How Kolam Became Mathematics

In 1974, a Madras research group published a paper in a computer-science journal treating kolam patterns as a formal language — one of the earliest moments a household threshold art entered the technical literature on its own terms. Half a century of research has followed, across mathematics, computer science, and anthropology. This essay surveys what that research found, and ends with the correction the field itself insists on: the mathematics was in the practice long before the mathematicians arrived.

What the mathematicians saw

The kolam offered formal study an unusually rich object. Its designs carry explicit symmetry — reflection and rotation held by eye. Its dot grids behave as coordinate systems, laid in counted rows. And its most distinctive family, the sikku kolam, has a property that mathematics finds irresistible: a single line that loops around every dot, crossing itself but never retracing, and closes back on its starting point. Regularity, countability, closure — the practice had built, by hand, exactly the kind of structure formal methods are made to describe.

Pattern as language

The 1974 paper by Gift Siromoney, Rani Siromoney, and Kamala Krithivasan took the linguistic route: if a kolam family’s members are generated by consistent construction rules, those rules can be written as a grammar — and the grammar then generates the family, including members no one has drawn yet. Their array-grammar work made kolam a reference corpus in the study of picture languages, and it established the research programme’s basic insight: kolam designs are not a heap of individual patterns but families produced by rules, which is precisely how practitioners hold them in memory.

The ethnomathematics reading

Where the Madras school formalised, Marcia Ascher’s work asked what the practice itself was doing mathematically. Her analyses read the sikku kolam through graph theory — the continuous line as a traversal problem, akin to the classic question of drawing a figure without lifting the pen — and placed the kolam within a comparative family of continuous-line drawing traditions across cultures. The point of the ethnomathematical reading is not that practitioners are secretly doing academic mathematics, but that mathematical ideas — symmetry, traversal, systematic extension of families — live inside cultural practices that never wrote them down.

Design systems and generation

Later computational work closed the circle by building generators. Yanagisawa and Nagata’s 2007 study set out a design system for kolam patterns — grid plus rule set producing valid designs — and work in this line has shown that even modest grids admit thousands of distinct patterns. The computer, given the practice’s own grammar, can enumerate the family’s possibility space; the practitioner, holding the same grammar in memory, navigates that space by taste and occasion. The two are doing recognisably the same thing by different means.

The direction of the description

Every serious study in this literature makes the same point, and this encyclopedia repeats it as a matter of accuracy rather than courtesy: the mathematics describes the kolam; it does not define, explain, or improve it. The properties the papers formalise — closure, symmetry, rule-governed families — were developed inside the practice, transmitted by observation and repetition across generations of artists, most of whom never used formal notation and none of whom needed it. The research is a description of an achievement, not the achievement itself. Kolam is evidence that sophisticated mathematical thinking can grow through artistic practice — which is, for the study of mathematics as much as the study of art, its deepest lesson.

Open questions

The research literature has since widened. A 2012 study built a global typology of the kolam subfamilies and an expanded gestural lexicon for the loop family — treating the artist’s hand-sequence itself as the notation — and a 2015 paper completed the constructive turn: a formal statement of the pulli kolam’s three mandatory rules and a five-step topological method generating every possible kolam for any arrangement of dots. Cognitive questions about memorisation and the classroom use of kolam in geometry teaching remain live directions whose specific claims await further verified sources; this encyclopedia reports them and withholds the details until the bibliography’s source gate passes them.

References

Gift Siromoney, Rani Siromoney & Kamala Krithivasan, “Array grammars and kolam”, Computer Graphics and Image Processing 3(1), 1974.
Marcia Ascher, “The Kolam Tradition”, American Scientist 90(1), 2002.
Marcia Ascher, Mathematics Elsewhere: An Exploration of Ideas Across Cultures, Princeton University Press, 2002.
Kiwamu Yanagisawa & Shojiro Nagata, “Fundamental Study on Design System of Kolam Pattern”, Forma 22, 2007.
Timothy M. Waring, “Sequential Encoding of Tamil Kolam Patterns”, Forma 27, 2012.
Venkatraman Gopalan & Brian K. VanLeeuwen, “A topological approach to creating any pulli kolam, an artform from South India”, Forma 30, 2015.