In October 2015 my time coaching arithmetic academics at a French highschool in Port Vila, Vanuatu, was coming to an finish. The principal invited me to share kava, a standard drink within the nation. As each social scientist in Vanuatu discovers, sharing kava is a fruitful alternative for studying. This beverage, which is made out of the roots of a tree of the identical identify, relaxes the drinker and loosens the tongue.
This primary encounter with kava was additionally my introduction to sand drawing. That night, one of many trainees took out a big board coated with very advantageous sand. After fastidiously flattening the floor, he drew a grid of horizontal and vertical traces. Then he started tracing furrows within the sand with out ever lifting his finger. When the artist completed, he defined within the language Bislama, “Hemia hem i wan fis i ronwe i stap unda ston from i kat wan sak,” which means “It is a fish that hides under a stone to escape the shark.”
The fluidity of the road, combined with the consequences of kava, plunged me right into a state of marvel. The approach jogged my memory of the traditional problem to attract a posh determine with a single stroke, with out lifting one’s pen or going over the identical line twice. It additionally known as to thoughts a “Eulerian graph” in arithmetic, which entails a path that traverses each edge precisely as soon as whereas beginning and ending on the similar level.
As I thought-about these concepts, an intern approached me and whispered, “Where is the mathematics in this drawing, teacher?” Although he couldn’t have recognized it, that comment would go on to form the following six years of my life, together with my doctoral work on sand drawing. One query notably impressed me: How have been such drawings created?
My investigation took me additional than I might have imagined. By watching professional sand artists, studying about their strategies, gathering drawings and historical past and exploring the work of Twentieth-century ethnologists, I’ve developed a mathematical mannequin of sand drawing. My work exhibits that these artworks might be modeled as the results of algorithms and operations of an algebraic nature. Certainly, mathematical language seems to be acceptable for describing the work of sand drawing consultants. Moreover, sand drawing can assist us perceive the relationships that Vanuatu societies keep with their surroundings.
[Read more about ethnomathematics]
A Conventional Artwork
Vanuatu is an archipelago with a inhabitants of some 315,000 folks unfold all through 83 islands. The nation has the best linguistic density on this planet, with 138 vernacular languages. The 2 official languages taught in class are French and English. Bislama, or bichlamar, an Anglo-Melanesian pidgin utilized in Vanuatu, is the frequent language.
Cultures range within the north and south of the nation and even inside the similar island. The sand drawing follow is widespread solely in some central islands, for instance. Though the custom is harking back to drawings achieved on soil in Tamil Nadu, India, it’s distinctive in some ways. In 2008 UNESCO categorized the sand drawing of Vanuatu as a part of the intangible cultural heritage of humanity.
My analysis relies on two discipline surveys that have been performed on Maewo Island in 2018 and Pentecost Island in 2019 and that notably targeted on drawings made by folks within the Raga area (pronounced “Ra-ra”) on northern Pentecost Island. These islands, together with Aoba Island, represent the province of Penama and are certain by frequent traditions, which enormously facilitated my analysis.
“Sand drawing,” or sandroing, as it’s recognized in Bislama, might be hundreds of years outdated. Historically, it consists of an individual drawing a steady, closed line with their finger in overwhelmed earth, sand seashores or ashes. (The phrases “continuous” and “closed” have the identical which means right here as in arithmetic: a drawing within the sand is just like the closed steady curve of a aircraft.) This drawn line is constrained by a composite grid of traces or dots. The grid might be rectangular or round.
Though it’s tough to know what number of designs are in use, it’s clear that, over time, new ones seem, and others disappear. A system very near mental property protects these drawings, making entry to this conventional data typically delicate and difficult.
These artworks are multidimensional of their significance. Some iconic drawings of animals, bugs or vegetation are carefully linked with the beliefs, cosmogonies, social group and even traditions of those societies—that are grouped collectively below the generic identify of kastom. The drawings also can assist narratives; they reveal the moral or political dimensions of societies in central Vanuatu. In lots of circumstances, every design bears a vernacular identify associated to those completely different facets.
At present these societies acknowledge this follow as a standard graphic artwork that helps folks recall ritual, non secular and environmental data. As well as, Jief Todali, a chief whom I met within the Raga area, defined 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.’”
Lastly, this ephemeral artwork—every drawing is erased as soon as it’s completed—stimulates storytelling. Practitioners typically pair their drawings with the telling of a story, and probably the most gifted ones are ready to do that whereas drawing. It isn’t unusual for them to enchantment to the creativeness of spectators by including particulars associated to their historical past, together with acquainted locations, characters, animals and even greens.
Specialists and Guidelines
There are completely different ranges of experience and follow. Some folks don’t follow sand drawing in any respect. Others know just a few fairly easy drawings. “Experts”—designated as such by the remainder of their society—have a formidable repertoire (as much as 400 drawings, in keeping with some). Whereas the primary ethnographies of sand drawing talked about that this artwork was reserved for males, that’s not the case right now. A number of girls I met had a excessive stage of experience.
From newbies to consultants, everybody follows a set of “rules.” As a result of these communities have an oral custom, there is no such thing as a written report, however throughout my discipline survey, I drew up a listing of rules which are adopted typically. All drawings start with a grid that gives assist and defines a set of nodes, or crossing factors, and contours.
An animation reveals a traditional turtle-shaped sand drawing design. Credit score: Alban Da Silva
Guidelines then point out the actions which are allowed. Extra exactly, an artist should (1) go from node to node with out crossing the identical path or chopping the grid apart from at its nodes and (2) should return to the start line with out lifting their finger.
There was a further rule that I found within the discipline that turned out to be essential to my mannequin, as mentioned beneath.
Marcia Ascher’s Instinct
I used to be not the primary particular person to acknowledge the resemblance of those guidelines to ideas from arithmetic. In reality, my thesis is a continuation of labor carried out within the Eighties by American mathematician Marcia Ascher, a pioneer of ethnomathematics. In sand drawings, she argued, there was a transparent connection to what mathematicians name graph theory and particularly to Eulerian graphs.
To understand how revolutionary Ascher’s perspective was, contemplate that earlier than her work and that of her contemporaries, students typically assumed that solely societies with writing might actually follow arithmetic. They constrained their investigations of mathematical data to textual sources and ignored many different practices seen in societies with oral traditions that didn’t use a written language.
However for the reason that introduction of ethnomathematics, some students have begun to overturn these assumptions. The shift undoubtedly started within the Forties, when mathematician André Weil demonstrated, in a now well-known appendix to anthropologist Claude Lévi-Strauss’s ebook The Elementary Constructions of Kinship, that the kinship guidelines of the Australian Yolngu adopted what are known as non-trivia group legal guidelines. Since then researchers have recognized mathematical rules in lots of different locations, together with sowing video games and divination in Madagascar, string video games on Papua New Guinea’s Trobriand Islands, textiles within the Andes and decorative window hangings on the island of Réunion.
Ascher’s work on sand drawing drew on that of a younger ethnographer, Bernard Deacon, who had collected and transcribed greater than 100 sand drawings from Malekula and Ambrym Islands in Vanuatu in 1926 and 1927. Ascher studied these figures and realized that these photographs may very well be described mathematically.
Every sand drawing may very well be considered a “graph” within the sense that it included vertices, or dots, related by edges, or traces. (This definition of graph is a particular one in arithmetic. In graph principle, graphs are easy figures made up of vertices and edges to map out the connections of many nodes in a community—similar to cities linked by roads, computer systems linked by the Web and even folks linked by social ties.)
Ascher noticed that within the sand drawings, the road traced within the sand was corresponding to a graph whose vertices included all of the crossings created within the sample. The sides have been all of the arcs between these vertices. These graphs have been additionally Eulerian, which means that the sand artist needed to go to every edge solely as soon as and needed to return to a place to begin. Ascher additionally documented the variety of edges per vertex, which mathematicians name the “degree” of the vertex. This was essential as a result of, in keeping with Euler’s theorem, created by mathematician Carl Hierholzer, a related graph is Eulerian provided that every vertex has a fair diploma.
The truth that Ascher by no means had the chance to go to Vanuatu to see the designers at work undoubtedly restricted her analysis. My very own analysis, due to this fact, might go additional. One specific statement of Ascher’s propelled my examine of how these photographs are created. By inspecting Deacon’s drawings, she had noticed that a few of these graphs may very well be damaged down into three or 4 substeps that, when linked collectively, shaped the ultimate drawing. Put one other method, every step produced a subdrawing that returned to the start line. When these subdrawings have been superimposed, the ultimate drawing was revealed. I known as this strategy of figuring out subdrawings “decomposition.”
These subdrawings raised a number of questions for my work: Is one of these decomposition doable for each drawing? How did these layers match collectively, and did the order matter? Above all, what did these layers symbolize for the sand artists?
To research, I sought to first refine the graph mannequin proposed by Ascher after which decide whether or not computer systems might generate an automatic breakdown of the sand drawings. Ascher had proposed modeling sand drawings with a set of patterns subjected to transformations (similar to rotation, translation, symmetry and inversion). However whereas listening to the creators of sand drawings, I noticed that transformations didn’t play a basic position within the execution of their designs. In reality, they used the identical time period to designate a motif, no matter its orientation on the grid. I due to this fact wanted one other method to describe the strategy of those sand artists.
The artists themselves helped me on this effort. Graphs are very helpful for modeling a set of relationships (edges) between objects (vertices). However in her work, Ascher didn’t contemplate the character of those relations. By emphasizing the nodes of the grid, the vertices of the graph and the curves on the edges, she ignored the methods by which the artist would transfer from one peak to a different. Whereas questioning the consultants, I noticed that, for them, the path of the motion is as essential because the nodes: the designer’s finger strikes from node to node, from an preliminary path to a ultimate path, in order that the nodes play completely different roles when they’re crossed, in keeping with one or the opposite of the 2 doable diagonals.
To mannequin the sand design with a graph, due to this fact, it’s doable to create a graph by which each node of the grid is handled not as one however fairly two vertices which are every assigned to a diagonal. We thus acquire a brand new graph—named Gmod—whose vertices aren’t the nodes S of the grid, as Ascher proposed, however as an alternative the pairs (S, d).
On this graph, d is the path taken, and every motion of the drawing that leaves from S within the path d and arrives in S’ within the path of d’ corresponds to an edge between the vertices (S, d) and (S’, d’). And this ensuing Gmod graph remains to be Eulerian!
A Theorem Found in Drawings
In 1912 mathematician Oswald Veblen recognized one other attribute of Eulerian graphs in what has since been known as Veblen’s theorem: a graph is Eulerian if and provided that it may be damaged down right into a union of disjoint cycles. In graph principle, the phrase “cycle” refers to a sequence of distinct consecutive edges whose begin and finish vertices are an identical.
It seems that the cycles of the Gmod graph correspond to these of a sand drawing, which may due to this fact be damaged down as a disjoint union of cycles.
Does this strategy distance us from sand drawers? I’d argue that it doesn’t. Quite the opposite, cycles can present keys to raised understanding their strategy. Of the 60 or so drawings that I’ve collected, I’ve observed that artists typically take breaks of their drawing after they full a cycle. Furthermore, when a sand drawer is pressured to seek out one other path (a rod in Bislama), they have an inclination to rearrange the everyday cycle order that they use of their drawing or to attempt to discover one other decomposition into cycles.
Lastly, some cycles have vernacular names, which means that they’re like constructing blocks for the artists. In reality, this concentrate on graph cycles additionally appears to echo the tales that accompany the drawings, which play a basic position in the best way Vanuatu societies perceive the world. In my analysis, I’ve additionally urged that some cycle decompositions is likely to be linked to the best way these societies conceive of their relationship with nonhumans.
These outcomes elevate questions concerning the universality of arithmetic and the shape that math takes in different cultures. They open up views for educating arithmetic as effectively. Since 2010 the acquisition of conventional data similar to sand drawing has been one of many targets of Vanuatu faculties, and it’s half of a bigger motion to decolonize schooling, very like efforts in Hawaii and within the French territory of New Caledonia. Within the present faculty curriculum, nevertheless, no hyperlink is made between sand drawing and arithmetic. To that finish, Vanuatuan Pierre Metsan, a doctoral pupil in schooling on the College of New Caledonia, is learning whether or not the follow of sand drawing might assist arithmetic instruction. We are able to sit up for what he learns from this investigation within the years to return.
This text initially appeared in Pour la Science and was reproduced with permission.
