In the Timeless Ballet of Wisdom, an Ancient Art Unravels the Veil of Mathematical Assumptions

In October 2015, as their time training mathematics teachers in a French high school in Port Vila, Vanuatu, neared its conclusion, our storytellers found themselves at a crossroads. The principal extended an invitation to partake in kava, a traditional drink synonymous with communal bonding in the country. In the enchanting realm of Vanuatu, every sip of kava served as a gateway to profound insights, a truth well-known to every social scientist who ventured there. Crafted from the roots of the eponymous tree, this elixir had the mystical ability to relax the drinker and set the tongue free.

This pivotal moment marked not just an introduction to kava but also an initiation into the world of sand drawing. That evening, under the Vanuatu stars, a trainee unfurled a sizable board covered with fine sand. With meticulous care, he fashioned a grid of intersecting lines on the canvas of sand before him. As the surface lay ready, he began etching furrows without lifting his finger, creating a mesmerizing pattern. In the rhythmic cadence of Bislama, he explained, "Hemia hem i wan fis i ronwe i stap unda ston from i kat wan sak," translating to "It is a fish that hides under a stone to escape the shark."

The fluidity of the lines, coupled with the effects of kava, transported our storyteller into a realm of awe. The technique echoed the classic challenge of sketching a complex figure with a single, unbroken stroke—a test of artistic finesse and mathematical ingenuity. It also resonated with the concept of an "Eulerian graph" in the realm of mathematics, where a trail traverses every edge exactly once, starting and concluding at the same point.

Lost in contemplation of these parallel worlds, an intern approached, curiosity etched on their face. In a hushed tone, they queried, "Where is the mathematics in this drawing, teacher?" Little did the intern know, this seemingly innocuous question would become the catalyst for the next six years of the storyteller's life—a journey that unfolded through doctoral pursuits exploring the intricacies of sand drawing. One inquiry, whispered under the Vanuatu night sky, sparked the unraveling of a captivating mystery: How were such drawings conceived?

Embarking on an unforeseen journey, our intrepid investigator delved into uncharted territories that surpassed even their wildest expectations. Immersed in the world of expert sand artists, they meticulously observed, absorbed the intricacies of their methods, and amassed a trove of drawings and historical insights. Venturing into the annals of 20th-century ethnologists' studies, our investigator stitched together a narrative, weaving threads of knowledge into a tapestry of understanding.

In the wake of this immersive exploration, a mathematical model of sand drawing emerged, a testament to the fusion of art and science. The investigator's groundbreaking work unveiled the hidden symphony within these ephemeral artworks, revealing them as the orchestrated result of algorithms and operations with an inherent algebraic nature. Surprisingly, the esoteric language of mathematics found resonance in describing the nuanced craft of sand drawing experts.

Beyond the confines of mathematical abstraction, the significance of sand drawing extended into the cultural landscape of Vanuatu societies. In a revelation as profound as the drawings themselves, our investigator demonstrated that sand drawing serves as a profound reflection of the intricate relationships these societies maintain with their environment. The grains of sand, meticulously arranged, whisper tales of cultural symbiosis and environmental harmony, offering a unique lens through which to comprehend the profound connections woven into the very fabric of Vanuatu life.

A Traditional art

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.

In 1912, the luminary mathematician Oswald Veblen unraveled an additional layer of insight within the realm of Eulerian graphs, birthing what is now celebrated as Veblen's theorem. According to this theorem, a graph achieves Eulerian status if and only if it can be dissected into a union of disjoint cycles—distinct sequences of consecutive edges with identical start and end vertices.

Surprisingly, the intricate cycles woven into the fabric of the Gmod graph find a counterpart in the delicate dance of sand drawings. Thus, the enigmatic art of sand drawing can be deciphered as a harmonious assembly of disjoint cycles, unlocking a new perspective on this ancient craft.

Rather than creating a schism between the observer and the sand artists, this approach becomes a bridge to deeper understanding. Each cycle in a sand drawing serves as a key, offering insights into the artist's methodology. Among the 60 collected drawings, a fascinating pattern emerges: artists occasionally pause their work upon completing a cycle. Moreover, when faced with the necessity of forging a new path (referred to as a "rod" in Bislama), sand drawers exhibit a tendency to alter the conventional cycle order or seek alternative decompositions into cycles.

The significance of cycles extends beyond mere methodology; some cycles bear vernacular names, implying they are foundational elements for the artists. This revelation aligns seamlessly with the rich narratives accompanying the drawings, narratives that hold a pivotal role in the worldview of Vanuatu societies. The focus on graph cycles echoes the profound stories embedded in the drawings, reflecting the symbiotic relationship between artistic expression and the cultural understanding of the world in Vanuatu.

In the tapestry of research, there's a subtle suggestion that specific cycle decompositions might be intertwined with how these societies perceive their connection with the nonhuman realms. As the colorful lines converge to create a butterflylike shape, the intricate dance of cycles in sand drawings unfolds, revealing a captivating synergy between mathematical theorems, cultural narratives, and the timeless artistry of Vanuatu societies.

In contemporary times, these societies collectively acknowledge a sacred tradition, recognizing it as a timeless graphic art form intricately woven into the fabric of their cultural memory. This artistry serves as a mnemonic device, a visual repository that allows individuals to summon the depths of ritualistic, religious, and environmental wisdom at a mere glance.

In the remote reaches of the Raga region, Chief Jief Todali, a custodian of ancestral lore, shared profound insights. He portrayed the artists not merely as creators but as oracles, emissaries of a bygone era. In the era predating the arrival of the tutoring, the white foreigners, the people of northern Pentecost communicated through the language of drawings etched on the earth with their fingers. In this symbiotic exchange, the rocks, stones, hills, valleys, wind, rain, and sea spoke on behalf of the people. However, the chief lamented a transformation: now, the roles are inverted. It is the people who articulate their thoughts, while the once-vocal earth, wind, rain, and sea have fallen into silence. The chief from the Raga region poignantly conveyed, "Now [the people] sometimes say, ‘We have to speak for the land because it can no longer speak for itself.’"

The transient nature of this ephemeral art, where each drawing fades into oblivion once completed, becomes a catalyst for storytelling. The practitioners, endowed with a unique blend of artistic and narrative prowess, seamlessly weave tales while sketching. As the drawing takes form, these storytellers enthrall spectators by invoking the imagination, embellishing their narratives with intricate details rooted in their history—familiar locales, characters, animals, and even whimsical vegetables come to life in the ever-evolving tapestry of their stories.