A Mathematical Wonderland of Alice, the Poincaré Disk, and a Turtle Named TurtleStitch

An Embroidered Poincaré Disk Using Variegated Blue Thread

At first glance, Lewis Carroll's Alice's Adventures in Wonderland and the Poincaré disk model of hyperbolic geometry seem to have little in common. One belongs to a world of talking rabbits, tea parties, and grinning cats. The other belongs to mathematics, where curved spaces and non-Euclidean geometry challenge our understanding of distance and perspective.

Yet the more I explored the Poincaré disk through TurtleStitch, the more I found myself following Alice down the rabbit hole.

My own journey into this curious mathematical world began last summer at the ICERM Illustrating Mathematics Reunion/Expansion. In a fascinating presentation, by Alba Málaga Sabogal from the Université de Lorraine, I was introduced to Voltaire Brossier's right-angled regular pentagon tiling of the hyperbolic plane on the Poincaré disk. Alba shared a physical model of the tiling. You can see her Poincaré disk in this ICERM video archive; at 10:47. Seeing that model sparked my own curiosity and inspired me to begin exploring the Poincaré disk through TurtleStitch. Although distorted in Euclidean appearance, each pentagon has five equal sides and five right angles in the hyperbolic metric. Seeing this model was my first glimpse of the surprising beauty of hyperbolic geometry.

The Poincaré disk model, introduced by the French mathematician Henri Poincaré in the late nineteenth century, provides a way to visualize hyperbolic geometry inside a circle. Although the entire hyperbolic plane lies within the disk, objects appear to shrink as they approach the boundary, creating a mathematical landscape that seems perfectly suited to Alice's Wonderland.

Three Interpretations of the Poincaré disk which have been sketched onto cardstock. 

The code was exported from TurtleStitch as an SVG and then sketched with the Silhouette machine.

Here is the TurtleStitch code - Poincaré disk - left, Poincaré disk - middle, Poincaré disk - right.  

Over the past year, I have created three interpretations of the Poincaré disk. Each of the above designs began as a TurtleStitch program and was later sketched on a Silhouette Cameo. What fascinates me most about the disk is that it represents a world where the familiar rules of geometry no longer behave as we expect. Straight lines become arcs, distances appear distorted, and objects seem to shrink as they approach the boundary. 

In many ways, this mathematical universe feels remarkably similar to Wonderland. 

Wonderland is a place where the rules of everyday life are suspended and replaced with a different set of rules. Alice repeatedly encounters situations that defy ordinary logic. She grows larger and smaller, time behaves strangely, and familiar assumptions no longer apply. Hyperbolic geometry asks us to do something similar. It invites us to leave behind the comfortable geometry of the classroom and enter a space where our intuition must be rebuilt.

One of the most beautiful features of the Poincaré disk is the way geodesics, the hyperbolic equivalent of straight lines, appear as circular arcs that meet the boundary at right angles. This remarkable property became the foundation of my TurtleStitch programs. By carefully coding these arcs, I was able to create embroidered visualizations of this extraordinary geometric world.

Recently, my interest in this world took on a more personal meaning. My five-year-old granddaughter performed in a ballet production of Alice in Wonderland. To celebrate her performance, I designed and built a three-dimensional paper diorama and a Wonderland-themed vase depicting Alice, the Mad Hatter, the White Rabbit, the Cheshire Cat, and a colorful parrot, the role my granddaughter danced.

As I looked at the diorama and vase, I couldn't help noticing the connection between the projects that had occupied my creative time: the embroidered Poincaré disks and the Wonderland diorama and vase. Each invites us to enter a world that challenges our expectations. Both encourage curiosity and exploration. And both remind us that there is beauty in looking beyond the familiar.

Lewis Carroll, after all, was not only an author but also a mathematician. While scholars continue to debate the extent to which Alice's Adventures in Wonderland reflects mathematical ideas of the nineteenth century, it seems fitting that Wonderland and geometry should occasionally cross paths. Perhaps it was inevitable that the Cheshire Cat would eventually find its way into one of my Poincaré disks.

I created two versions of the Poincaré disk featuring the Cheshire Cat sitting mischievously inside. Reflecting Alice's ever changing world, the Cheshire Cat appears in two sizes, one large and one small. 

Both pieces were stitched as hot pads, combining mathematics and whimsy with a practical purpose. They can be used to protect surfaces from hot pots and dishes.

For the Mathematically Curious

Each design began as a TurtleStitch program built around a key geometric fact: in the Poincaré disk, the hyperbolic equivalent of a straight line appears as a circular arc that meets the boundary circle at a right angle. To draw these arcs, I calculated the radius and sweep angle of each arc from a chosen angular step, the spacing between successive points where neighboring geodesics meet the boundary circle. By varying that spacing, I could create patterns of different densities, ranging from closely woven networks to the spare, cusp-like structure of the Cheshire Cat disk.

All three designs share the defining features of the Poincaré disk model: the boundary circle represents points at infinity, geodesics appear as circular arcs meeting the boundary at right angles, and objects of equal hyperbolic size appear progressively smaller as they approach the edge.

Wonderland meets Mathematics

My embroidered/sketched Poincaré disks, my granddaughter's Wonderland diorama and vase may appear to be entirely different creations. Yet they share a common theme: the joy of discovering that imagination and mathematics are not separate worlds at all. Sometimes they meet in the most unexpected places. 

And sometimes, all it takes is a turtle, a needle, a sketch pen and a curious rabbit to lead the way....a fitting reminder that mathematics and Wonderland are never very far apart.