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What Is Knit Theory?

Knit Theory is the practice of knitting or crocheting a hyperbolic form as represented in hyperbolic geometry. These forms represent planes of space that are constantly curving away from themselves at all points. Knitting the planes was a practice first developed by Latvian mathematician Daina Taimina to use as university classroom models during her work as an adjunct associate professor at Cornell University in Ithaca, New York

Hyperbolic Geometry
- Hyperbolic geometry was first theorized by mathematician Carl Gauss in 1816. All of the standard Euclidean laws apply, except for the Parallel Postulate. That essentially states that the third angle of a triangle will always equal less than the sum of the two base angles. A hyperbolic plane has a constant negative curvature. Thus, in Euclidean geometry, two parallel lines are straight, but in Hyperbolic geometry, those lines curve in towards one another and are still considered parallel. So, a triangle would constantly be curving in on itself and the angles practically don't exist to be measured.
Physics of Hyperbolic Geometry
- Hyperbolic geometry is non-Euclidean geometry, which means that the planes discussed can not actually be mapped in standard Euclidean N-dimensional space. Hyperbolic planes are curved inwards on themselves at all points, while Euclidean planes are 2-D and 3-D and space does not curve. The simplest way to think of hyperbolic geometry is to imagine an infinite number of lines curving inwards at a single given point.
Knitting vs. Crocheting
- A hyperbolic form needs an exponentially increased number of stitches added to each new row of yarn. This represents how hyperbolic space expands exponentially. The final form will closely resemble a piece of ruffled coral or a sea anemone. When knitting, it can be difficult to place the required number of stitches into larger rows since the length of knitting needles is limited. Crocheting uses only one needle and the stitches are completed one at a time, so there's less worry about holding all of the stitches on the needles. This makes it much easier to add new stitches to each row of a yarn form.
Hyperbolic Patterns
- Crochet Coral Reef at crochetcoralreef.org offers patterns for a number of hyperbolic forms. They can probably be duplicated through knitting, if the knitter starts with just a few stitches and stops knitting while the model is still small. Instructions for a hyperbolic plane and a pseudosphere are included in the patterns. A paper published by Cornell mathematics professors David W. Henderson and Daina Taimina in the Spring 2001 issue of "Mathematical Intelligencer" also details how to crochet a hyperbolic plane, as well as create a plane out of paper at math.cornell.edu.

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