Adventures in Mathematical Knitting
Rendering mathematical surfaces and objects in tactile form requires both time and creativity
Getting the Math Right
The way an object is constructed, in any art or craft, highlights some of the object’s properties and obscures others. Modeling mathematical objects is no different: It requires that we make choices as to which mathematical aspects of the object are most important. When it’s possible to do so, I knit objects so that a particular set of properties is intrinsic to the construction.
Most of the objects I make have both topological and geometric aspects. That is, these objects have an overall shape that is preserved when the objects are bent or stretched, and they have a specific form and structure in space. Sometimes the topology takes precedence and sometimes the geometry; this difference dictates whether and where curvature is placed on the object. Often I knit surfaces. These are 2D and mainly smooth, which means that the constructions should have no seams, visible or otherwise. An ideal model of a surface should have no edges or bound stitches. In reality, yarn must have two ends. Similarly, a knitted piece may be made from more than one piece of yarn, but the transition from one ball to another should be invisible. These characteristics are part of a topological model for knitting, which I described in a 2009 paper in the Journal of Mathematics and the Arts.
Another consideration in constructing knitted mathematical objects is surface texture. For example, an abstract Klein bottle’s inside adjoins its outside, so the texture of a knitted Klein bottle should be the same on the two physical sides; there should be no way of telling which real side is being looked at, and no way of identifying a transition between different parts of the knitted piece. For some other objects, the two sides are mathematically distinct and so their textures should be distinguishable.
There’s more to consider—one can also indicate mathematics using a design knitted into the object. As a simple example, one can use stripes to indicate Möbius bands within a Klein bottle.
Most knitted-in designs are mathematically challenging because of the discretization problem: A smooth line or patch of color drawn on a piece of paper or electronically must be knitted as a sequence of discrete stitches. This harkens back to the mesh shown in Figures 3 and 4. Computer scientists who work on visualization of 3D objects have developed algorithms for imposing a mesh on an idealized object. A finer mesh gives a smoother look, and in fact the use of very fine meshes is what produces realistic computer-generated imagery. In knitting, creating a finer mesh requires both a thinner yarn and substantially more time to complete the project. A great application of meshes to knitting appears in a 2012 SIGGRAPH paper, in which Cem Yuksel, Jonathan M. Kaldor, Doug L. James and Steve Marschner explain how they use a mostly rectangular mesh to produce highly realistic virtual knitted garments.
Discretizing a pattern is not as simple as imposing a mesh. Knitted stitches are not square in proportion; they are close to 5:6 in aspect ratio. This fact has to be taken into consideration during design; if it is not, the finished object will be elongated. I usually make a swatch of fabric in the yarn I’m planning to use, measure to see what the stitch aspect ratio will be and then create graph paper with appropriately proportioned rectangles. Next one must determine the overall shape—where there will be increases and decreases, short rows and other construction elements—and restrict the graph paper appropriately. The challenge comes in accurately placing designs or colors on that shape. Here are some examples.
First, consider the torus. A torus is a surface that is essentially a hollow doughnut shape—it is formed by rotating a circle in 3D space about an axis. A torus knot is a closed path that can be drawn without crossings on the torus, so that when the rest of the surface is removed, the path is knotted. On a “flat” torus (similar to the “flat” Klein bottle drawn as a rectangle in Figure 5), a torus knot can be drawn as a straight path with a constant slope. Converting this to knitting requires changing the slope of the path with the curvature of the torus in a consistent way. Luckily, there is a knitted torus construction that minimizes the changes needed. Knitted torus knots are shown in Figure 10.
A further challenge is knitting a line or curve into an object with a bumpy surface texture. Each stitch has to be knit in a single color of yarn (because one stitch is the minimal mesh size), and the surface texture dictates that adjacent stitches interlock colors. Thus, to show a clear line, the line or curve has to be at least two stitches wide. Compare the graph drawn on both Möbius bands in Figures 8 and 11; one has a finer mesh than the other.
Knitting multiple line segments into a curved surface is also quite difficult. In another project, I knitted the complete graph K7 into a torus, pictured in Figure 9. This project required the use of 7-fold symmetry. To make it possible to distinguish between the line segments emanating from a dot, each had to have different slopes, and those slopes had to be modified with the construction of the torus. Additionally, the dots were of nonzero height and width, which threw off slope calculations further.