Stitches and STEM: Part 1
Hooked on Mathematics
By Faith
I love crochet. I did not pick it up to solve complex equations or test the strength of materials. I just wanted to add something to my list of many hobbies, and it only took me a few stitches to get hooked.
Instead of strictly following patterns, I quickly found myself making my own items from scratch. From bulky bags to tiny keychains, I enjoy reverse-engineering projects into simple shapes that can be pieced together. One of my earliest free-handed projects was a sling bag, and my latest is an airpods case.
Breaking down items into simpler shapes and logical steps comes naturally to me, and I never really questioned why the geometry felt so intuitive, until this blog assignment gave me the chance. As a STEM student, I completely missed the link between this craft and my academic skills. But as it turns out, mathematics and crochet are a lot more intertwined than I imagined. Because pieces are built one stitch at a time, crochet is perfect for turning algorithms and surfaces into physical forms. A huge community of mathematics and crochet enthusiasts has actually been out there this whole time, and I am excited to finally join the conversation!
Syntax of a Stitch
The most basic stitch in crochet is called a single crochet which can be thought of as 1x1 square. Pieces are built by stacking these squares side-by-side and row-by-row. If I want to build a straight, flat wall, I just work one new stitch directly into the stitch below it, like so:
where the green square is the new stitch added.
This is the standard way to add a single crochet, which keeps the row count steady (at 6 in this case).
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If I want the fabric to grow wider, I can do an ‘increase’. By working two stitches into a single space below it, I add one new block to my total count. This can be visualised as a hexagon, in which at the base sits the old stitch and the top two sloped edges fan out to become the new stitches.
Conversely, if I want my row to become narrower, I can do a ‘decrease’. This merges two stitches from the previous row into one single stitch. Visually, it is exactly like an upside-down hexagon, pulling the two stitches (the two sloped edges) inward and reducing the row count by one.
These basic functions set the foundation for different topological structures that can be crocheted.
Increments of Increases and Negative Curvatures
When I increase the stitches by a fixed number each row, I end up with a flat surface that has a sloped edge. The slope neatly matches the linear graph of the stitch count against the number of rows. For example, in my first swatch, I strictly increased the number of stitches by 1 each row.
But what if I do an increase every n number of stitches instead of doing a fixed number of increases per row? This adds a significantly greater number of stitches every row, causing the stitch count to grow exponentially. The fabric runs out of room in the 2D plane and extends outwards. This yields a negative curvature, where the surface curves inward in one direction and outward in the other at the same time (like a pringle chip).
A Flat Circle
We all know from secondary school maths that the circumference of a circle is 2πr, where r is the radius. To make a circle, most crocheters approximate the circle as a hexagon and start with 6 stitches. Because the length of a single crochet is 1 unit, if we estimate π as 3, the first row must have a circumference of 2 × 3 × 1 = 6 units and thus 6 stitches.
When I move onto row 2 using blue yarn, the circumference becomes 2 × 3 × 2 = 12 units. Because I now need 12 stitches, I must work an increase into every single stitch. On row 3, the circumference is 2 × 3 × 3 = 18 units, so I scale back and only increase every other stitch using orange yarn.
If you look closely at this pattern, you will notice that to force the fabric into a perfectly flat circle, the increments of my increases must actually decrease as the number of rows goes up.
Hyperbolic Planes
I wanted to see how fast the flat circle could warp. On the fourth and fifth row, I broke the pattern and increased in every stitch. As expected, the new stitches could not lay flat and the surface extended into the 3D space. This ruffled surface is special because every single point on it curves outwards with a negative curvature, creating what mathematicians call a hyperbolic plane. For a long time, these complex planes were difficult for mathematicians to model accurately. However, by simply executing a repetitive increase algorithm with yarn, I can build a perfect, tactile model of hyperbolic geometry right in my hands!
Non-Orientable Surfaces
Crochet is also the perfect medium for visualizing topological structures that are otherwise mind-bending to think about. For example, a standard sheet of paper has two distinct sides and four edges. But a Möbius strip has one continuous side and edge. This may be hard to grasp, so I crocheted one by making a long strip and sewing the ends together after twisting it. To prove the math to myself, I ran my finger continuously along the surface of the finished piece, and without crossing an edge, I ended up right back at my starting point!
Infinite 2D and 3D Fractals
This brings me back to my initial realisation that crochet is essentially a physical algorithm. Because pieces are built one stitch at a time, we can execute physical equations to generate fractals. A fractal is a never-ending geometric pattern that repeats itself at every scale. This means if you zoom into a tiny part of the pattern, it will look like the bigger whole.
By incorporating a technique called interlocking crochet, crafters can replicate fractal patterns such as the famous Sierpiński triangle.
In fact, many lay crafters have been unknowingly creating and appreciating these complex mathematical concepts by incorporating them into projects such as doilies.
Because stitches can easily expand into 3D space, these algorithms do not have to stay flat. Crafters also make 3D fractals. By stacking repeating increases on top of each other, the yarn spirals out into a 3D fractal structure
A Scientific Tool
By experimenting with basic stitch increments, I have realised that crochet is far more than just a hobby. It is a highly effective scientific tool.
This craft is incredibly beneficial for STEM students and visual learners who might struggle with abstract mathematical concepts on a whiteboard. Trying to picture or calculate the exact negative curvature of a hyperbolic plane in your head can be daunting. But when you translate those complex equations into physical stitches, you can literally hold the math in the palm of your hand! It makes intimidating topology accessible and, most importantly, fun. Whether you are a crafter designing your own patterns, or a student trying to wrap your head around dynamic surfaces, all you really need is a ball of yarn and a willingness to experiment.
