Adventures in Mathematical Knitting
Rendering mathematical surfaces and objects in tactile form requires both time and creativity
Knitting as Geometry
In a discrete geometric model for knitted fabric, plain knit stitches form a rectangular grid, or mesh, with one stitch sitting inside each rectangle (see Figure 2). We shape knitted fabric primarily by using increases and decreases. True to their names, increases add to the number of stitches in a row, and decreases lessen the number of stitches in a row. Both processes create mathematical curvature in the knitted fabric. Figure 3 shows an increase (angles and stitch sizes are not to scale). Interestingly, making a decrease has the same effect on the mesh; a decrease looks like an increase if the fabric is held upside down.
Rows and columns of stitches draw the eye along paths around the surface. Think of a sphere knitted from pole to pole: The rows and columns of stitches mimic latitude and longitude lines.
All knitting is the generation of global structure via choices made in local stitch creation. A set of stitches appears to create a coordinate system (a grid in two-dimensional space). Because shaped fabric is not mathematically flat, however, any such system is only consistent locally, for that small patch of stitches.
An object that has consistent coordinates locally, but perhaps not globally, is in mathematical terms a manifold. Manifolds have a dimensional restriction: Every patch on a manifold must have the same number of coordinate dimensions. A pullover sweater represents a manifold—unless it has sewn-on pockets, because where a pocket joins the sweater, there are three different coordinate directions (up-down, left-right onto the pocket and left-right under the pocket). A great deal of mathematical research concerns manifolds.
Most (but not all) knitted mathematical objects represent manifolds. In particular, although knitted fabric is of course three-dimensional, it represents two-dimensional things in the same way that paper represents an ideal 2D sheet. That means knitting can stand in for 2D objects—or for the boundaries (“skins”) of 3D objects. Some mathematical objects are 2D but cannot be represented in three dimensions without either intersecting themselves or having holes added. (These are objects that can be immersed, but not embedded, in real 3-space.) Klein bottles are in this class of objects.