A small group of teenagers were led by an older woman as they sang and danced. They stopped outside people’s houses and waited for presents to be brought out to them.
30. The Iroquois celebrated a winter festival similar to trick-or-treating. A small group of teenagers were led singing and dancing around the village by an older woman. They stopped outside people’s houses and waited for presents to be brought out to them.
Scientists believe that pumpkins originated in North America about 9000 years ago. The oldest pumpkin seeds have been found in Mexico and date back to somewhere between 7000-5550 B.C.. Pumpkins (along with other forms of squash) were a historically important food staple among Native Americans.
Pumpkins (along with other forms of squash) were a historically important food staple among Native Americans. They would grow the squash along river banks next to maize and beans, a planting technique that was called the “Three Sisters Method,” which allowed the three crops to sustain each other. Corn served as the trellis upon which the beans could climb; beans were nourished by the sunlight and kept the corn stalks stable on windy days, while also nourishing their soil; and pumpkins sheltered the corn’s shallow roots and prevented weeds from taking hold.
Where do pumpkins come from originally?
Pumpkins are a member of the gourd family, which includes cucumbers, honeydew melons, cantaloupe, watermelons and zucchini. These plants are native to Central America and Mexico, but now grow on six continents—all but Antarctica.
In Vanuatu, before making a sand drawing, the artist traces a grid that can be circular (pictured) or rectangular.
Credit:
Alban Da Silva
In October 2015 my time training mathematics teachers at a French high school in Port Vila, Vanuatu, was coming to an end. The principal invited me to share kava, a traditional drink in the country. As every social scientist in Vanuatu discovers, sharing kava is a fruitful opportunity for learning. This beverage, which is made from the roots of a tree of the same name, relaxes the drinker and loosens the tongue.
This first encounter with kava was also my introduction to sand drawing. That evening, one of the trainees took out a large board covered with very fine sand. After carefully flattening the surface, he drew a grid of horizontal and vertical lines. Then he began tracing furrows in the sand without ever lifting his finger. When the artist finished, he explained in the language Bislama, “Hemia hem i wan fis i ronwe i stap unda ston from i kat wan sak,” meaning “It is a fish that hides under a stone to escape the shark.”
The fluidity of the line, mixed with the effects of kava, plunged me into a state of wonder. The technique reminded me of the classic challenge to draw a complex figure with a single stroke, without lifting one’s pen or going over the same line twice. It also called to mind a “Eulerian graph” in mathematics, which involves a trail that traverses every edge exactly once while starting and ending at the same point.
As I considered these ideas, an intern approached me and whispered, “Where is the mathematics in this drawing, teacher?” Though he could not have known it, that remark would go on to shape the next six years of my life, including my doctoral work on sand drawing. One question particularly inspired me: How were such drawings created?
Sand drawings are complex figures drawn with a single finger stroke. Credit: Alban Da Silva
My investigation took me further than I could have imagined. By watching expert sand artists, learning about their methods, collecting drawings and history and exploring the work of 20th-century ethnologists, I have developed a mathematical model of sand drawing. My work shows that these artworks can be modeled as the result of algorithms and operations of an algebraic nature. Indeed, mathematical language turns out to be appropriate for describing the work of sand drawing experts. Furthermore, sand drawing can help us understand the relationships that Vanuatu societies maintain with their environment.
Vanuatu is an archipelago with a population of some 315,000 people spread throughout 83 islands. The country has the highest linguistic density in the world, with 138 vernacular languages. The two official languages taught in school are French and English. Bislama, or bichlamar, an Anglo-Melanesian pidgin used in Vanuatu, is the common language.
A map shows a portion of the archipelagic country of Vanuatu. Ethnomathematician Alban Da Silva studied sand drawing in the nation’s province of Penama, specifically on the islands of Maewo and Pentecost. Credit: Rainer Lesniewski/Alamy Stock Vector (map); Amanda Montañez (green and purple outlines and key)
Cultures vary in the north and south of the country and even within the same island. The sand drawing practice is widespread only in some central islands, for example. Although the tradition is reminiscent of drawings done on soil in Tamil Nadu, India, it is unique in many ways. In 2008 UNESCO classified the sand drawing of Vanuatu as part of the intangible cultural heritage of humanity.
My research is based on two field surveys that were conducted on Maewo Island in 2018 and Pentecost Island in 2019 and that particularly focused on drawings made by people in the Raga region (pronounced “Ra-ra”) on northern Pentecost Island. These islands, along with Aoba Island, constitute the province of Penama and are bound by common traditions, which greatly facilitated my research.
“Sand drawing,” or sandroing, as it is known in Bislama, is probably thousands of years old. Traditionally, it consists of a person drawing a continuous, closed line with their finger in beaten earth, sand beaches or ashes. (The words “continuous” and “closed” have the same meaning here as in mathematics: a drawing in the sand is similar to the closed continuous curve of a plane.) This drawn line is constrained by a composite grid of lines or dots. The grid can be rectangular or circular.
Although it is difficult to know how many designs are in use, it is clear that, over time, new ones appear, and others disappear. A system very close to intellectual property protects these drawings, making access to this traditional knowledge sometimes sensitive and challenging.
These artworks are multidimensional in their significance. Some iconic drawings of animals, insects or plants are closely linked with the beliefs, cosmogonies, social organization or even traditions of these societies—which are grouped together under the generic name of kastom. The drawings can also support narratives; they reveal the ethical or political dimensions of societies in central Vanuatu. In many cases, each design bears a vernacular name related to these different aspects.
A sand drawing evokes a fish hiding under a stone to escape a predator. Credit: Alban Da Silva
Today these societies recognize this practice as a traditional graphic art that helps people recall ritual, religious and environmental knowledge. In addition, Jief Todali, a chief whom I met in the Raga region, explained to me that the artists are spokespeople: “Before the arrival of the tuturani [the white foreigners], the people of northern Pentecost did not know how to speak. They expressed themselves through drawings that they traced on the ground with their fingers. Instead of people, the rocks, the stones, the ground of the hills and valleys, the wind, the rain, the water of the sea spoke. But now the situation is reversed. It is the people who speak, and the earth, the wind, the rain and the sea are silent. Now [the people from the Raga region] sometimes say, ‘We have to speak for the land because it can no longer speak for itself.’”
Finally, this ephemeral art—each drawing is erased once it is finished—stimulates storytelling. Practitioners generally pair their drawings with the telling of a tale, and the most gifted ones are able to do this while drawing. It is not uncommon for them to appeal to the imagination of spectators by adding details related to their history, including familiar places, characters, animals and even vegetables.
EXPERTS AND RULES
There are different levels of expertise and practice. Some people don’t practice sand drawing at all. Others know a few rather simple drawings. “Experts”—designated as such by the rest of their society—have an impressive repertoire (up to 400 drawings, according to some). While the first ethnographies of sand drawing mentioned that this art was reserved for men, that is not the case today. Several women I met had a high level of expertise.
From beginners to experts, everyone follows a set of “rules.” Because these communities have an oral tradition, there is no written record, but during my field survey, I drew up a list of principles that are followed in most cases. All drawings begin with a grid that provides support and defines a set of nodes, or crossing points, and lines.
An animation reveals a classic turtle-shaped sand drawing design. Credit: Alban Da Silva
Rules then indicate the movements that are allowed. More precisely, an artist must (1) go from node to node without crossing the same path or cutting the grid other than at its nodes and (2) must return to the starting point without lifting their finger.
There was an additional rule that I discovered in the field that turned out to be critical to my model, as discussed below.
MARCIA ASCHER’S INTUITION
I was not the first person to recognize the resemblance of these rules to concepts from mathematics. In fact, my thesis is a continuation of work carried out in the 1980s by American mathematician Marcia Ascher, a pioneer of ethnomathematics. In sand drawings, she argued, there was a clear connection to what mathematicians call graph theory and especially to Eulerian graphs.
To appreciate how revolutionary Ascher’s perspective was, consider that before her work and that of her contemporaries, scholars generally assumed that only societies with writing could truly practice mathematics. They constrained their investigations of mathematical knowledge to textual sources and ignored many other practices seen in societies with oral traditions that did not use a written language.
But since the advent of ethnomathematics, some scholars have begun to overturn these assumptions. The shift undoubtedly began in the 1940s, when mathematician André Weil demonstrated, in a now famous appendix to anthropologist Claude Lévi-Strauss’s book The Elementary Structures of Kinship, that the kinship rules of the Australian Yolngu followed what are called non-trivia group laws. Since then researchers have identified mathematical principles in many other places, including sowing games and divination in Madagascar, string games on Papua New Guinea’s Trobriand Islands, textiles in the Andes and ornamental window hangings on the island of Réunion.
Ascher’s work on sand drawing drew on that of a young ethnographer, Bernard Deacon, who had collected and transcribed more than 100 sand drawings from Malekula and Ambrym Islands in Vanuatu in 1926 and 1927. Ascher studied those figures and realized that these images could be described mathematically.
Each sand drawing could be thought of as a “graph” in the sense that it included vertices, or dots, connected by edges, or lines. (This definition of graph is a special one in mathematics. In graph theory, graphs are simple figures made up of vertices and edges to map out the connections of many nodes in a network—such as cities linked by roads, computers linked by the Internet or even people linked by social ties.)
Ascher observed that in the sand drawings, the line traced in the sand was comparable to a graph whose vertices included all the crossings created in the pattern. The edges were all the arcs between those vertices. These graphs were also Eulerian, meaning that the sand artist had to visit each edge only once and had to return to a starting point. Ascher also documented the number of edges per vertex, which mathematicians call the “degree” of the vertex. This was important because, according to Euler’s theorem, created by mathematician Carl Hierholzer, a connected graph is Eulerian only if each vertex has an even degree.
Credit: Alban Da Silva/Pour la Science, restyled by Amanda Montañez
The fact that Ascher never had the opportunity to go to Vanuatu to see the designers at work undoubtedly limited her research. My own research, therefore, could go further. One particular observation of Ascher’s propelled my study of how these images are created. By examining Deacon’s drawings, she had observed that some of these graphs could be broken down into three or four substeps that, when linked together, formed the final drawing. Put another way, each step produced a subdrawing that returned to the starting point. When these subdrawings were superimposed, the final drawing was revealed. I called this process of identifying subdrawings “decomposition.”
These subdrawings raised several questions for my work: Is this type of decomposition possible for every drawing? How did these layers fit together, and did the order matter? Above all, what did these layers represent for the sand artists?
To investigate, I sought to first refine the graph model proposed by Ascher and then determine whether computers could generate an automated breakdown of the sand drawings. Ascher had proposed modeling sand drawings with a set of patterns subjected to transformations (such as rotation, translation, symmetry and inversion). But while listening to the creators of sand drawings, I realized that transformations did not play a fundamental role in the execution of their designs. In fact, they used the same term to designate a motif, whatever its orientation on the grid. I therefore needed another way to describe the approach of these sand artists.
The artists themselves helped me in this effort. Graphs are very useful for modeling a set of relationships (edges) between objects (vertices). But in her work, Ascher did not consider the nature of these relations. By emphasizing the nodes of the grid, the vertices of the graph and the curves at the edges, she overlooked the ways in which the artist would move from one peak to another. While questioning the experts, I observed that, for them, the direction of the movement is as important as the nodes: the designer’s finger moves from node to node, from an initial direction to a final direction, so that the nodes play different roles when they are crossed, according to one or the other of the two possible diagonals.
At each node or crossing point within this sand drawing, the finger takes one of two diagonal paths (purple and blue). Credit: Alban Da Silva/Pour la Science, restyled by Amanda Montañez
To model the sand design with a graph, therefore, it is possible to create a graph in which every node of the grid is treated not as one but rather two vertices that are each assigned to a diagonal. We thus obtain a new graph—named Gmod—whose vertices are not the nodes S of the grid, as Ascher proposed, but instead the pairs (S, d).
Credit: Alban Da Silva, restyled by Amanda Montañez
In this graph, d is the direction taken, and each movement of the drawing that leaves from S in the direction d and arrives in S’ in the direction of d’ corresponds to an edge between the vertices (S, d) and (S’, d’). And this resulting Gmod graph is still Eulerian!
Credit: Alban Da Silva, restyled by Amanda Montañez
A THEOREM DISCOVERED IN DRAWINGS
In 1912 mathematician Oswald Veblen identified another characteristic of Eulerian graphs in what has since been called Veblen’s theorem: a graph is Eulerian if and only if it can be broken down into a union of disjoint cycles. In graph theory, the word “cycle” refers to a sequence of distinct consecutive edges whose start and end vertices are identical.
It turns out that the cycles of the Gmod graph correspond to those of a sand drawing, which can therefore be broken down as a disjoint union of cycles.
Does this approach distance us from sand drawers? I would argue that it does not. On the contrary, cycles can provide keys to better understanding their approach. Of the 60 or so drawings that I have collected, I have noticed that artists sometimes take breaks in their drawing when they complete a cycle. Moreover, when a sand drawer is forced to find another path (a rod in Bislama), they tend to rearrange the typical cycle order that they use in their drawing or to try to find another decomposition into cycles.
Finally, some cycles have vernacular names, which suggests that they are like building blocks for the artists. In fact, this focus on graph cycles also seems to echo the stories that accompany the drawings, which play a fundamental role in the way Vanuatu societies understand the world. In my research, I have also suggested that some cycle decompositions might be linked to the way these societies conceive of their relationship with nonhumans.
Credit: Alban Da Silva
These results raise questions about the universality of mathematics and the form that math takes in other cultures. They open up perspectives for teaching mathematics as well. Since 2010 the acquisition of traditional knowledge such as sand drawing has been one of the objectives of Vanuatu schools, and it is part of a larger movement to decolonize education, much like efforts in Hawaii and in the French territory of New Caledonia. In the current school curriculum, however, no link is made between sand drawing and mathematics. To that end, Vanuatuan Pierre Metsan, a doctoral student in education at the University of New Caledonia, is studying whether the practice of sand drawing could support mathematics instruction. We can look forward to what he learns from this investigation in the years to come.
This article originally appeared in Pour la Science and was reproduced with permission.
The Basque were among the first European fishermen to visit North America. Writers and historians have debated whether the Basques may have reached North America before Christopher Columbus's arrival in 1492. Jean-Pierre Proulx in "Basque Whaling in Labrador in the 16th Century", published by the Canadian Parks Service 1993, claims the earliest known archival documents referring to Basques in North American date to the early 16th Century, beginning in 1517, and that there are no oral traditions in the Basque country of a pre-Columbian exploration of the New World. Fifteenth and 16th century Basque seafarers were not explorers or colonists in the traditions of some other European nations. Indeed, rather than publicize their voyages, the Basques desired to keep their discoveries secret in order to protect them from competitors. Others claim that some Basque records show that the Basque had settled in Newfoundland before 1300 and claim that Spanish records show that taxes on whalers in Newfoundland and Labrador were paid in 1372. Documents translated in 1968 by B. Robertson state that in 1514: "the fishermen of the island of Brehat paid tithes (of Newfoundland fish) since sixty years" (since 1454). It is known that from the 12th to the 15th century that the Basques pursued an intensive whale hunt during winter months in their home waters, the Bay of Biscay. Late in this period the Basque began expanding their activities northward, reaching Iceland by the year 1412 according to one writer. Folk sagas of Greenland and Iceland tell of voyages to a new world, long before the voyages of Columbus and Cabot. Since no extensive archaeological studies were conducted in the Placentia area, it is difficult to determine just when the Basque fishermen came here. By the time permanent settlements were established along Newfoundland's south coast, the migratory ship fishery from Europe was more than a century old. The French Basque, who entered the fishery in 1520's were now dominant. This record shows a copy of a ship's log in 1565 on a voyage to Placentia. Two English captains reported 60 Basque vessels in the harbour of Placentia (a Basque place name) in 1590's. Little is known of the pre-planter phase but by 1600 the Portuguese and the Spanish Basque, prominent early in the century, had largely abandoned the waters. It is likely the Basque named Placentia for their own Placentia which is a sea level town at the base of Pyrenees mountains. Caesar camped there on his way to Gaul. Both communities are 'level discs within a womb of hills'. The Basque fishery extended from southern Newfoundland into the Gulf of St. Lawrence to Gaspe' and Labrador, and along the shores of Cape Breton and southern Nova Scotia. Bayonne and St. Jean de Luz dominated this vast territory; Ciboure and Cape Breton were secondary ports. Pasajaes in northern Spain had been an important port of departure for the French Basque fleet through the 16th century. Following the collapse of the Spanish fishery in the wake of the Armada, the French Basque redirected their voyages north to Bordeaux and LaRochelle, where they packed up supplies and sometimes crew en route to southern Newfoundland. This explains in part the mixture of Basques, Bordelais and Rochellais across the Atlantic. The Basque fishery, was well established at Placentia. The expanse beaches permitted the fishermen to dry their catches. The decision by France to start a settlement at Placentia led to numerous disputes over boundaries and access to the beach as the harbour was crowded with its mix of residential, migratory and military fisheries. This frustration was displayed in The Basque Revolt of 1690. Several tombstones, the earliest dated 1676, indicate that there was a Basque cemetery on the Placentia beach near the present Anglican church.
The theory that Basque whalers and other fishermen from towns on the Cantabrian coast had traveled to Newfoundland (Canada), around the year 1375, long before Christopher Columbus did so, has little historical evidence and only one certainty: the Spanish left a deep footprint in northwestern Canada. Thus, when French navigator Jacques Cartier named Canada and claimed these new territories – the Newfoundland – for the French Crown, he wrote down a surprising finding in his letters: «In those remote waters I found a thousand Basques fishing for cod».
Around the year 1001, ‘The Icelandic Viking Sagas’ locates the expeditions of the explorer Leif Ericson in Helluland, Markland and what he
called Vinland (“Pasture Land”). And archaeological investigations have indeed confirmed the existence of a Norse settlement, ‘L’Anse aux Meadows’, in Newfoundland, which was declared World Heritage by UNESCO in 1978. In any case, this Viking presence in America, which even genetic studies have endorsed, was ephemeral, and in no case were there settlements on the American mainland.
Map of the whales of the North Atlantic, in 1592.
The Viking raids were supposedly succeeded by the Basques. According to the strictest version of the legend, the Basques arrived in Newfoundland around the 14th century and decided to keep it a secret to avoid sharing the area’s prodigious fishing grounds with other fleets. Between myth and reality, it is related that when French explorers came into contact with the natives of Newfoundland, they greeted them with the formula “Better priests!” (“The priests are better!” in Basque), which Basque sailors used as a response if someone asked them about their health.
Like it’s some kind of quest for the holy grail several decades before Columbus, Portuguese navigators also undertook the voyage to Bacalao Island (also called “Bachalaos”), diffusely represented on 16th-century maps in the vicinity of Newfoundland. So the Portuguese Joao Vaz Royal Court would have reached the vicinity of Newfoundland in 1472, and it is even speculated that it bordered the banks of the hudson river and the San Lorenzo.
Throughout the following century, different European fishermen of both whales and cod settled permanently in Newfoundland. According to the doctoral thesis of Caroline Ménard ‘Galician fishing in Newfoundland, XVI-XVIII centuries’ (Santiago de Compostela University, 2006), the beginning of cod fishing among the Basques, Bretons and Normans in this region is usually placed in the year 1504. The French were followed by the Portuguese, and then the Galicians. The first trip to Newfoundland undertaken by a Galician was recorded in 1517, in the town of Pontevedra specifically, and recorded in a rental contract that brought together a merchant from Pontevedra, Fernando de la Torre, with a sailor from Betanzos, Juan de Betanços, so that the latter would serve the former in a campaign to fish for cod, for a salary of about five gold ducats.
From that year on, commercial fishing, cultural and possibly genetic exchanges were very frequent between Galician, Basque (Biscayan and Gipuzkoan) and Basque fishermen. Newfoundland Native Americans. In 1527, an English ship arrived in Newfoundland and encountered some 50 Spanish, French, and Portuguese vessels fishing. The Spanish factories scattered along the coasts of Newfoundland, Labrador and the Gulf of Saint Lawrence gathered up to 9,000 people in some seasons and constituted the first great industry in the history of North America.
The main source of profit for the whale was in the fat of the animal, later converted into an oil which they called saín
Originally a large center for cod, the Island of Newfoundland evolved into a preferred target for whalers. The whaling tradition in the Bay of Biscay It dates back to the Middle Ages and was an important driving force for the coastal towns. The main source of profit was in the fat of the animal, later converted into an oil that they called saín. This product was used for lighting and burned without giving off smoke or smell. Likewise, the bones served as construction material for making furniture. The meat was hardly consumed in Spain, but it was salted and sold to the French.
As a result of the exhaustion in the Catabric of these animals, which came here only during their calving period, it was inevitable that these fishermen would make the leap in search of other fishing grounds. In the decades from 1530 to 1570, the whaling business registered its heyday. The fleet came to be made up of some thirty ships, manned by more than two thousand men, who captured some four hundred whales each year.
The Footprint In Newfoundland
The annual itinerary of the fishermen began with their departure from the Iberian Peninsula in the second week of June. The Atlantic crossing took about 60 days, reaching Newfoundland in the second half of August, in time to intercept the whales on their autumn migration from the Arctic Ocean to the South Seas. The hunt lasted until the end of the year, when the arrival of winter covered the waters of the bay with ice and made navigation very complicated. That is why only the boats that had not managed to catch a good catch stayed in North America during the winter season. The return trip was usually shorter, between 30 and 40 days, thanks to favorable currents and winds.
Island of Newfoundland, seen from a satellite.
As the 17th century progressed, the decline of both cod fishermen and whalers from the Peninsula accelerated. The entry onto the American scene of French, English, Danish, and Dutch sailors, among others, severely compromised activity in Newfoundland. The King of France It came to prohibit the Spanish from fishing in its waters, refusing to provide them with passports and preventing French sailors from embarking on Spanish ships, a practice that had been carried out because the French were necessary for the work of trenching cod. The Treaty of Utrecht, which marked the passage of Newfoundland from French to English hands, was the final blow to an industry that was no longer as profitable as it used to be.
Without a strong naval presence in the area, Spanish fishermen depended on agreements with the French and English, which made things even more difficult for them. With a great demand for cod in Spain, English fishermen rose in a few years as the most important suppliers of these fish, which entered the Peninsula through Galicia and whose waters did not meet the needs of the country. The last thing the British wanted was for Galicians to go to Newfoundland to buy the merchandise they sold in Spain.
Both Basques and other peninsulars left a deep mark on the inhabitants of the Island of Newfoundland. Many of the current names of cities and other places are of Spanish origin. As an example, the city Port aux Basques it is present on maps of 1612; Port-au-Choix is a distortion of Portuchoa, “little port”; and Ingonachoix (Aingura Charra) translates as “bad anchoring”. Galician references can also be found in the toponymy. The name of Ferrol appears on a map of Newfoundland from 1674 to distinguish the northern tip of the island.
A Montana Man Has The Oldest DNA Native To America, And It Changes What We Know About Our Ancestors
Native American man's DNA traced back 17000 years Darrell “Dusty” Crawford, whose Native American Blackfoot name is Lone Bull, looks over his results with fascination. He’s taken a DNA test with an outfit called Cellular Research Institute (CRI) and learned a lot about his heritage. What he doesn’t know yet, though, is that the conclusions will also have implications for all Native Americans.
