Thursday, January 12, 2012

Crochet Mathematics: Hyperbolic Crochet

Think "Crochet Coral Reef" whenever you see the words "hyperbolic crochet.".

Making hyperbolic crochet pieces is not terribly difficult; however, to get spheres is a little more challenging. The base idea is to increase each row in a set pattern. An example would be crocheting in 3 stitches then putting 2 stitches in the next. As along as you increase in a row in a set pattern and every row after that using that same pattern, your ruffly, hyperbolic creation will emerge from your hook.

But let's look at hyperbolic crochet and where it came from.

Daina Taimina is a name synonymous with hyperbolic crochet. Taimina had seen models attempted in paper. While good, she knew that if they could be created in paper, the same models would be more successful if created using crochet - and in 1997 she created her first hyperbolic crochet piece. I have seen it written that crochet is truly the only way to successfully represent hyperbolic space - doesn't that make you feel special as a crocheter?

But what is it? Why is it significant? Why does it get such a fancy, completely enigmatic name?

When a person begins learning geometry, they are usually first exposed to Euclid Geometry. This "high school geometry" is based on postulates established by Euclid, a man considered the Father of Geometry. Euclid established a deductive system in which his approach to geometry was about proofs through axioms. Some of his concepts include, but are not limited to 1) the idea that the shortest distance between two points is one unique straight line and 2) the sum of all angles in any triangle will equal 180 degrees (which consequently is the same degree of a line).

But the most important of Euclid's postulates that conflicts to a postulate in hyperbolic space is this: in Euclid's 2D world, if you have a line and a point outside of the line (not included in the line), there is only ONE line that can be drawn through the point that would never touch the first line.

But because we do not live in a 2D world, this is not necessarily true.

Hyperbolic space is a non-Euclidean geometric space that views space more as a saddle, a space that is not flat. This type of geometry, also called Lobachevskian Geometry, named after the Russian mathematician who helped further the concepts, has a different postulate than Euclid when it comes to parallel lines. While Euclid said that only one line could be drawn through a point and never touch another pre-existing line, the postulate for hyperbolic space states that two or more lines can be drawn through a single point and never intersect with another predefined line.

What a big difference, eh!? In fact, the complete opposite. Everything regarding hyperbolic space postulates looked great until someone came along and said "prove it."

Lettuce Sea Slug
The pro-"saddle space" mathematicians knew they had to be right. They knew that this "saddle space" existed - they could see it! Think of a leaf of lettuce. Or a Pringle's potato chip..... or a sea slug - these are all hyperbolic shapes... and clearly - they exist.

But a slug doesn't usually agree to being used as a chalkboard.

What we see in these shapes is that they can move - the saddle shape is not always the same. Grasping the ruffling, saddle-shaped space with our 2D minds and concepts is very difficult.

And what is one of the best features of crochet? It's flexible. The old paper models were not near as flexible as what crochet is. Crochet can move fluidly much like our little friend the lettuce sea slug.

Taimina's model using crochet is a response to "prove it" when it comes to the parallel postulate for hyperbolic crochet. As you can see in this image, courtesy of SciTechBlog and Daina herself, you can see the postulate is true.

The postulate states that if given a line (the one across the bottom) and one point, there are two or more lines that can intersect through the point but never cross the original given line. As you can see, the bend of all three lines, because of the hyperbolic shape of the space, can go through the same point and never cross the line on the bottom.

Crochet gives us a visual of this postulate. This means on every leaf of lettuce and every tiny sea slug, we have this same postulate. But we can bend and shape the crochet, we can create the lines (as shown in the image above) so others can see what we are saying - and we can see that it is true.

Pretty darn amazing, eh?

As it turns out, coral is also hyperbolic shaped, which is why the creation of the Community Reef for the Smithsonian, a project driven by Margaret Wertheim, became an easy task in reference to what needed to be done, not the volume of what was done.

Sites Referenced:
Smithsonian Institute Website:

Hyperbolic Crochet Image:

Information about Daina Taimina's hyperbolic crochet:

Euclidean and nonEuclidean Geometry:

Sea Slug Image:

Margaret Wertheim Video talking about hyperbolic space and the reef project:


florencefrazier2002 said...

Very interesting!  I love the fluidity of her creations.

Andrea Morrison said...

Wowza!  That's good stuff.  I love those photos and who knew there was so much to be learned about crochet and what you can do with it.  I have to admit, until I started knitting, I thought it was only about afghans but I am learning about all the beautiful projects that can be created.

Cris said...

Its amazing what she has created. I really cannot wait to go see the satellite exhibit for the Community Reef Project.

Cris said...

It is easy to think that knitting and crochet are "just crafts" or "fiber arts" but there are some deep seeded concepts. I mean, even after I finish something I can sit and look at it and think... this is just one long string.... one, big long string twisted together and it looks like this! It's pretty cool :)

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