# When the Ground Speaks of Mathematics: Sand Drawing in Vanuatu through the Lens of 4E cognition and Ethnomathematics.

Alban Da Silva

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# When the Ground Speaks of Mathematics: Sand Drawing in Vanuatu through the Lens of 4E cognition and Ethnomathematics*

Alban Da Silva, Laval University

<figure><img src="image_1.png" /><figcaption>Figure 1: the *Tutel* drawing (Vanuatu Cultural Center, 2016)</figcaption></figure>

## 1 Mathematical cognition: Enlarging the contexts, concepts, and methods

### 1.1 Embracing cultural and historical diversity

For the beginning of the 2000s, mathematical cognition (MC) has garnered increasing interest. The early developments in this field can be notably perceived in the influential book *Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being*, in which Núñez and Lakoff posed two foundational questions.

1. What mechanisms of the human brain and mind allow us to formulate mathematical ideas and reason mathematically ?

2. Is brain-and-mind-based mathematics all that mathematics is ? [...] (Lakoff and Núñez, 2000, p.1).

The second is a topic of ongoing and active philosophical debates. Sometimes deemed universal and endowed with an objective reality independent of human cognitive experience, sometimes reduced as a purely product of brain and/or

<footnote>*First revision submitted August 2025</footnote>
mind, there does not emerge an epistemological consensus from these two ex-
treme positions (Hacking, 2014; Caveing, 2004; Changeux and Connes, 1989).
Although some have argued that this is primarily a matter for mathematicians
(Courant et al., 1996), this perspective has been enriched by significant contribu-
tions from historians of mathematics and ethnomathematicians who have high-
lighted the historical (Chemla and Keller, 2017; Joseph, 2011) and cross-cultural
variability of mathematical practices (Vandendriessche and Petit, 2017; Gerdes,
2000; Ascher, 1991; Pinxten and others, 1987). In fact, as Marcia Ascher, one of
the pioneers in ethnomathematics, stated :

Mathematics has no generally agreed-upon definition; it means to some
whatever was included in their school or college courses and to others
whatever is done by the Western professional class called mathemati-
cians. Some attempts to define mathematics emphasize its objects of
study and others its methods; some definitions are extremely narrow
and others exceptionally vague and broad. (Ascher, 1991, p.2-3)

Therefore, the Núñez and Lakoff's initial questions opens three critical avenues
that frame this article's theoretical foundation:

1 - What constitutes the mathematical object of inquiry in MC ?

Our work contributes to studies of mathematical practices as they manifest across
diverse cultural contexts and historical periods, a perspective that remains un-
derexplored in MC studies. Although the difficulty of integrating historical per-
spectives is evident - the access to past thinkers being limited to textual sources
- the neglect of cultural perspectives is less understandable (Beller and Jordan,
2018). Few comparative studies have been carried out involving indigenous pop-
ulations (in Munduruku in Amazonia or Aboriginal peoples of Australia) (But-
terworth and Reeve, 2008; Pica et al., 2004), but they would be substantially
enriched by incorporating a comprehensive understanding of their specific cul-
tural contexts. Several studies underscore how cognitive processes are deeply
shaped by cultural and linguistic contexts, concepts such as "numbers" or "time"
- often deemed universal - exhibiting significant cultural variation (Bender and
Beller, 2012; Núñez and Sweetser, 2006; Gordon, 2004). But they mainly focus
on numerical cognition (Gilmore et al., 2018; Alcock et al., 2016). This article
aim(s) to show that algebraic concepts such as group and group action could en-
rich our understanding of MC. We address this challenge through the lens of
ethnomathematics, a framework that we detail in section 1.3.

2 - Which historical, cultural and social data are relevant for MC ?

Given the diverse conceptions of culture, we draw on an anthropological approach grounded in thick description (Geertz, 1973). More precisely, we claim that the ways in which societies inhabit and experience the world (Descola, 2005) illuminate how mathematical concepts have emerged around the world. This echoes the MC as an "enculturation process", drawing on the insight that the plasticity of the brain allows the redeployment of older neural circuits to new culturally specific functions (Pantsar, 2020; Menary, 2015; Gallagher, 2015). Based on the example of an oral tradition society, this article aims to account for the dynamic interplay between cognitive processes and all cultural data that mediate mathematical thinking.
## 3 - Which theoretical frameworks of cognition for MC ?

We claim that MC is situated because it emerges from the dynamic interaction between body, environmental affordances, and cultural context. This stance is aligned with Merleau-Ponty's phenomenological approach for whom "mathematics [...] is not a self-contained realm of pure thought, but the outcome of the interaction of man and his environment" (Merleau-Ponty et al., 2013, p.330). In fact, given that the vocabulary of cognition in nearly all cultures is fundamentally rooted in bodily and spatial metaphors (e.g. "grasping an idea," "weighing options", etc.), mathematical terminology follows this same pattern (Lakoff and Johnson, Mark, 1999; Lakoff and Johnson, 2008). These observations call for a deeper investigation of how the body participates in mathematical cognition, whether through body-related lexical fields that facilitate mathematical thinking (Winter and Yoshimi, 2020), or through bodily anchoring that fundamentally structures mathematical concepts (Lakoff and Núñez, 2000). Our study addresses this issue through the lens of 4E cognition, a framework that encompasses not only the role of the body in cognitive processes but also other dimensions that we detail in the next section.

## 1.2 4E cognition and external representations

### The 4E framework

Over the past 40 years, the cognitive sciences have undergone a shift that can be referred to as "cognitive ecology": a paradigm shift toward understanding cognitive phenomena holistically within their environmental context. This approach extends traditional brain-centered computational models to a bidirectional and dynamical approach between cognition and the environment (Varela et al., 2017; Hutchins, 2010; Bateson, 2000). Building upon this foundation, the promising 4E framework<fnref n="1" /> brings together different research perspectives in philosophy of mind. These four "Es" refer to dimensions of cognition that go beyond the limits of the brain. In the particular case of this article, we will address the roles of the body (Embodied cognition), the sociocultural context (Embedded cognition), the external tools and resources (Extended cognition), and the dynamical brain-body-environment interaction (Enactive cognition). The 4E framework challenges in various ways some central constructs of traditional cognitive science that compare the brain to a device that converts an input (a stimulus) into an output (a reaction). In this perception-action process, the intermediate step lies in processing information through an encoded system of symbols and syntactic rules, called mental (or internal) *representations*. This "language of thought" (Fodor, 1975) is biologically located or distributed within the neural architecture of the brain (Pezzulo et al., 2017). Nevertheless, this approach proves insufficient in some cases as exemplified in the next section.

### A first example

In this article, we will use the term "representation" in an external and ecological sense rather than a purely internalized one. To set an example, let us take an example related to numerical cognition. The models that prevail in neuroscience

<footnote n="1">¹For brevity, 4E will refer to the 4E theoretical framework or the 4E approach (as is common in recent literature).</footnote>
to represent numerical distributions in the brain are linear (or logarithmic) uni-
directional scales (Dehaene et al., 2008; Fias et al., 2011). But this does not seem
to apply in some cultures, particularly those in which there are no exact words
for numerical quantities beyond a certain limit (Núñez, 2017). Even in those in
which such limitations do not exist, such as the *Sia Raga*<fnref n="2" /> society in which the
present study has been carried out, the unidirectional scale model does not seem
to echo ethnographic observations. In fact, in this society, each number is rep-
resented by a drawing (figure.2,14b and a video available here) that corresponds
to an algorithm on a grid *n* × *n* (following rules that will be developed further).

<figure><img src="image_2.png" /><figcaption>Figure 2: Number five (Gailima in Raga language), a drawing on a two-dimensional grid 5 × 5.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 3: The *Bus-nuts* (*Barringtonia edulis*) - [Image credit: Dr Giuseppe Mazza (Monaco Nature Encyclopedia)]</figcaption></figure>

While we will not develop this algorithm further here, it should be noted that
ethnographic data suggest that it is deeply inspired by the growth of an endemic
fruit (figure.3) that wraps around its branch. The algorithm itself, consisting
of successively wrapping quarter-circles diagonally (Da Silva, 2022, p.389-402),
leans toward a bidirectional rather than unidirectional representation. Moreover,
according to *Sia Raga*, drawings and numbers are two facets of the same reality.
This ontological status indicates that the environment (the disposition of fruits
in trees) contributes significantly to the representation of numbers within the
*Sia Raga*.

<footnote n="2">²The word *Raga* refers to the language, while *Sia Raga* refers to the people or the society</footnote>
## External representations

To align with the 4E perspectives, the term "representation" will be employed in an embodied, extended, embedded and enactive sense. By this we mean that a representation is not purely internal constructs but rather emergent properties of the dynamic system comprising the embodied agents, their external tools, their environment<fnref n="3" /> and their sociocultural contexts. They are not merely mnemonic aids, but active participants in cognitive tasks. A representation may possess a tangible existence, exemplified by fruits cited in the previous example, which aligns with an ecological conception of cognition situated in a landscape of affordances, namely the "possibilities for action" (Gibson, 2014a; Kiverstein and Rietveld, 2018). It can also be the written symbols or figures utilized in geometry by mathematicians<fnref n="4" /> (Mesquita, 1998; Goldin and Shteingold, 2001). It can also take the form of external rules, constraints, relations, or structures embedded in the environment, such as the way these fruits wrap around the branches or the underlying structures in a myth.

Just as one might compare different forms of numerical representation (sand drawing algorithms and symbolic notation), this article juxtaposes the mathematician's formal representations with indigenous ones, examining their points of congruence.

## 1.3 Ethnomathematics as an avenue for 4E empirical evidences

Several MC studies, even if they do not explicitly refer to the 4E tenets, incorporate sociocultural contexts in the development of numerical and geometrical concepts (Beller and Jordan, 2018; Núñez, 2017; Bender et al., 2015; Saxe, 2012; Butterworth and Reeve, 2008). Surprisingly, ethnomathematics, a field dedicated to exploring the cultural variability of mathematics and its relations to education, has remained largely separate from this endeavor. Few studies clearly integrate ethnomathematical and cognitive science methods, although potential benefits have been suggested (Bender and Beller, 2018; Chemillier et al., 2007). In fact, for more than 40 years, ethnomathematical research has investigated the cultural variability of concepts previously considered universal, such as numbers (Owens et al., 2017; Urton, 1997; Vellard, 1988) and geometric concepts (Zaslavsky, 1999; Pinxten and others, 1987; Gerdes, 1999). During the past decade, enriched by ethnographic and computational approaches, ethnomathematics has illuminated the mathematical aspects of various practices, including Malagasy divination (Chemillier et al., 2007), Malagasy sowing games (Tiennot and others, 2014), Trobriand string games (Vandendriessche, 2015), Andean textile friezes (Desrosiers, 2012), Réunionese lambrequins (Tournès, 2022) and the practice of sand drawing (SD) in Vanuatu (Da Silva, 2022; Devylder, 2022; Vandendriessche and Da Silva, 2022; Ascher, 1988). The seminal work of Marcia Ascher on SD is an ideal starting point for questioning the role of mathematical models in this kind of study which are less an observer point of view than the

<footnote n="3">³To avoid reproducing the nature/culture dualism that contemporary anthropology has shown to be peculiar to Western societies (Descola, 2005), we take care to specify that the term 'environment' is used only for descriptive purposes. In fact, as anthropologist Tim Ingold has highlighted, in certain societies (such as the one discussed in this article) no distinction is made between humans and non-humans, which means that the word "environment" has no vernacular equivalent (Ingold, 2021).</footnote>

<footnote n="4">⁴We recognize that some authors use the concepts of “signs” (in a semiotic sense) and “representations” interchangeably in mathematics (Mainali, 2021). However, it has been shown to be a widely shared trend in cognitive psychology and philosophy of mind (Favela and Machery, 2025).</footnote>
result of a fine-tuning between ethnographic data and the search for mathemat-
ical concepts that will come as close as possible to the point of view of the actors
themselves<fnref n="5" />. According to Ascher, the predictive power of such models can bring
into light structures that "may otherwise espace us" :

Keep in mind that tracing procedures are hand motions that leave tran-
sient marks in the sand. One way to talk about them or write about
them is to refer to them by symbols. The symbols are ours, not theirs.
But, as often happens in mathematics, if we assign our symbols with
care, we can highlight logical and structural relationships that may
otherwise escape us. It is the Malekula who created the tracing sys-
tems but it is we who are introducing symbols in an attempt to capture
and convey the structure of their systems. (Ascher, 1988, p.48)

Although Ascher does not explicitly conceptualize it as such, her work em-
ploys two different types of external representations: mathematical formaliza-
tions (symbols, formulas, figures and underlying logical structures) and indige-
nous practices documented through ethnography (such as the Malekula<fnref n="6" /> tracing
systems). This article adopts this comparative perspective, which remains un-
derexplored in ethnomathematical research.

**1.4 Outline of the article**

While empirical studies traditionally separate ethnographic content from method-
ological considerations, it is more common in anthropology to blend these consid-
erations through monographic description. Section 2 therefore adopts this per-
spective, documenting the rules, terminologies and cultural embedding of sand
drawing practice among the Sia Raga while offering an overview of the methods.
In Section 3, we develop our mathematical framework, introducing group actions
and the pivotal concept of cycles that decompose any drawing into fundamental
building blocks. Section 4 examines how these mathematical structures ground
cognitive processes of memory and creation, revealing striking convergences be-
tween the concept of cycles and indigenous representations. Finally, in Section 5,
we discuss how sand drawing exemplifies 4E cognition in mathematical activities
and explore its potential in mathematics education.

<footnote n="5">⁵Although this approach is occasionally labeled as "ethnomodeling" by certain scholars (Rosa and Orey, 2011; Magnate, 2025), we prefer to use the term "modeling," as it is traditionally employed in anthropology.</footnote>

<footnote n="6">⁶Malekula is an island in the Vanuatu archipelago. Between 1924 and 1926, A.B. Deacon, a young ethnographer, collected the first ethnographic data on the sand drawing practice from this island. Based on these materials, Ascher published the first ethnomathematical study on the subject in 1988. My research, carried out 35 years later, represents the second study of this kind.</footnote>
**2 Methodological and ethnographic framework**

**2.1 Sand drawing in Vanuatu culture**

<figure><img src="image_4.png" /><figcaption>Figure 4: Sand drawing is practiced in the center islands of Vanuatu. *Sia Raga* area is located in the North of Pentecost island. Map: GISGeography.com (freely modified).</figcaption></figure>

Known for its great cultural diversity, traditional ways of life and high linguistic density, Vanuatu<fnref n="7" /> is an archipelago made up of 83 islands spanning more than 600 km. Different cultural realities can be observed from north to south, and while there are traits common to all Vanuatu societies, there are also a number of cultural variations, including sand drawing<fnref n="8" />. In a context of oral tradition, there are few written documentations on sand drawing except ethnographic studies (Devylder, 2022; Da Silva, 2022; Baron, 2020; Layard, 1942; Deacon, 1934),

<footnote n="7">⁷Vanuatu, formerly New Hebride, was administered by a Franco-British colonial condominium from 1906 until its independence in 1980, when it became the Republic of Vanuatu. With a population of around 315,000, Vanuatu has the highest linguistic density in the world, with 138 vernacular languages (François et al., 2015). Outside of the port-Vila capital, the way of life has remained very traditional, with subsistence farming as the main source of income.</footnote>

<footnote n="8">⁸The term 'sand drawing' is inherited from the colonial past and vernacular terms are closer to 'writing' and/or 'painting,' as the north Pentecost term *Uli-Uli* used to refer to the way of writing in western societies</footnote>
books of enlightened enthusiasts (Cabane, 2016, 1997).

<figure><img src="image_5.png" /><figcaption>Figure 5: Practitioners from Avatvotu (North-Pentecost), 2019</figcaption></figure>

Probably thousands of years old, this practice was inscribed on the UNESCO's Representative List of the Intangible Cultural Heritage of Humanity in 2008. It consists of drawing symmetrical figures on the ground by tracing a continuous line with a finger that generally does not lift off the ground during the execution and by not passing over a part that has already been drawn. After the drawing is finished, the index finger is, in a very large number of cases, returned back to the starting point. This ephemeral art - the drawing is erased once it is finished - stimulates storytelling and is usually concluded with the telling of a myth or a tale including diverse theme such as special places, characters, animals, chieftainship, kinship system, social organizations, etc. As is the case in non-naturalist societies, that is, those that do not operate a discontinuity between humans and nonhumans (Descola, 2005, p.176), the relationships are reversed. Thus, from an indigenous perspective, the drawings are not the property of those who produce them, but a manifestation of the words of the earth: "sand drawings are the way the ground speaks" (Taylor, 2008, p.1) - a quote that inspired the title of this article.

Drawings can be the object of local reappropriation, in which case they can bear completely different names (figure.6), and very different drawing procedures can be observed<fnref n="9" />. As illustrated in this article, it has a strong connection with rituals, mythological and environmental knowledge.

<footnote n="9">⁹We will not address this issue but I have demonstrated that the number of possible "paths" for completing a drawing increases exponentially with the size of the grid (Da Silva, 2022).</footnote>
<figure><img src="image_6.png" /><figcaption>(a) *Nimbingge*, a sand drawing relating to the yam<fnref n="10" />, collected by Deacon in Malekula between 1924 and 1926.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(b) *Vatu*, North Pentecost (2019). "A rocky islet that has been bombed during the second world war".</figcaption></figure>

<caption>Figure 6: The same drawing in two different names, place and time.</caption>

Although the simplest drawings are freely circulating (most of them known to children), it is hard to say exactly how many of these drawings are still used today, with new drawings appearing in some societies and disappearing in others<fnref n="11" />. The community acknowledges a drawer's expertise according to its ability to execute or create drawings that cannot be learned merely by reproducing gestures. Only a small number of experts (we will call them *practitioners*) are considered such. They can be of any age: young designers creating new designs or elders recognized for their encyclopedic knowledge of the practice. In the north-Pentecost, they have a special social status that places them in the position of guardians of collective memory and bearers of the values of customs.

It is crucial to acknowledge that mathematics, as a formalized discipline, is not an indigenous category in the Vanuatu Islands. Rather, it is a subject that is typically encountered within the context of formal education and is accessible only to those fortunate enough to receive schooling. Therefore, there is no perceived relationship between SD and mathematics from an emic point of view. So my presence on Pentecost Island, as a mathematician interested in SD, was the object of curiosity and astonishment among practitioners.

## 2.2 Inquiry and data collection method

### Experimentation with practitioners

Between 2018 and 2020, I collected more than 120 drawings across three islands, 50 recorded with *n* = 5 advanced practitioners from the *Raga* area. After developing expertise through Deacon's works (Deacon, 1934), learning Bislama

<footnote n="10">¹⁰The yam is one of the most widely cultivated tubers (along with water taro) throughout Melanesia, and its ritual importance has been cited in several anthropological studies. (Lanouguère-Brunneau, 2000; Muller, 2009)</footnote>

<footnote n="11">¹¹The conservation of traditional knowledge, and in particular that related to sand drawing, is the subject of a sustainable development policy by the Vanuatu authorities. The Vanuatu Cultural Center of the capital Port-Vila plays a key role, offering demonstrations, courses, and festivals that bring together the best experts from each cultural area.</footnote>
pidgin and *Raga* language<fnref n="12" />, I gained community trust to access this protected traditional knowledge. The recorded sessions (figure 7b) lasted 2-3 hours, divided into three stages.

1. The practitioner executes the drawing, potentially accompanied by a narrative (in *Raga* language, or Bislama);

2. An exchange with the researcher wherein the practitioner elucidates the drawing process, ideally in relation to the different step of the narrative.

3. The researcher attempts to reproduce the drawing, intentionally or unintentionally making errors (figure. 7a). This step proved useful in establishing the list of "rules" further detailed.

## Tacit knowledge

The data collection has focused on vernacular language and terminology, plus various relevant manifestations: gesticulation, hesitation in finger movements, practitioner surprise. This nonverbal dimension is particularly significant since sand drawing knowledge is not always explicitly verbalized for transmission. This exemplifies "tacit knowledge" that, while "not readily expressible in sentential logical forms" (Bloch, 2018, p.13-14), is deeply rooted in contextual, cultural and practical applications (Gascoigne and Thornton, 2014). This concept is fundamental to anthropology, showing how knowledge forms comprehensive world-views and classification systems — what Lévi-Strauss called "the science of the concrete" (Lévi-Strauss, 1962). This perspective converges with the philosophical framework of our article, thought and knowledge being intimately connected to motor actions that ground "meaning, concepts, logic, and inferential patterns in our bodies and bodily activities" (Johnson, 2007; Sheets-Johnstone, 2011).

<figure><img src="image_8.png" /><figcaption>Figure 7: Sessions with practitioners.</figcaption></figure>

<footnote n="12">¹²Supplemented by linguistic documentation (Hardacre, 1924; Yoshioka and Leona, 1992; Vari-Bogiri, 2011; Duhamel, 2020) and anthropological studies (Yoshioka, 1987; Taylor, 2008).</footnote>
## 2.3 Oral rules

Practice follows rules that, while rarely explicitly articulated, are consistently observed. Current rules remain consistent with those documented a century ago by early ethnographers, aligning with Dan Sperber's concept of a "cultural attractor". Despite the potential for alternative rules, other logical possibilities remain unexplored<fnref n="13" /> (Sperber, 1996). This adherence to traditional norms suggests solid cultural transmission mechanisms that preserve core principles over time, making the detailed model robust.

By integrating the corrections of practitioners applied to our deliberate errors, we have been able to establish that a SD performance can be split into two distinct steps. Moreover, the tracing follows seven fundamental rules that are usually respected.

### Step 1 - The grid

Before starting, the practitioner traced a grid, usually rectangular, on a sand or ash-covered ground (figure. 8), which plays a role as a framework. Within *Sia Raga*, the lines (horizontal or vertical) are *Gai*, the term for the long wooden sticks used to build the walls of *imwa* (huts). Vertical/horizontal lines are differentiated by the terms *Gaina/Gai vol volo*, again referring to the vocabulary of hut construction, with posts (horizontal) differentiated from horizontal ties (figure. 9). At the time of my survey, line crossings in Bislama were referred to as *joen paen* ("junction point"), which leads to calling them *nodes*. Grids come in many shapes and sizes (rectangles, combinations of rectangles, concentric circles, etc.), and their shapes seem limited only by the creativity of experts. Nevertheless, in this article we will confine ourselves to the case of rectangular grids.

<figure><img src="image_9.png" /><figcaption>Figure 8: Ground preparation and layout of a rectangular grid, Pentecost 2019.</figcaption></figure>

### Step 2 - The tracing

Once the grid is complete, the practitioner draws a line with the tip of a finger, the characteristics of which are usually as follows.

* **Rule 1: The line is continuous.** The finger does not lift off the ground while the drawing is in progress. Nevertheless, it may occasionally lift to

<footnote n="13">¹³Sand-drawing offers a myriad of unexplored possibilities. For example, alternative rules could have been devised, such as allowing multiple passes over the same segment.</footnote>
<figure><img src="image_10.png" /><figcaption>Figure 9: Grid terminology borrowed from *imwa* (traditional hut) construction vocabulary.</figcaption></figure>

add ornamentation or simply to pause, but in this case it generally starts again from the point where the finger left the ground;

* **Rule 2: The line only intersects the grid at its nodes.** In other words, it runs from one grid node to another;

* **Rule 3: At each node, the line lies on one of the two 45° or -45° diagonal lines.** In other words, the tangent to the curve at a node lies in one of these two directions<sup>14</sup> (figure 10).

<figure><img src="image_11.png" /><figcaption>Figure 10: Each node crossed by the finger is crossed in one of the two red directions.</figcaption></figure>

* **Rule 4: No change in direction at nodes.** The finger continuously crosses the node without changing direction (figure 11).

<footnote n="14"><sup>14</sup>The word 'direction' here refers to the line on which the curve lies, without distinguishing between the two possible senses.</footnote>
<figure><img src="image_12.png" /><figcaption>Figure 11: Starting from A to B to traverse into the red or brown line, but not in green one (that would make a break direction).</figcaption></figure>

* **Rule 5: The line does not pass twice on the same "route".** The practitioner avoids tracing over an existing section continuously. However, crossing points can be created. This is the case for grid nodes, which are often crossed multiple times.

* **Rule 6: Return to the starting point.** Once the drawing is complete, the index finger returns to the starting point in the same direction.

* **Rule 7: Symmetry.** In a large number of cases (over 90% of the SD Raga corpus), the designs are symmetrical.

## 2.4 A specific vocabulary for certain patterns

Specific terms are commonly used to name certain drawing patterns (table. 1)<fnref n="15" />. The vocabulary sometimes alludes to the shape of the movement employed, as is the case with *gabau*, which refers to "a wing." The arc is designated *bul taiwage* ([bool-tie-wa-ray]), etymologically suggesting a contraction of *bulu* ("to join") and *taiwage* ("a parcel of land"), thus reflecting the joining of two grid nodes. Patterns can have multiple names, such as *livon gahi*, which can also be denoted by a term related to the wings of a butterfly. The term *Gagaravwe*, which refers to loops, is borrowed from the construction vocabulary, as is the term for grids. This term describes the knots of vines used to bind vertical and horizontal wooden beams in the construction of *imwa* ("traditional houses") figure 12). The term *gai balasi* results from the combination of *gai* ("a stick, a mark") and the verb *balas* ("to partition"), the pattern being a symbol etched on the ground to denote a restricted area.

<figure><img src="image_13.png" /><figcaption>Figure 12: *Gagaravwe*: the knots used to tie a roof structure.</figcaption></figure>

<footnote n="15">¹⁵All the translations have been verified by Raga language specialist Marie Duhamel (Duhamel, 2020), from Fiji University.</footnote>
The term *huri* designated for "circle" warrants further elaboration. As a pol-
ysemous term, it serves as an action verb meaning "to follow" or "to participate"
but can also mean "the husband of the sister of my father" (see section.4.1). This
terminology appears to imply that the act of tracing a circle facilitates the ability
to "follow" the path without deviation (figure.13).
<table><caption>Table 1: Some vernacular terms for patterns</caption><thead><tr><th>Name (in Raga)<br>Translation</th><th>Motion</th><th>Picture</th></tr></thead><tbody><tr><td>Gabau - a wing</td><td>(0, 1, -i) ou<br>(0, 2, -i)</td><td></td></tr><tr><td>Bul taiwage -<br>Bulu = To join</td><td>(1, 0, -i)</td><td></td></tr><tr><td>Gagaravwe - A<br>type of knot</td><td>m = (0, 0, -i)</td><td></td></tr><tr><td>Huri - To follow</td><td>(1, 0, -i)<sup>4</sup></td><td></td></tr><tr><td>Gai Balasi</td><td>A mark of<br>prohibition</td><td></td></tr><tr><td>Livon gahi - Ant's<br>teeth</td><td></td><td></td></tr></tbody></table>
<figure><img src="image_14.png" /><figcaption>Figure 13: The *huri* action allows to continue the path</figcaption></figure>

**2.5 A vocabulary invariant to rotations**

The designer's position fundamentally differs from that of a student performing geometry exercises manipulating and orienting a sheet as desired while sitting at a desk. In fact, he is typically kneeling or squatting and remains facing the drawing - this point proved crucial in designing the mathematical model - even using their body in extension (figure. 14).

<figure><img src="image_15.png" /><figcaption>Figure 14: Extended position of the practitioner facing the drawing</figcaption></figure>

Besides, we can observe that the SD terminologies (section.2) are the same what-
ever the designer's relative orientation and the direction in which these proce-
dures are executed. Consequently, the model must incorporate these specific
nuances. For example, the pattern *Gai Balasi* in North Pentecost is named as
that, whatever his orientation or his position on the grid (figure. 15) .
<figure><img src="image_16.png" /><figcaption>Figure 15: *Gai Balasi* three times.</figcaption></figure>

**3 A mathematical model of sand drawing**

We are setting up the modeling tools that will recruit the two fundamental concepts of *group* and *action*<fnref n="16" /> of a group *G* on a set *X*. A group *G* is a set with an internal associative operation and an identity element such that every element has an inverse in it (e.g, the set of relative numbers $\mathbb{Z}$ or the Euclidean transformations form a group). The concept of group action gives meaning to the notation $g.x^{17}$, where *g* is an element of a group *G*, *x* is an element of a set *E*, and $g.x$ remains an element of *E*.

The use of complex numbers proves to be highly advantageous for geometric reasoning, significantly simplifying the calculations. For more fluidity, we detail the building of the group *G* on an example, the generalization being easy (Da Silva, 2022).

**The set X of all pairs (node/direction)**

Traditionally, the nodes of the grid are expressed as complex numbers within a Cartesian coordinate system ($i, j$), as determined by the position of the drawer (figure. 16). To denote each of the four directions, it is preferable to avoid using the same coordinate system, as this would unnecessarily complicate the calculations. Instead, we have opted to arbitrarily assign the notation "1" to one direction, with the others being represented as $i, -1, -i$ by rotating 90° counterclockwise (Fig. 17).

<footnote n="16">¹⁶The concept of group actions emerged in the 18th century from Lagrange's work on polynomial equations. This theory has provided mathematicians with a precise vocabulary to solve a wide range of problems (Kleiner, 1986).</footnote>

<footnote n="17">¹⁷To avoid any confusion, please note the dot product is classically used to represent an external product (and is not the Euclidean scalar product). Two properties must be verified : $g.(g'.x) = (gg').x$ and $e.x = x$ where *e* is the neutral.</footnote>
<figure><img src="image_17.png" /><figcaption>Figure 16: The Cartesian coordinate system ($\vec{ij}$)</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Figure 17: The four possible directions associated with the complex numbers $1, i, -1, -i$</figcaption></figure>

The ethnographic study revealed that practitioners not only focus on moving their fingers from one node to another, but also pay careful attention to the starting and ending directions. To be consistent with this observation, the set $X$ will be all possible pairs $(S, d)$ where $S \in \mathbb{Z}[i]$ and $d \in \{1, i, -1, -i\}$:

$$X = \mathbb{Z}[i] \times \{1, i, -1, -i\}$$

### The motions group $G$

The figure.18 illustrates a motion (we momentary use this term with no more explanation) between two nodes of the grid, the starting/arrival direction being $d$ and $d'$. The angle between $d$ and $d'$ being $-90^\circ$ (in anticlockwise orientation), we can write $d' = -i \times d$ (complex multiplication). Moreover, since $S'$ is reached from $S$ by moving $-2$ units along the x-axis and $1$ unit along the y-axis, we have $S' = S + (-2 + 1.i)$.
<figure><img src="image_19.png" /><figcaption>Figure 18: The motion $m = (-2, 1, -i)$</figcaption></figure>

Now, let us observe the same motion at the figure. 19 which depicts the same motion but oriented at four different positions. Whether the relation

$$d' = -id \quad (1)$$

still holds, the relation $S' = S + (-2 + 1.i)$ is no longer correct. The reason is that we forgot to consider the particular position of $d$ in the figure. 18. To be correct in any position, simple reasoning shows that we have to write

$$S' = S + (-2 + 1.i)d \quad (2)$$

<figure><img src="image_20.png" /><figcaption>Figure 19: The same motion $m = (-2, 1, -i)$ in three other position</figcaption></figure>

The two relations (1) and (2) can be compacted into one:

$$\underbrace{\begin{pmatrix} S' \\ d' \end{pmatrix}}_{\in X} = \underbrace{\begin{pmatrix} 1 & -2 + 1.i \\ 0 & -i \end{pmatrix}}_{\in G} \cdot \underbrace{\begin{pmatrix} S \\ d \end{pmatrix}}_{\in X} \quad (3)$$

From now on, the matrix $\begin{pmatrix} 1 & -2 + 1.i \\ 0 & -i \end{pmatrix}$ will simply be observed as $m = (-2, 1, -i)$, this triplet being referred to as *motion*. More generally, the group of all possible
motion $(\alpha, \beta, \gamma)$ is the set

$$G = \left\{ \begin{pmatrix} 1 & \alpha + \beta \cdot i \\ 0 & \gamma \end{pmatrix}, (\alpha, \beta, \gamma) \in \mathbb{Z} \times \mathbb{Z} \times \{1, i, -1, -i\} \right\}$$

To identify the value of $(\alpha, \beta, \gamma)$, we read by placing the motion in a position where $d = 1$, and by counting the distance moved along the x-axis and y-axis and calculating the angle $(d, d')$ (see more examples table.2).

## A group action

It is easy to prove that $G$ has a (non-commutative) group structure<fnref n="18" />, the neutral element being $e = (0, 0, 1)$ and each element $m$ having an inverse (figure.20).

<figure><img src="image_21.png" /><figcaption>Figure 20: A motion and its inverse</figcaption></figure>

The relation (3) is easy to generalize to prove that the group $G$ of motions acts on the set $X$. The word "action" is used both in the mathematical context and also perfectly describes the physical action of the artist's finger as it moves. between two nodes, respecting the starting and arrival directions (figure.21).

<footnote n="18">¹⁸</footnote>

<footnote n="1">1. The matrix product of two motions is still a motion¹⁹;</footnote>

<footnote n="2">2. The neutral element is $e = (0, 0, 1)$, that is $m \cdot e = e \cdot m = m$ for all motion $m$</footnote>

<footnote n="3">3. Each motion $m$ has an inverse : $(\alpha, \beta, \gamma)^{-1} = (-\frac{\alpha}{\gamma}, -\frac{\beta}{\gamma}, \frac{1}{\gamma})$ that verify $m \cdot m^{-1} = m^{-1} \cdot m = e$.</footnote>
<figure><img src="image_22.png" /><figcaption>Figure 21: Finger action: the tracing transforms one pair (node,direction) into another.</figcaption></figure>

**The role of the curve in this model**

Focusing on the relative positions of $(S, d)$ and $(S', d')$, we temporarily forgot the
role of the curve. By analogy with the mathematical concept of *vector representation*, we can assign a functional role to curves. A vector $\vec{AB}$ is uniquely defined
by its magnitude (length $AB$) and direction (from point $A$ to point $B$). Any arrow with identical magnitude and direction is an alternative representation of
the same vector. In our model, the curve connecting $(S, d)$ and $(S', d')$ functions
analogously to the arrow in vector representation, and any curve that preserves
the same arrangement of these pairs represents the same motion. As illustrated
in figure 22, changing the representation does not alter the global structure of a
drawing, and this property is valuable for creativity. While we cannot fully explore
this phenomenon here, some practitioners leverage it to create new drawings or
to add a personal touch to existing ones.
<figure><img src="image_23.png" /><figcaption>Figure 22: All these curve play, in the model, the role of representatives of the same motion (0, 1, i)</figcaption></figure>
## 3.1 Words, patterns and cycles

### SD as a word

We denote the action of two consequent motions $m_1$ then $m_2$ as $m_1m_2$ (figure.23), and more generally, as $m_1m_2...m_N$ for the concatenated sequence of movements $m_1, m_2, ..., m_N^{20}$. Each drawing can be encoded as a words $m_1m_2...m_N$ (not unique), without considering the initial node and direction chosen by the drawer<fnref n="21" />

<figure><img src="image_24.png" /><figcaption>Figure 23: This sequence of two motions can naturally be written $m_0m_1$ with no reference to the start ($A_0, d_0$), nor to the intermediate points $A_0, A_1,...$</figcaption></figure>

### Factorization

A sequence $m_km_{k+1}...m_p$, extracted from a word, will be called a *pattern*. Given a list of patterns $M_1, M_2, ... M_n$, we can factorization and patterns.

$$M = \underbrace{(m_1m_2...m_k)}_{M_1} \underbrace{(m_{k+1}m_{k+2}...m_l)}_{M_2} \cdots \underbrace{(m_{p+1}m_{p+2}...m_n)}_{M_N} = M_1M_2...M_n$$

### Cycles

Cycles are closed-loop patterns that return to their starting point in the same direction. From the model point of view, $m_km_{k+1}...m_p$ is a cycle if the matrix product $m_k.m_{k+1}...m_{p-1}.m_p$ yields the identity motion (0, 0, 1). That means that such elements are global stabilizers of the action, that is, a cycle g verify $g.x = x$ for all $x \in X$.

<footnote n="20">²⁰Please note the difference between $m_1.m_2$ and $m_1m_2$. The first indicates the product, the second being the concatenation of the two symbols $m_1$ and $m_2$.</footnote>

<footnote n="21">²¹Although we will not address the specifics here, this tool was used to generate all the illustrations, excluding the photographs. The PYTHON codes are available here: https://github.com/nablanabla/Thesis</footnote>
<figure><img src="image_25.png" /><figcaption>Figure 24: A cycle, the starting and arrival point/direction ($S, d$) and ($s', d'$) are equal.</figcaption></figure>

The patterns *huri* (a circle) and *Livon gahi* (table. 1) are examples of cycles. Although *Gagaravwe* (a knot) is a closed loop, it is not a cycle because it does not return to its starting point in the same direction. We will see further that the research of cycles is at the core of creation of new drawings, the algebraic formalism offers an interesting perspective on it. Starting from a motion *m* and a given node/direction *x*, producing a cycle consists of finding the inverse of *m*, because $m^{-1} \cdot (m \cdot x) = x$ (figure. 20).

**A fundamental theorem**

This concept of cycle is all the more crucial that the following theorem<sup>22</sup> holds :

**Theorem :** Any sand drawing can be decomposed<sup>23</sup> into a disjoint union of cycles.

By disjoint, we mean that the cycles do not share any point except from the nodes of the grid (e.g., figure. 30).

This theorem has at least two consequences :

1. Just as the generators of a group are considered the 'building blocks' that underpin its structure, cycles can be viewed as the fundamental components in the construction of a SD.

2. As there are many ways to traverse the cycles one after the other, there exist multiple methods to trace a SD. In fact, practitioners are interested in alternative drawing procedures, particularly imagining new drawings or simply reappropriating them.

Armed with these modeling tools, we can now turn our attention to the cognitive aspects of SD.

<footnote n="22"><sup>22</sup>The Veblen's theorem of Eulerian graph theory (Veblen, 1912) applied to a labeled graph labeled on the motion group construction, is the cornerstone of the proof, which is nevertheless beyond the scope of this article (Da Silva, 2022, p.162)</footnote>
# 4 External representations and cognitive process

## 4.1 Cognitive grounding of cycles

We argue that the cycle concept is grounded in vernacular reality for three reasons. First, by returning to the starting point, the cycle execution echoes the action verb *huri*, which means "continuing the path". Second, cycles based on complex algorithms often serve as a "first layer" in drawings (Da Silva, 2023). Third, anthropologist J.P Taylor demonstrated that cycles are fundamental to *Sia Raga* culture, serving as a conceptual lens for understanding their interconnection with past, present and future, as well as social relationships, environment and cosmology (Taylor, 2008). Notable evidence includes the significance of the pig tusks. Manipulated through ancestral processes into spiral shapes (Jolly, 1984; Bedford, 2018), they are central to *Sia Raga* imagery, "by which they conceive exchange relations [that are] not simply a repetitive process, but also productive and regenerative across space and time" (Ibid., p.105-106). Taylor's comparison of cycles to "circles in space" or "spirals" is particularly relevant to our model. With motion described through three spatial coordinates (α, β, γ), cycles indeed resemble circles in space rather than mere closed curves. A pig with full-circle tusks is called *livoala*, derived from *alo* (sun), which also means "to circle", "to encircle" or "to circulate" (Ibid., p.121). This aligns with our definition of the cycle as closed curves that add detail to drawings. The kinship system provides another example: *huri* also refers to "the husband of the sister of my father" (figure 26), a pivotal role that ensures lineage continuity through cycles of alliances, marriages, gifts, and ritual or economic support between extended families (Löffler, 1960).

<figure><img src="image_26.png" /><figcaption>Figure 25: Pig's tusk from Vanuatu. Credit: Australian Museum, Registration number E81623, photo: Carl Bento © Australian Museum, Sydney. (permission asked on 25/07/16)</figcaption></figure>
<figure><img src="image_27.png" /></figure>

Figure 26: Circular representations of Sia Raga relationship terminology (after Taylor, 118, figure 20) and the pivotal role of *Huri* for *ego*.

## 4.2 Memory

### Narratives as representations

Since practitioners do not document drawing procedures, how do they memorize them? By *memory*, we refer to long-term memory in its broadest sense, without distinguishing between declarative memory (events, facts, places) and nondeclarative memory (motor skills, automatic procedures), due to their potential intertwining. Using drawings *Vatu* (figure 6b) and *Vata Vwaga* (figure 28), we argue that narratives are public external representations supporting memory processes. Although practitioners likely recruit private representations, we focus on collective memory distributed among individuals, following the extended cognition framework (Wilson, 2005; Wertsch, 2009). The narratives associated with sand drawings are often myths, cosmogonies, or historical accounts. Lévi-Strauss's structuralist work shows that these are shared knowledge whose narrative expression varies individually, but whose structure remains robust and reveals how societies organize their worldview (Lévi-Strauss, 1971).
## Two examples : Vatu and Vata Vwaga

<figure><img src="image_28.png" /><figcaption>Figure 27: The drawing Vatu written as $M^4$</figcaption></figure>

The translation of Vatu would be "an emerged coral reef," with the prefix Vat- commonly used in words related to "rocks." In sia Raga, each coral reef of this type has special significance and is central to mythical, cosmogonic, or historical narratives<fnref n="24" />.

This drawing relates to an actual event<fnref n="25" /> that occurred during World War II, when Vanuatu served as a rear for the US army. During the tracing of figure 27, a practitioner explained: "This is a rocky islet that was bombed during World War II." As he indicated the four symmetric parts of the drawing, he added, "A plane flew over it, dropped a bomb, and broke it into four pieces." This narrative leads us to describe the drawing procedure as $M^4$, with the pattern $M$ (figure 27) repeated four times. Although $M$ lacks a vernacular name, this notation echoes the story associated with the drawing. In North Pentecost, contrary to

<footnote n="24">²⁴This is particularly true for the drawing *Vatangele*, which everyone must know to enter paradise upon passing away (Yoshioka, 1987; Taylor, 2008).</footnote>

<footnote n="25">²⁵This story was recounted to me by several individuals in various locations, lending credence to its authenticity.</footnote>
observations in North Ambrym<sup>26</sup>, this appears to be the only known drawing procedure. An external representation involved in the memorization process is the image of the "rock broken into four pieces" mentioned in the narrative.

<figure><img src="image_29.png" /><figcaption>Figure 28: The drawing "Vata Vwaga".</figcaption></figure>

This phenomenon is not limited to a single drawing and can be observed in more complex ways. The drawing *Vata Vwaga* (whose translation would be "the pirogue rock") recounts the survival story of two fishermen whose boat crashed on a rock. They survived by back-floating (their bodies forming a cross) and drifted back to their village. In this case, the multiple stages of the narrative (boat crash, body position, and return home) are reflected in the drawing's multiple layers (in blue, red, and green, respectively). Various drawing procedures exist for this figure, as demonstrated in *this animation* and *this one*, the last drawing being drawn by the practitioner Julia.

Before jumping to the following question, let us have two remarks :

1. Observing the figure.27, it is not trivial to predict *a priori* that a fourfold repetition of pattern *M* will perfectly entangle.

2. Since the pattern *M* has no symmetry properties, it is unlikely that this drawing has spontaneously been created as a repetition of the pattern *M*.

In the next section, we address the following question : How has the drawing *Vatu* been created ?

**4.3 Creation**

The concept *creation* is quite more problematic to address. Quite entangled with
the concept of intelligence (Sternberg, 1997), creativity has been perceived di-
versely and controversially in psychology, intelligence science, and cognitive sci-
ence (Wang, 2009). Moreover, in the context of the societies of Vanuatu, it is
always difficult to clearly identify the origins of the drawings, and the practice
itself is undoubtedly the fruit of a slow evolution. The creation of the drawings
undoubtedly followed the same trajectory, each drawing being enriched with new
details over the generations of experts, the word *creation* encompassing this evo-
lutionary process.

<footnote n="26"><sup>26</sup>We have identified alternative tracing procedures for a similar drawing named *Peng* in North Ambrym, related to the "green turban" (a marine gastropod)(Vandendriessche and Da Silva, 2022).</footnote>
## The "graft" operation

The analysis of various corpora led us to postulate the existence of *operations* that are used in the design of new drawings. Although there is no widely recognized equivalent in vernacular language to convey this concept, substantial clues indicate that these operations play a crucial role in the creative process (Da Silva, 2022, p.171-242). We give a brief overview of them by focusing on the operation *graft*, which involves the addition of cycles by "grafting" one cycle onto another.

<figure><img src="image_30.png" /><figcaption>Figure 29: Diagrammatic representation of a grafting operation</figcaption></figure>

Starting with an existing cycle, the process involves adding a second cycle at a junction point, which is a couple (node, direction) in the drawing (figure.29b<fnref n="27" />). The tracing is then adjusted as follows: As the finger traverses cycle 1 along edge 1, the route changes at the junction point. At this point, the path branches onto cycle 2, which then loops back to the junction point. From there, the original route of cycle 1 resumes, eventually returning to the starting point. This operation can be iterated as much as possible. Note that cycle 2 can have more than one common point with cycle 1, offering new ways of traversing the cycles. A graft can be done at each common node between two cycles **that share the same direction** (to respect Rule 4: "no change in direction at each node"), as exemplified by the figure.11.

## Break into cycles to reconstruct the creation process

The figure.30 illustrates the breaking of the drawing *Vatu* into nine cycles, colored and enumerated from 0 to 8.

<footnote n="27">²⁷The cycles depicted here are as closed simple curves without embedding them in a specific SD (figure.29a). Such diagrammatic representations are commonly designed to closely mirror the knot or graph operations (Starikova and Giaquinto, 2017; Casati, 2013; McLeay and Piggins, 1998).</footnote>
<figure><img src="image_31.png" /><figcaption>Figure 30: The drawing Vatu broke into a cycles decomposition</figcaption></figure>

<figure><img src="image_32.png" /><figcaption>Figure 31: A possible creation process</figcaption></figure>

Cycle 0 serves as the starting and ending point. Due to its proximity to construction vocabulary, it functions as an "armature" onto which four circles (huri) are grafted to form the second pattern in figure 31. Four additional "Livon Gahi" cycles (table 1) are then added to complete the drawing. Given the numerous shared (node, direction) couples across cycles, multiple traversal paths exist with possible cycle reorderings. Observations show that some drawers use alternative
methods<sup>28</sup> to $M^4$.

# 5 Discussion and conclusion

## 5.1 Sand drawing in the 4E framework

### A bodily performance

Influential contemporary cognitive psychologists have highlighted the active participation of gestures (especially with hands) in speech (Goldin-Meadow, 2005; McNeill, 2019) and the memory process (Stevanoni and Salmon, 2005; Iani, 2019), aligning with an embodied conception of MC (Núñez, 2006, 2012; Tran et al., 2017). This takes on particular significance when we consider motions whose vernacular names derive from verbs of bodily action or craft terminology (Table 2). Moreover, using the index finger to draw resonates with recent research suggesting fingers offer cognitive advantages over pencils, providing richer sensorimotor experiences and improving visual attention and memory performance (Xie and Zhou, 2024). Developmental evidence shows that finger writing, combined with manipulative abilities, represents an early stage in infant development that precedes the use of conventional writing implements for representational drawing (Tanaka et al., 2003). For example, in Japanese educational contexts, the practice of kusho writing characters in the air with the index finger has been shown to facilitate the acquisition of kanji (Itaguchi et al., 2015; Fernandes and Araújo, 2021).

These investigations resonate with embodied theories of cognition and help explain the performance of practitioners, who engage not merely their fingers, but their entire bodies in the drawing process. Often adopting extended physical positions, they make SD a bodily performance that cannot be separated from the cognitive abilities it requires.

### Sand and ash as affordances

The use of the ground as a drawing surface (rather than paper or a computer screen) exemplifies extended, embedded and enacted cognition, and thus deserves particular attention. The first implication concerns the ephemeral nature of the drawings. As demonstrated in section 4, this ephemerality leads to memorization strategies and external representations rooted in the practitioner's immediate environment. This exemplifies the enacted dimension of cognition within a cultural context (Varela et al., 2017), particularly through the concept of "affordance", the term whose original meaning is "the ability of humans (and animals in general) to guide their behavior by perceiving what the environment offers in terms of potential actions" (Gibson, 2014b; Luyat and Regia-Corte, 2009). The SD rules listed in section 2 provide a meaningful illustration of this concept. If we consider SD as a system that has emerged from the enactive coupling between practitioners, the medium (sand), and social interactions within and between groups, an important question arises: How have these rules emerged? These rules are not purely internal constructions but have emerged from dynamic interactions between successive generations of practitioners and environmental affordances. The physical properties of sand or ash, for example, explain Rules 4

<footnote n="28"><sup>28</sup>Two animations, *Animation 1* and *Animation 2* demonstrate different methods by two North Ambrym practitioners, *drawer 1* and *drawer 2*. The drawing *Peng* in North Ambrym is equal to *Vatu*, except a pattern is drawn beforehand.</footnote>
and 5 (traversing grid nodes without changing direction/not passing two times on the same route). A directional change at nodes would create a stationary point of zero velocity, causing the sand or ash to compact in an unsightly manner that disrupts the drawing's aesthetic quality. In the same way, passing twice on the same route would deform the initial tracing. Rule 3 (crossing at 45 or -45°) remains less well understood and would require further investigation to elucidate its cultural grounding.

## 5.2 The mathematical dimension of SD

In our introductory remarks, we emphasized the cross-cultural and cross-historical variability of mathematical practices. Building on this foundation, we examine here the mathematical "nature" of SD by putting into perspective the various external representations of practitioners and those of the researcher who has modeled SD practice (Table 3). Our aim is to emphasize that they take the form of tacit knowledge in the *Raga* tradition.

<table><caption>Table 3: Putting representations into perspectives</caption><thead><tr><th>Researcher representations</th><th>Indigenous representations</th></tr></thead><tbody><tr><td>Cartesian plane Z[i]</td><td>The grid drawn on the ground<br>The construction vernacular terminologies<br><i>Gaina/Gai Volovolo/Jooan Paen</i></td></tr><tr><td>The set Z[i] × {1, i, -1, -i}</td><td>The rule 3 ("direction to direction")</td></tr><tr><td>The motions G = Z × Z × {1, i, -1, -i}</td><td>The mix of rules 2 and 3<br>("node to node, direction to direction")</td></tr><tr><td>The group structure of G</td><td>All patterns m<sub>1</sub>m<sub>2</sub>...m<sub>p</sub> carrying a vernacular name,<br>used as a "new" motion</td></tr><tr><td>A representation of a motion (α, β, γ)</td><td>All various curves that are used in the corpus<br>while corresponding to the same values α, β and γ</td></tr><tr><td>Cycles,<br>Cycles as global stabilizers of the action</td><td>Cycles/spiral ontology in *Raga* life :<br>marriage, pig tusks, kinship system</td></tr><tr><td>Given m, finding m<sup>-1</sup></td><td>Finding new cycles to create new drawing</td></tr><tr><td>Factorization M<sup>p</sup></td><td>Tales or myths that involve p identical steps</td></tr><tr><td>The action of the group G on the set X</td><td>Physical action of the finger on the grid</td></tr></tbody></table>

## SD as diagrams, diagrams as mathematics

The juxtaposing of the researcher's formalism with practitioners' rule-following actions raises a fundamental epistemological question, underlaid by the classical philosophical tension between thought and action. To what extent can these two kinds of representation be complementary facets of the same mathematical reality? This issue has traits similar to those encountered in the history of mathematics, particularly when historians examine textual records where diagrams<sup>29</sup>

<footnote n="29"><sup>29</sup>A diagram in mathematics is a visual representation or graphical illustration used to depict mathematical concepts, relationships, structures, or processes.</footnote>
function as substitutes for verbal exposition. Chinese and Indian mathemati-
cal traditions, for example, extensively employed diagrams to construct "proofs
without words" (Chemla, 2018; Keller, 2005). Similarly, the mathematician Rot-
man<fnref n="30" />, relying on Gilles Châtelet's work, has suggested that diagrams serve not
merely as components of category theory's language but as instruments of defi-
nition and proof themselves (Rotman, 2012). In his view, the arrows traced with
a pencil represent not mere illustrations but the incarnation of the mathematical
idea itself. In this respect, SD constitute visual productions whose underlying
logical structure can be perceived in Table 3. As external representations of the
fundamental pillars of Sia Raga culture - including myths, rituals, and kinship
systems - we argue that SD function analogously to mathematical diagrams and
to what ethomathematician Marcia Ascher called "traces of mathematical ideas"
(Ascher, 1991).

**Group (in) action**

We discuss here the hypotheses that group actions could constitute a new avenue
to explore for research in MC, expression in *Raga* culture taking the form of tacit
knowledge that underlie the practice of SD.

Our hypothesis could be corroborated by empirical evidence from recent neu-
roscience findings that have highlighted a phylogenetically ancient cognitive ar-
chitecture (Izard et al., 2022; Spelke, 2011; Dehaene and Brannon, 2011). In fact,
to facilitate the sensorimotor experience of Euclidean 2D or 3D geometric forms
and evaluate their equivalence, humans inherently act on them through the
group of Euclidean transformations (translations, symmetries, and rotations)<fnref n="31" />.
We claim that this group action could constitute an example of a more general
ability to act on a set through a group structure. Let us take the toy example of
moving an object from *A* to *B*, then from *B* to *C* (figure.32). According to these
findings, we "intuitively" understand that 1) this task is equivalent to moving
from *A* to *C* (with the vector **AC** representing this motor action mathematically)
and 2) that the operation is reversible by returning from *C* to *A*. This does not
mean that we "do" mathematics when moving an object, but rather that there
exists tacit knowledge, widely shared across cultures, that makes statements 1)
and 2) universally true. Extending this framework to our analysis, we assert that
SD offers an instance of a group action that takes the form of tacit knowledge
within the *Raga* culture. Even if this knowledge is only manifested through a
small group of experts, *Sia Raga* pay more attention to the performance than to
the final artifact. From an indigenous perspective, SD is a practice whose value
lies less in the drawing itself than in the dynamics of the movements that enable
its construction. Therefore, all the external representations in Table.3 exemplify
the group of motions in action within a cognitive ecology.

<footnote n="30">³⁰To demonstrate that category diagrams are embodied in gestures, Rotman used the ordinal 5 category diagram as the basis for a choreography. https://vimeo.com/80409215</footnote>

<footnote n="31">³¹Although these works do not emphasize that this can be expressed as a group action, the Euclidean group is discreetly mentioned in a footnote in (Izard et al., 2022, p.2).</footnote>
<figure><img src="image_33.png" /><figcaption>Figure 32: The colored motor action of an object and the grey a mathematical representation of the action.</figcaption></figure>

**5.3 Educational perspectives**

**Mathematics in motion**

Recent scholarship has increasingly recognized the importance of embodied cognition in education (Shapiro and Stolz, 2019), with empirical studies compelling educators to understand mathematical concepts as dynamic, fluid phenomena inseparable from the material world and learners' embodied experiences (De Freitas and Sinclair, 2013, 2014; Tran et al., 2017). For example, children have been shown to perform significantly better - both faster and more accurately - on mathematical problems when allowed to use gestures (Alibali and DiRusso, 1999; Goldin-Meadow et al., 2001, 2009). Moreover, these gestures seem to function not only as mnemonic tools but also actively help generate new mathematical ideas and their transmission between learners (Walkington et al., 2019) or between the teacher and students (Liu and Takeuchi, 2023; Nemirovsky and Ferrara, 2009). These insights have inspired innovative "Embodied design" approaches, such as "Graspable Math", a digital environment where learners manipulate algebraic terms such as Lego blocks (Abrahamson et al., 2020). The authors argue that it enables students to "feel the logical structure in [their] hands", which particularly resonates with the present study. The same idea can be found in the use of Rubik's cube (Turner and Gold, 1985) to "manipulate with hands" group theory concepts such as 3-cycles or algebraic relations such as $(gh)^{-1} = h^{-1}g^{-1}$ (Cornock, 2015).

Such perspectives could require careful consideration. Critically, as Elizabeth De Freitas and Nathalie Sinclair have pointed out, rethinking the place of embodied cognition in education would require one to "de-essentialize the body as traditionally conceived" (i.e., merely as a biological entity) (De Freitas and Sinclair, 2014, p.16). In cross-cultural contexts, an anthropological approach would be essential to recognize the body as a complex social and cultural construct whose symbols, uses, and experiences are shaped by norms, beliefs, and social representations (Lock, 1993; Csordas, 2002; Mascia-Lees and Mascia-Lees, 2011). Our research contributes to this emerging framework by demonstrating how mathematical thinking manifests itself through culturally specific embodied practices. As we have stressed the role of the finger as a tool for drawing and this role in memory process, it could be a promising avenue to explore in an educational context. While finger use in numerical cognitive tasks has at-
tracted substantial research attention (Fischer and Brugger, 2011; Bender and Beller, 2012; Wasner et al., 2014; Prete and Tommasi, 2020), finger drawing remains relatively understudied, perhaps less common in contemporary Western practices.

## Use the potential of diagrammatic representation in cultural context

The concepts of "visual thinking in mathematics" and "visualization in mathematics" have drawn increasing interest (Presmeg, 2006; Giaquinto, 2007), alongside the related notion of "diagrammatic reasoning in mathematics" (Giardino, 2017). Within this framework, drawings are conceptualized as "we-spaces", that is, not merely externalizations of thought, but integral components of a dynamic process of individual or collective reasoning (Andrade et al., 2022). Building on this perspective, many educators argue that visualization can be understood as a mathematical language and mode of thinking that rely on a bidirectional relationship: visual thinking requires an expressive language, and this language must be grounded in concepts to be meaningful (Nardi, 2014; Clements, 2014). Nevertheless, they contend that its integration into teaching — as "proofs and definitions without words" — presents challenges, opportunities, and significant epistemological shifts for curriculum design. Our study tends to show that these pedagogical artifacts should not be decontextualized from their cultural embeddings, exemplified by recent developments in Vanuatu's education system<fnref n="32" />.

## Mathematics outside of the box : artifact from other culture

Following the example of SD, can artifacts from diverse cultures serve as pedagogical tools in mathematics education? We propose an approach that considers materials not as constraints but rather as an "affordance space". More precisely, the embedding of the learner in "an (abstract) range of possibilities provided by any active movement in body or change in environment", is conducive to the emergence of mathematical concepts (Gallagher, 2015). This approach has recently been developed through the "materialized action" of fiber making (Peppler et al., 2025) or the "material reasoning" in folding paper (Friedman and Rittberg, 2021). Just as the mathematical dimensions of SD arise from a space of affordances (material, bodily and cultural), such activities foster mathematical reasoning among students engaged with different types of materials (yarn, needles or paper) and sensory and physical elements (visual perception, muscular tension, rhythmic movement) (Mendoza Straffon, 2024; Peppler et al., 2022). Although our study aligns with these perspectives, the authors recognize that these activities "are not representative of all learners" and suggest that these material approaches could be supplemented by ethnomathematical methodologies (Peppler et al., 2025, p.23). Since one of the primary objectives of ethnomathematics is to restore meaning to educational content that has been removed from its cultural context (Eglash, 2000; Bishop, 1988; D'ambrosio, 1985), a potential outcome of this type of work would be to provide new approaches to teaching and expanding mathematics content in postcolonial contexts, thereby fostering the reappropriation of endangered traditional knowledge (Beatty and Clyne, 2020; Verran, 2018; Lipka et al., 2014). In this respect, textile work offers promising perspectives in

<footnote n="32">³²Since 2010 in Vanuatu, primary and secondary education curricula have incorporated the acquisition of traditional knowledge, such as SD, as one of the school's objectives. These teachings are never disconnected from broader indigenous classification systems: agriculture, belief systems or relationship with nature</footnote>
multicultural contexts, as recently demonstrated with students from the Cook
Islands in New Zealand (Hunter and Hunter, 2024).

This article suggests that SD can serve as an intermediary between cultures,
leveraging their mathematical potential while avoiding their reification. An ongo-
ing experiment in New Caledonian classrooms <fnref n="33" /> seeks to implement the mathe-
matical aspects of SD to enhance an algorithmic course in grades 7-8. Through
block programming, students are instructed to program motions as blocks and
construct specific SD such as *Vatu*. Upon completion, students are encouraged
to identify alternative drawing methods, which allows to introduce the concept of
cycles and Hierholzer's algorithm (Hierholzer and Wiener, 1873). This classical
graph theorem facilitates a comparison with smartphone games of the "one-touch
drawing" variety (Figure 33), which involve discovering a path in a Eulerian graph.
Although such research appears promising, it is imperative that educators strive
to remain closely aligned with indigenous perspectives, thereby advocating for the
integration of anthropological and ethnomathematical training in teacher educa-
tion programs. For example, it would be fruitful to present the mathematical
definition of cycles while exploring their cultural and existential significance in
the lived experience of *Sia Raga*. Additionally, as previously noted, the use of
computers or tablets to execute the drawing process is not a neutral cognitive
transposition; thus, these experiments require careful consideration.

<figure><img src="image_34.png" /><figcaption>Figure 33: Finding a new path in an SD can be compared to find an Eulerian path in a graph</figcaption></figure>

**5.4 Limitations and future considerations**

This work has been carried out according to traditional ethnographic methods,
but some aspects of this work could invite legitimate criticism, notably the small

<footnote n="33"><sup>33</sup>New Caledonia, a French territory currently in a decolonization process, has a population that is half Melanesian, with cultural affinities to Vanuatu. This creates a highly multicultural educational context.</footnote>
sample size of the practitioners (N = 5). Although at least 30 less-skilled partic-
ipants could have been recruited (including children and women in particular),
in the cultural context of Vanuatu, the manifestations of shared knowledge are
distributed across specific times, places, and individuals. Restricting the sample
size has led to a virtuous phenomenon: at some point, the expert participants
feel sufficiently confident in the researcher to free up some of their time to take
part in the experiments.

In addition, some hypotheses will require further investigation and interdis-
ciplinary research that expands the 4E framework. As anthropologist Maurice
Bloch emphasized, one of the main challenges in addressing cognition in oral tra-
dition societies is to address it not as an individual process, but as a distributed
phenomenon among the group (Bloch, 2012). In our study, while mathematical
decompositions into cycles compensate for our inability to trace back to the cre-
ators, the processes of memorization would need to be studied in greater depth
using methods from cognitive psychology that focus on the cultural and collective
dimensions of memory and cognition (Hirst and Manier, 2008; Hutchins, 2010;
Wang, 2016). A promising avenue involves examining the "graft" operation to
determine how it reflects these memory processes. Future research directions
should also explore the potential of fine-grained concepts in cognitive ecology
such as action-perception loops (Gibson, 2014a; Gordon et al., 2011). With each
practitioner's gesture simultaneously modifying the perceptual environment and
guiding subsequent actions, the mathematical concepts presented in this article
could emerge not just from the affordances of materials but from the dynamic
coupling between learner actions and environmental feedback. Moving away from
representation-based approaches to embodied perspectives, this framework has
recently provided a fine analytical lens for observing and interpreting the learning
process in geometry (Shvarts et al., 2021; Pittalis and Drijvers, 2023).

Finally, while acknowledging the critiques and controversies surrounding the
4E framework (Alexander, 2025; Wołoszyn and Hohol, 2017; Goldinger et al.,
2016), we align with recent scholarship that questions its scientific validity through
empirical evidence from music learning, improvisation, and creation (Forbes and
Cantrell, 2023; Torrance and Schumann, 2019; Van Der Schyff et al., 2018). By
fostering a dialogue between ethnomathematics and the 4E framework, we hope
to have opened new avenues for understanding the ontogenesis of mathematical
ideas across cultures. Several richly documented mathematical knowledge sys-
tems would deserve future investigation, including the algebraic logic underlying
kinship systems (Rauff, 2016; Bush, 1963; Guilbaud, 1970; Lévi-Strauss, 1949)
and the creation of continuous ground patterns such as Indian kolams (Govin-
daraj and Mahendran, 2018; Gopalan and Vanleeuwen, 2015; Waring, 2012; Ishi-
moto, 2006). Another promising direction involves the integration of activities
such as sand drawing for students with cognitive disorders, similar to musical
training approaches that demonstrate long-term cognitive benefits for patients
with dyscalculia (Ribeiro and Santos, 2020; Rodriguez et al., 2019). We hope
that our study will encourage future research in this direction.
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