Knitting and Math
What is knitting?
Knitting is a technique for producing a two-dimensional fabric made from a one-dimensional yarn or thread. Knitted fabric consists of consecutive rows of loops, called stitches. As each row progresses, a new loop is pulled through an existing loop. The active stitches are held on a needle until another loop can be passed through them. This process eventually results in a final product, often a garment.
Different yarns and knitting needles may be used to achieve different end products by giving the final piece a different colour, texture, weight, and/or integrity. Using needles of varying shape and thickness as well as different varieties of yarn can also change the effect.
The yarn in knitted fabrics follows a meandering path (a course), forming symmetric loops (also called bights) symmetrically above and below the mean path of the yarn. Extra curvature can be introduced into knitted garments without seams, as in the heel of a sock; the effect of darts, flares, etc. can be obtained with short rows or by increasing or decreasing the number of stitches.
If they are not secured, the loops of a knitted course will come undone when their yarn is pulled. To secure a stitch, at least one new loop is passed through it. Although the new stitch is itself unsecured ("active" or "live"), it secures the stitch(es) suspended from it. A sequence of stitches in which each stitch is suspended from the next is called a wale. To secure the initial stitches of a knitted fabric, a method for casting on is used; to secure the final stitches in a wale, one uses a method of binding off. During knitting, the active stitches are secured mechanically, either from individual hooks (in knitting machines) or from a knitting needle or frame in hand-knitting.
- the stitch- formed by wrapping the yarn around the needle and drawing it through the previously knitted loop (called overlap).
- length of yarn- formed by the lateral movement of the yarns across the needles (called underlap).
- sequential formation and interlinking of loops in an axial direction on a lateral array of needles with at least one separate thread being supplied to each.
- produces parallel rows of loops simultaneously that are interlocked in a zig zag pattern
- in order to connect the stitches, each needle deflects yarn laterally between other needles
- knitting needle draws new yarn loop through the knitted loop formed by another end of yarn in the previous knitting cycle
The first step to knitting is casting on. In every cast on method, a slip knot is the foundation for success.
To make a slip knot:
- Unwind a strand of yarn at least 8 inches long from the ball. Hold this in your left hand.
- With your right hand, wrap the ball end of the yarn clockwise around your forefinger and middle finger.
- Pull a loop of the ball end of the yarn through the loop of yarn around your fingers, and
- Drop the yarn off the fingers of your left hand while still holding on to the loop with your right hand.
- Gently pull the tail end until a knot forms at the bottom of the base of the loop.
- Lastly, slip the knot onto a knitting needle.
Now you are ready to cast on. Be sure that the tail of your slip knot is at least three times the width of what the end product measures. With the needle in your right hand and the long tail end hanging to the left and the ball end hanging to the right, close your bottom three fingers of your left hand around the yarn and spread apart the two strands of yarn. Bring your fingers through these strands from behind, being sure that the tail end is over your thumb and the ball end is over your forefinger. 
To cast on:
- With the needle in your right hand, scoop the strand of yarn that runs across your palm to the bottom of your thumb.
- Wrap the yarn on your left forefinger around the front of your needle, counterclockwise.
- Bring the loop of yarn that's on your left thumb up and over the tip of your knitting needle.
- Pull your thumb out and tighten the cast-on stitch.
Do steps 1-5 until you have casted on the desired amount of stitches.
Knit and Purl Stitches
The two most commonly used stitches are knit and purl stitches. There are two basic ways to create these loops. The action of inserting your needles through the bottom of a loop and pulling a new loop down and through the first loop is called 'knitting'. And the action of inserting your needles from the top of a loop and pulling a new loop up and through the first stitch is called 'purling'.
- knit stitches look like “V”’s stacked vertically
- purl stitches look like wavy horizontal lines across the fabric
- by alternating the two you can create vertical stripes (ribbing)
To finish off your now 3-dimensional piece of fabric, you must bind off your work.
- Starting at the beginning of a row, knit a stitch (let's call it blue). Then, knit a second stitch (let's call it red).
- Slide the tip of the left needle under blue's front leg.
- Then let blue leapfrog red by lifting blue up and over red and then letting it drop off the tip of the right needle.
- Only Red is left on the right needle.
- Repeat steps 2 and 3 over and over again, knitting a stitch and then leapfrogging the previous stitch over it, until all your stitches have been bound off and you are left with only one stitch. Cut the yarn about 6 inches from the end and pull it through the last stitch, tightening gently.
For detailed pictures and instructions on how to complete a basic knit, see How to Read Knit Patterns
Math and Handicraft
The math of handicraft was long dismissed as merely a cute trick or an inconsequential coincidence. Now, however, handicraft has begun to come into its own as a legitimate tool for mathematical research. This is especially true of knitting and crochet, which, thanks to the efforts of a new group of researchers, are now receiving a great deal of attention from the world of theoretical mathematics. Yackel and Osinga, along with Sarah-Marie Belcastro of Smith College and Daina Taimina of Cornell University, form the core of the group looking at the intersection of math and craft. Some of them are using craft to help answer math problems, while others are using math to answer knitting problems.
It is believed that the partnership between math and craft dates back to the invention of geometry, where the repetitive patterns seen in ancient baskets and weavings first hinted at a mathematical subtext to the world at large. Later, Alan Turing, the theorist and computer scientist, was often seen knitting Möbius strips and other geometric shapes during his lunch break.
The modern interest in math and craft began in 1997 when Taimina devised a plan for crocheting a hyperbolic plane. Hyperbolic planes  are spaces of negative curvature (imagine the shape of a riding saddle) where all lines curve away from each other. Hyperbolic planes are fairly common in nature, appearing everywhere from the frills on a sea slug to growth patterns of coral to the way the brain folds.
Taimina realized that she could crochet a durable model of the hyperbolic plane using a simple rule: Increase the number of stitches in each row by a fixed factor, by adding a new stitch after, for instance, every two (or three or four or n) stitches. Given two points, all that's necessary is to grab each point and gently pull tight the fabric between them. The line can then be marked, for future reference, by sewing yarn along it.