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?
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
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.
Taimina had to study hyperbolic geometry to obtain her degree in math at the University of Latvia, and she constantly lamented the trouble she had with the twisted equations and the non-visual way the class was taught. It wasn't accessible, and she felt it required too much imagination when she tried to draw the figures onto 2D space. Years later, she was forced to teach hyperbolic geometry, and she wanted to develop a way to make the subject more understandable to her students than it was for her. Unfortunately, the only 3D models mathematicians had tried to develop at the time were flimsy paper.
Taimina figured out that crocheting was a viable method, and adding stitches at a constant ratio became a tangible, handheld method of physically creating a hyperbolic surface.
One could also experiment with different types of surfaces. The greater the ratio of stitches, the more negatively curved the surface gets. (The one pictured above was crocheted by Claudia Carranza with a 3:1 stitch ratio, and took about a year to complete.)
Here is a video from the University of Auckland in New Zealand that discusses the crocheting process. I find it especially interesting how they chose to display their creations on hyperbolic steel frames.
Hyperbolic Geometry and Parallel Lines
Taimina's model is able to illustrate many important properties of hyperbolic surfaces, most notably the fact that any given line in hyperbolic space has an infinite number of parallel lines through a given point. This is mind-blowing, so give me a second to lay the groundwork.
To understand this, let's first look at flat space with zero curvature.
On a flat piece of paper, every line you could draw has exactly one parallel line through a given point:
See how the line that passes through Point P that is parallel to Line m is pre-determined? That is, there is only one line that meets these conditions. Take a second to think about why that makes sense.
It turns out this statement is not actually one you can prove from other statements. It is so basic that is actually considered one of the five axioms (or postulates) on which all the geometry you know relies. It is called Euclid's Parallel Postulate.
For more background, the father of geometry, Euclid, who lived in the 4th and 3rd centuries BC, wrote a book called The Elements, in which he proved all of the statements he knew about geometry from five basic assumptions. Quoted from Wolfram, they are:
A straight line segment can be drawn joining any two points.
Any straight line segment can be extended indefinitely in a straight line.
Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
All right angles are congruent.
If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
The last one here is equivalent to the Parallel Postulate, and can be rephrased as 'Given any straight line and a point not on it, there exists one and only one straight line which passes through that point and never intersects the first line, no matter how far they are extended.'
If the parallel postulate seems to be an outlier among these axioms, that's because it is. The other four statements seem like much more intuitive assumptions, but the fifth is a bit of a mystery. The parallel postulate is entirely necessary for us to understand anything in our basic sense of geometry, but yet it seems a lot more complex than the idea that "all right angles are congruent."
This disparity confused mathematicians for millennia after Euclid.
In the 1830s, Nikolai Lobachevsky and János Bolyai independently discovered that there are entirely logically-consistent fields of geometry that do not satisfy Euclid's parallel postulate (but do satisfy the other four axioms stated above). These are called non-Euclidean geometries.
Spherical geometry is one of them. In positively-curved space, there are zero lines through a given point parallel to a given line. We consider a line in spherical geometry to be a great circle on the sphere, and any two great circles MUST intersect, so there is no way for two lines to be parallel.
Hyperbolic geometry is another non-Euclidean geometry, and here, any given line has an infinite number of lines through a given point parallel to the original line. I'll let this video demonstrate it.
This is Daina Taimina's fantastic TEDx talk. I recommend watching the entire thing to fully understand the idea here, but for the immediate task at hand, the timestamp is 5:10 for the parallel postulate.
Coral Reefs and Environmental Activism
As you can imagine, the constant rate of adding stitches to a surface is actually a relatively simple rule, and it can be emulated by many structures in nature, including coral reefs. Coral reefs use the exponential increase of their surface caused by the hyperbolic structure to increase their surface area, allowing them to exchange more nutrients with their environment. (Hyperbolic surfaces are also found in sea slugs, kale leaves, flowers, and cacti.)
Today, the concept of crocheting hyperbolic surfaces for display has become viral, and mathematicians and artists alike have latched onto the idea of displaying crochet as coral reefs around the world.
Margaret Wertheim, a renowned science writer, has done much to popularize the hyperbolic crocheting project, celebrating both the awe of coral reefs and the mathematics behind the structures themselves. These international exhibitions have another purpose, as they have been used to spread awareness for environmental activism; they often model the contrast between the untouched vibrant, brilliant coral and the coral bleached by rising temperatures.
Here is an image of one of Wertheim's coral reef exhibitions at the Museum Kunst der Westkünste in Alkersum.
Here is a video about the biology of how a coral reef grows:
And here is Margaret Wertheim's viral TED talk:
Hyperbolic figures naturally encode exponential growth, as the curve's edge expands at a constant, exponential rate. In fact, the image below is similar to something that Taimina originally saw in her hyperbolic geometry class, and it was what gave her the 'a-ha' moment where she was able to visualize a crochet pattern within this noise.
It turns out that this exponential model can also model the spread of COVID-19. This is one of my favorite videos at the moment because it is such a brilliant, visual way of explaining something that is affecting all of us.
See if you can think about how her explanation directly relates to hyperbolic surfaces.
The Visual (and Feminist) Math Movement
Throughout her career, Taimina has faced many stereotypes surrounding the integrity of the work. She says that others have often told her, "Mathematicians do not crochet; they do mathematics." Many mathematicians still live with the idea that math is something that needs to be gatekept and that it is a subject that should only be appreciated and studied by those who can discern order from solely the equations themselves.
With Taimina's influence, crocheted hyperbolic structures have begun to be incorporated as an outreach tool in math festivals, math circles, and classrooms worldwide, teaching students everywhere that math is creative and accessible and empowering more children to reach for the stars in STEM. In short, the success of Taimina's work has illustrated how vitally important visual and artistic connections are in math. She is part of a major movement to reshape how mathematicians approach complex ideas, creatively incorporating new tools far beyond equations, formulas, and numbers. As you have seen in many of my articles, visual representations and new avenues promoting greater accessibility have become a major part of the popular math movement: from the Domino Computer to the Collatz Conjecture visualization to the giant Menger Sponge (which can be located here, here, and here in my GLeaM articles).
The fight for inclusivity, for what was once broadly considered "untraditional" in mathematics, has become central to my life's mission, and I'm honored to share pieces of that mission with all of you through this online community.
Moreover, crochet itself comes with a lot of stereotypes about women, as it, according to Taimina, is seen as a handicraft that women enjoy only to while away idle time. This made it extremely difficult for her groundbreaking ideas to be published in a journal because their connection to feminity somehow made them appear less credible. Though Taimina eventually convinced a publisher to include it, her work was initially not featured the way it should have been. Three years later, crochet made it on the cover of that magazine—but only when a pair of men used a computer program to create another crochet structure to model chaos.
There is still much ground to cover when it comes to incorporating the ideas of women into the mathematical canon. By now, Taimina has received much recognition, but it was an uphill battle to publicize something that might have more quickly adopted if it had been introduced by a man.
I believe visual representations and the feminist movement for math go hand in hand, and we need to keep creating more space for all types of people (and all types of ideas!) to be welcome in math, a subject we have all come to cherish.
As Daina Taimina herself said, "Dare to make unexpected connections." You'll face plenty of obstacles in life, but all it takes is a little creativity to begin to revolutionize your field.
Further Resources and Conclusion
Here are some amazing resources I used to write this article today, which you can use to learn more about the topic:
I love discussing this topic, so feel free to reach out if you'd like to have more dialogue about the mathematical connections within crocheting, coral reefs, COVID-19, and feminism. Also, as all of you know, I am offering online meetings for mentoring if you would like to connect with me. Together we can find even more creative connections... Join GLeaM and become a part of our movement!
Note: GLeaM is now moving into a monthly article schedule. The April edition of GLeaM will likely be published in mid- to late April, and of course, all members and subscribers will be notified over email when the article is published!