# Spinning a Mathematical Yarn: A Woman's Fight for Inclusivity by Crocheting Coral Reefs

Updated: Apr 18

Hello everybody! Welcome back to GLeaM! Today, I thought I'd tell you a particularly riveting story about a woman named Daina Taimina who managed to revolutionize a complex field of mathematics by discovering a way to visualize it. She also showcased her work at international art exhibitions and sparked environmental activism surrounding coral reefs—all in one sweep. About ten years ago, Taimina discovered how to visualize a then-very-abstract concept called hyperbolic geometry through crochet, and the idea has caught on like wildfire in the mathematical world.

Today, we'll look at hyperbolic geometry itself and then discuss Taimina's story. We'll also see how one mathematical idea—no matter if the mathematical world dismisses it at first—can make a profound and lasting impact on the world.

__What is hyperbolic geometry?__

__What is hyperbolic geometry?__

Hyperbolic geometry** **is a term that describes surfaces that are constantly *negatively *curved. To introduce this, I'll start with the idea of a saddle point, a point at which part of a surface curves downward while another part curves upward, just like in this image:

Along one axis, you can see that the surface looks like an *upward-facing* curve while along the other, it looks like a *downward-facing* curve. For the most part, saddle points serve as the minimum point as you move along one axis and the maximum point along another. Can you see it?

For more clarity, mathematicians often consider more than just two directions when thinking about a saddle point, and the formal definition states that a saddle point is "a critical point that is not a local extremum" (among some other conditions) if that means anything to you! You'll learn a lot more about them when you get to Multivariable Calculus in college, but for the moment, all you need to realize is that they're little saddles!

If you'd like another visual, think about sitting on a saddle on a horse. Your legs fall on the part of the surface that curves towards the ground, while the front and back of the saddle curve upward:

In hyperbolic (or negatively-curved) space, * every *point is a saddle point. Though the idea of a saddle can be relatively easy to grasp, expanding our concept to think about every single point meeting this condition seems rather crazy. After all, the saddle point itself relies on it serving as a maximum in one direction and a minimum in another, and that is a concept that seems, at first glance, to inherently require other non-saddle points to which we can compare each saddle point.

If you're having trouble thinking about this, you're not alone. In fact, it originally seemed almost impossible for mathematicians to effectively visualize what a hyperbolic space looks like in its entirety; as the idea of combining all of these saddle points into a single surface is far from intuitive. This is where the hero of this story will come in.

Before launching further into the narrative itself, let's take a sidestep to think about constantly *positively *curved space to give even more context to the situation. Here, instead of every point curving in *opposite *directions, the curvature moves in the *same *direction (either curving upward or downward) no matter which way you turn. This is just like what happens at the local min or local max points labeled below.

Think about a point with entirely positive curvature as at the bottom of a bowl. From that point, all surfaces of the bowl curve upward towards your face as you eat your cereal:

A surface with entirely positive curvature is defined similarly to hyperbolic space: *every *point on the surface acts like a local minimum or maximum; *every *single point is like a mini bowl! Does this seem as hard to visualize? Take a second to see if you can picture a 3D surface with entirely positive curvature. Hint: it's something relatively simple!

Following the approach of many of my 'solve-as-you-go' articles, I'll leave an image here to keep you from seeing the answer until you think about it. Jot down some ideas, and only continue reading when you're ready!

**It's a sphere!** Every single point on a sphere looks like a bowl, so a sphere is the typically cited example of positive curvature. For that reason, positively-curved space is often called **spherical **geometry. Spherical geometry is the "opposite" of hyperbolic geometry, and understanding it is extremely important to many mathematical problems involving our planet.

(Also, I did choose the particular painting above for a reason. __Wassily Kandinsky__ depicts circles in this famous painting, and circles are just 2D spheres!)

There's so much more we could unpack about positive versus negative curvature, so you're welcome to reach out or comment below if you'd like to discuss it more. Here's a quick visual summary if it helps, where we're thinking about how different curvatures manipulate triangles. (Flat—or zero—curvature is between positive and negative curvature.)

For the moment, let's think about what we just discovered.

**Spherical geometry and positive curvature are easy to visualize. But, somehow, their direct opposite—hyperbolic geometry and negative curvature—are out of our grasp. **

The ease at which we can consider spheres makes it even more baffling that hyperbolic space is so elusive to mathematicians. It seems like a simple extension of something we already know, but yet, for decades, mathematicians had no real way to visualize it with a truly accurate diagram—other than through increasingly abstract equations. If you read __my previous article__ about art and math, you'll remember that visuals are extremely important for understanding complex ideas. The ideas of hyperbolic geometry, therefore, hinged on some form of artistic breakthrough.

__Crocheting Hyperbolic Space__

__Crocheting Hyperbolic Space__

Into this intellectual void stepped one extremely creative, mathematically-minded woman, Daina Taimina, of Cornell University. With remarkable insight, she figured out how to create a visual of a hyperbolic "pseudosphere"—with nothing other than crochet.

This process allows the crocheter to focus on the curvature at each individual stitch, and allows the structure itself to emerge. One can use a crochet pattern that has a stitch ratio that expands: adding 2 stitches for every 1 stitch, 3 stitches for every 2 stitches, or 4 stitches for every 3 stitches, for example, causes the surface to naturally take on a negative curvature. It is very important that the ratio remains the same throughout the process for the surface to be truly hyperbolic.