in the family.
A logarithm is a function, usually from the positive real numbers to the real numbers (but some really cool things happen when you extend it to the complex plane).
Here are some properties:
Logarithms are continuous, monotone (always increasing or decreasing, but never changing direction), surjective (give me a real number. There is some number whose logarithm is it), and have the very cool relation that log(x)+log(y) = log(xy) (which implies that log(1)=0).
Ways of defining them:
The more “high school math class” way of defining them is “the inverse function of an exponential function.”
Many methods to defining the class are to define a single logarithm, and then prove that all logarithms are the defined one times a constant.
Another way of defining them is through the properties “continuous and log(x)+log(y)=log(xy)”
A third method is using calculus and saying “its derivative is 1/x” or “log(x) = the integral from 1 to x of 1/y dy,” which are equivalent.
Pulque is a viscous, milk-colored, alcoholic beverage produced by fermenting the sap obtained by the maguey plant. Until the 19th and 20thcentury, it was probably the most widespread alcoholic beverage in Mexico.
In ancient Mesoamerica pulque was a beverage restricted to certain group of people and to certain occasions. The consumption of pulque was linked to feasting and ritual ceremonies, and many Mesoamerican cultures produced a rich iconography illustrating the production and consumption of this beverage. The Aztec called this beverage ixtac octli which means white liquor. The name pulque is probably a corruption of the term octli poliuhqui, or over-fermented or spoiled liquor.
Pulque Production
The juicy sap, or aguamiel, is extracted from the plant. An agave plant is productive for up to a year and usually the sap is collected twice a day. Neither fermented pulque nor the straight aguamiel can be stored for long time; the liquor needs to be consumed quickly and even the processing place needs to be close to the field.
The fermentation starts in the plant itself, since the microorganisms occurring naturally in the maguey plant start the process of transforming the sugar into alcohol. The fermented sap was traditionally collected using dried bottle gourds, and it was then poured into large ceramic jars where the seeds of the plant were added to accelerate the fermentation process.
Among the Aztecs/Mexica, pulque was a highly desired item, obtained through tribute. Manycodices refer to the importance of this drink for nobility and priests, and its role in Aztec economy.
Pulque Consumption
In ancient Mesoamerica, pulque was consumed during feasting or ritual ceremonies and was also offered to the gods. Its consumption was strictly regulated. Ritual drunkenness was allowed only by priests and warriors, and commoners were permitted to drink it only during certain occasions. Elderly and occasionally pregnant woman were allowed to drink it. In the Quetzalcoatl myth, the god is tricked into drinking pulque and his drunkenness caused him to be banished and exiled from his land.
According to indigenous and colonial sources, different types of pulque existed, often flavored with other ingredients such as chili peppers.
Pulque Imagery
Pulque is depicted in Mesoamerican iconography as white foam emerging from small, rounded pots and vessels. A small stick, similar to a straw, is often depicted within the drinking pot, probably representing a stirring instrument used to produce the foam.
Images of pulque-making are recorded in many codices, murals and even rock carvings, such as the ball court at El Tajin. One of the most famous representations of the pulque drinking ceremony is at the pyramid of Cholula, in Central Mexico.
The Mural of the Drinkers
In 1969, a 180 feet long mural was discovered by accident in the pyramid of Cholula. The collapse of a wall exposed part of the mural buried at a depth of almost 25 feet. The mural, dubbed the Mural of the Drinkers, portrays a feasting scene with figures wearing elaborate turbans and masks drinking pulque and performing other ritual activities. It has been suggested that the scene portrays pulque deities.
The origin of pulque is narrated in many myths, most of them linked to the goddess of maguey,Mayahuel. Other deities directly related to pulque were the got Mixcoatl and the Centzon Totochtin (the 400 rabbits), sons of Mayahuel associated with the pulque’s effects.
Sources
This glossary entry is a part of the About.com guide to Mesoamerica , and the Dictionary of Archaeology.
Bye, Robert A., and Edelmina Linares, 2001, Pulque, in The Oxford Encyclopedia of Mesoamerican Cultures, vol. 1, edited by David Carrasco, Oxford University Press.pp: 38-40
Taube, Karl, 1996, Las Origines del Pulque, Arqueología Mexicana, 4 (20): 71
Mayahuel, The Aztec Goddess of Maguey
Mayahuel was the Aztec goddess of maguey, as well as one of the protectors of fertility. This deity played an important role in ancient Central Mexico, since it is associated with the origin of pulque.
Mayahuel Myth
According to the Aztec myth, the god Quezalcoatl decided to provide humans with a special drink to celebrate and feast and gave them pulque. He sent Mayahuel, goddess of maguey, to the earth and then coupled with her. To avoid the rage of her grandmother and her other ferocious relatives the goddesses Tzitzimime, Quetzalcoatl and Mayahuel transformed themselves into a tree, but they were found out and Mayahuel was killed. Quetzalcoatl collected the bones of the goddess and buried them, and in that place grew the first plant of maguey. For this reason it was thought that the sweet sap, the aguamiel, collected from the plant was the blood of the goddess.
A different version of the myth tells that Mayahuel was a mortal woman who discovered how to collect aguamiel, and her husband Pantecalt discovered how to make pulque.
Mayahuel Imagery
Mayahuel was also defined as “the woman of the 400 breasts”, probably referring to the many sprouts and leaves of maguey and the milky juice produced by the plant and transformed into pulque. The goddess has many breasts to feed her many children, the Centzon Totochtin or “the 400 rabbits”, who were the gods associated with the effects of excessive drinking. In codices, Mayahuel is depicted as a young woman, with many breasts, emerging from a maguey plant, holding cups with foaming pulque.
Sources
This glossary entry is a part of the About.com guide to Aztec Gods, and the Dictionary of Archaeology.
Miller, Mary, and Karl Taube, 1993, The Gods and Symbols of Ancient Mexico and the Maya: An Illustrated Dictionary of Mesoamerican Religion. London: Thames & Hudson.Taube, Karl, 1996, Las Origines del Pulque, Arqueologia Mexicana, Vol.7, N. 20, p.71.
The Domestication History of Agave Americana or Maguey
Maguey or Agave americana is a native plant from Mexico, and it is now cultivated in many parts of the world. This plant, also known as the century plant or American aloe, is neither an aloe nor a cactus, as it is sometimes erroneously believed, but instead is a member of the Agavaceae family. Maguey is one of the many species of agave plants that exist in the Americas. They grow in semi-arid environments from the sea level to an altitude of about 9,000 feet.
Archaeological evidence from Guitarrero Cave indicates that agave was used at least as early as 12,000 years ago by Archaic foraging groups, to obtain fibers for clothing, bags and to make tools. There is no direct evidence of the process of domestication of agave, and only an handful of species, of the hundreds existing in nature, have been fully domesticated.
Agave americana grows in the semi-arid highlands of Mexico and has been used for many purposes both in pre-Hispanic as well as Colonial and modern times.
Despite its importance in ancient Mesoamerican societies, very little is known about the process of domestication of this species.
Agave Products
In ancient Mesoamerica, maguey was first collected and then cultivated and used for a variety of purposes. From its leaves people obtained fibers to make ropes, textiles, as well as construction materials, and fuel. Its thorns were an important tool used as perforators in bloodletting rituals. However, the most important product obtained from maguey was a mildly alcoholic beverage called pulque, obtained by the fermentation of aguamiel, (“honey water” in Spanish), the sweet, milky juice extracted from the plant.
Maguey Fibers
To make textiles, maguey fibers must be obtained by processing the leaves. The leaves are cut from the body of the plant and the spines removed. The leaves are then cooked in an oven to make then tender. Once they are cooked, the leaves are used as food or scraped over a pounding slab to obtain long threads to make ropes or to spin to weave textiles.
Mescal
The word mescal (sometimes mezcal) comes from two Nahuatl terms melt and ixcalli which mean “oven-cooked agave”. To produce mescal, the ripe maguey plants are cut down and the leaves removed. The core, or head of the plant is then baked into earth ovens. Once the agave core is cooked, it is ground to extract the juice. The juice is then put into containers and let there to ferment. When the fermentation is complete, alcohol (ethanol) is separated from the non-volatile elements through distillation to obtain the pure mescal.
Archaeologists are still debating if this alcoholic beverage was known in pre-Hispanic times or if it was an innovation of the Colonial period. Distillation was a well-known process in Europe, derived from Arabic traditions, whereas the evidence from pre-Hispanic Mesoamerica and the accounts from the early Contact period about this process are not straightforward.
However, recent investigations in the site of Nativitas, Tlaxcala, Central Mexico, are providing an interesting set of evidence about the possibility of mezcal production in pre-Hispanic times.
Archaeological Evidence for the Use of Maguey
Because of their organic nature, products derived from maguey are rarely identifiable in the archaeological record. Evidence of maguey use comes instead from the technological implements used to process and store the plant and its derivatives. Stone scrapers with plant residue evidence from processing agave leaves are abundant in Classic and Postclassic times, along with cutting and storing implements. Such implements are rarely found in Formative and earlier contexts.
Ovens - probably used to cook maguey heads - have been found in archaeological sites, such as Nativitas in the state of Tlaxcala, Central Mexico, at Paquimé, Chihuahua, and La Quemada, Zacatecas. At Paquimé, remains of agave were found in place within one of these subterranean ovens. In the site of Nativitas,archaeologists identified several large jars, possibly used to store the maguey sap during the fermentation process, or used as distillation devices.
In Western Mexico, ceramic vessels with depiction of agave plants have been found in several burials, dated to the Classic period. These elements underscore the important role that this plant played in the economy as well as social life of the community.
The Aztecs/Mexica had a specific patron deity for this plant, the goddess Mayahuel.
Finally, many Spanish chroniclers, such as Bernardino de Sahagun, Bernal Diaz del Castillo and fray Toribio de Motolinia stress the importance that this plant and its products had within the Aztec empire.
Sources
This entry on Agave is a part of the About.com guide to Ancient Mesoamerica, and the Dictionary of Archaeology.
Parsons, Jeffrey R. and Parsons Mary H., 1990, Maguey Utilization in Highland Central Mexico: an archaeological ethnography. Anthropological Papers, Museum of Anthropology, University of Michigan ; no. 82. Ann Arbor.
Rakita, Gordon F.M.,2006, Emergent Complexity, Ritual Practices, and Mortuary Behavior at Paquimé, Chihuahua, Mexico, in Religion in the Prehispanic Southwest, edited by Christine S. VanPool, Todd L. VanPool, and David A. Phillips, Jr., AltaMira Press, Lanham
Serra, M. Carmen and Carlos A. Lazcano, 2010, The Drink Mescal: Its Origin and Ritual Uses, in Pre-Columbian Foodways. Interdisciplinary Approaches to Food, Culture, and Markets in Ancient Mesoamerica, edited by John Staller and Michael Carrasco. Springer
Mmmmm 😋
Vector space problem. I would like to get many friend who like mathematics too as much as I like.
If you have a chance to proof and make this problem share your answer, and thanks for reading.
