Does a sea slug understand hyperbolic math?
Celebrating mathematics as material play
What does it mean to “know” mathematics?
We usually think of math in terms of equations and symbols, but all over the world we find embodied practices encoding rich mathematical knowledge within material making processes.
African artisans, for instance, have been incorporating fractals into textile designs, hair braidings, village layouts and elsewhere, for perhaps a thousand years – long before Western mathematicians began to explore these ‘fractionally-dimensioned’ forms in the late 19th century.1
Using nothing more than a stick or a finger, Polynesian artisans draw remarkable diagrams in sand enacting university level graph theory.
Islamic tiling masters long ago found all possible tessellations of the plane, and even discovered aperiodic tiling patterns as early as the 16th century,2 whereas Western mathematicians only came to these surreal constructs in the 1960’s – most famously Sir Roger Penrose, whose name is now associated with “Penrose tiles”. [For a post on more recent developments in aperiodic tiling patterns see here.] 3
Celtic, Chinese, and Mayan craftspeople have been making knots for centuries – in some cases millennia – preempting the European mathematical study of knots by thousands of years.
All these are examples of an important underlooked category of mathematical thinking in which engagement with mathematical objects is done not though symbols, but via tactile material play. In an essay for the just-published catalog of the exhibition “Seeing the Unseeable: Data, Design, Art” I look at this tradition and propose a category of what I want to call “vernacular mathematics.”4
The term is inspired by historian Pamela H. Smith’s fruitful concept of “vernacular science”. Smith, a Columbia professor and Renaissance expert, studies early modern Europe with a focus on the ways in which artisanal making processes were instrumental in helping to develop what we now call ‘science.’5
Smith speaks of “imitation as knowledge making,” and her research zeroes in on techniques from the 15th through 19th centuries that were developed to represent and imitate nature. How were pigments created to emulate the distinct qualities of animals, flowers, beetles, and so on? What “lifecasting” techniques were invented to capture the soft bodies of lizards, snakes, and roses? What methods did pre-modern artisans concoct to make imitations of precious stones such as jasper and jade, or coral contrefaict, a faux version of branched red coral?
Smith’s work extends far beyond the usual modes of academic scholarship, for at Columbia she founded the “Making and Knowing Project,” setting up a laboratory where she and her students re-create recipes and processes described in centuries-old texts. Here they actually make faux jasper. They cast lizard toes. They taxidermy rats.
Smith argues that early modern artisans who developed these techniques should be seen as partners in the evolution of science, for their explorations helped shed light “on how nature works.”
In addition to passing along knowledge in their workshops, these “literate artisans” began to publish their findings in books and pamphlets, thereby creating “an early kind of technical writing” which helped to “lay the groundwork for how we think about scientific knowledge today.”[i]
To recognize the contributions of these non-academic material thinkers, Smith proposes the term “vernacular natural history” – to be understood as a parallel practice to the usually-told story of science as the progress of increasingly academic conceptual ideas.
As a corollary to Smith’s notion of “vernacular natural history,” I propose we start to value the many practices of “vernacular mathematics” – processes of material making also carried out by non-academic, often artisanal communities, which also embody deep mathematical ideas.
Though these practices may not have influenced the formal canon of Western mathematics – as Smith says happened with respect to the canon of Western science – nonetheless these makers were, and still are, working with rigorous mathematical content.
Among my favorite examples of vernacular math are Polynesian sand drawings, complex patterns created by drawing with a stick, or finger, in the sand with the constraint that one must do the entire thing in one go, without ever lifting the stick and without going twice over any part of the line. Such images are instantiations of what mathematicians call graphs – some being composed from multiple levels of subgraphs nested in complicated counterintuitive ways – a topic of university level study.6
Knots are another topic of tertiary level, even elite-research-level, math. Yet Mayan and Chinese artisans, and Celtic stone makers have been exploring these forms in some cases for thousands of years.
In the Crochet Coral Reef project I do with my sister Christine Wertheim, we explore the mathematics of non-Euclidean geometry using the artisanal craft of crochet.
The straight-edged Euclidean geometry we typically learn in school doesn’t apply to corals and kelps and many other reef organisms. These ancient lineages are biological manifestations of an alternative structure known as hyperbolic space which distinguishes itself to our eyes in swoops and curves. Although human mathematicians spent centuries trying to prove anything like this was impossible, reef creatures have been making hyperbolic surfaces in the fibers of their being since the Silurian Age.
Does a sea slug or a brainless head of coral “know” hyperbolic math? I want to say that in some sense it does.
The question of how humans might imitate this fleshy embodiment of a “pathological” geometry was no easy problem to solve. To do so, artisans had to work out how to embody hyperbolic math – and it turns out we can do so with crochet. Using a female-coded craft we can generate hyperbolic coraline frills, realizing in yarn an idea that shook mathematics to its core and was later used by Einstein in his general theory of relativity.7
Ladies crocheting doilies have long been exploring the frontiers of hyperbolic space.
The discovery of “hyperbolic crochet” is attributed to Cornell mathematician Daina Taimina who created soft-bodied hyperbolic surfaces as pedagogical tools for college-level geometry classes. Taimina’s brilliance was to identify how a humble craft could be employed to emulate a structure mathematicians had struggled to visualize for two hundred years. With crochet, she crafted models they could see and feel and manipulate in their hands. A crafter can even stitch lines onto such surfaces to demonstrate bizarre properties of hyperbolic space, such as the angles of a triangle adding up to less than 180˚and the circumference of a circle measuring more than 2πr.8
But if Taimina was the first to recognize the mathematics embedded here, she wasn’t the first to construct such shapes — women crocheting doilies have been making hyperbolic surfaces for at least a hundred years.
In the collection of doilies owned by my sister and I, we have an exquisite piece of lacework from the 19th century with cascading layers of ever more dense hyperbolic frills. Also, a selection of 1940s pattern books for “Ruffled Doilies” featuring dozens of patterns for hyperbolic edgings spelled out in stitch algorithms incorporating subroutines and other staples of computer-coding techniques.
The artisans who wrote out these patterns—and the women who reproduced the objects in their homes—had a clear understanding of how hyperbolic surfaces behave. Theirs was, and is, a mature form of mathematical “knowledge making.”
Thus, in parallel to the academic study of hyperbolic geometry going on in university math departments in the latter half of the 19th century, wives and maids at home were also developing an understanding of non-Euclidean concepts. Using what Smith calls “sensory tools of embodied experience,” ladies crocheting doilies have long been exploring the frontiers of hyperbolic space.
And this tradition of handcrafted exploration continues today in our Crochet Coral Reef project, with thousands of women all over the world creating a taxonomy of non-Euclidean forms through the humble tools of hook and yarn.
*
For a deep dive into this tradition, see Ron Eglash’s African Fractals: Modern Computing and Indigenous Design (New Brunswick, NJ: Rutgers University Press, 1999). The diagram at the start of this piece is from Eglash’s book.
See Metin Arik, “Mathematical Mosaics, Islamic Art and Quasicrystals,” talk given at Crystallography for the Next Generation conference, Hassan II Academy of Science and Technology, Rabat, Morocco, April 23, 2015, https://www.iycr2014.org/__data/assets/pdf_file/0016/111706/Session2_Arik.pdf
For a stunning visual record, and fascinating analysis, of Islamic tiling patterns, see Jean-Marc Castéra’s remarkable book, Arabesques: Decorative Art in Morocco (Courbevoie, France: ACR Edition, 1999).
The exhibition “Seeing the Unseeable: Data, Design, Art” was held at Art Center College of Design (Pasadena) from October 2024 - February 2025, as part of the Getty Center’s PST-ART: Art and Science Collide initiative.
Pamela H. Smith, From Lived Experience to the Written Word: Reconstructing Practical Knowledge in the Early Modern World (Chicago: The University of Chicago Press, 2022).
For more on Polynesian sand-graphs and other ethnomathematical practices, see Marcia Ascher’s pioneering work Ethnomathematics: A Multicultural View of Mathematical Ideas.
The discovery of hyperbolic geometry opened the door to a wider exploration of geometric possibility and led Bernard Riemann to develop a generalized description of geometric surfaces or “manifolds.” This Riemannian geometry underlies the general theory of relativity that describes the curving structure of space-time.
Daina Taimina, Crocheting Adventures with Hyperbolic Planes: Tactile Mathematics, Art and Craft for all to Explore, 2nd. ed. (Boca Raton, FL: CRC Press, 2018).
This is FABULOUS! And yet another example of how limited the standard definition of “intelligence” is…