Developing our three-dimensional footing: a first look at the mathematics of socks

Feet are anatomically complex. Many toes. Lots of joints. Bones and curves and odd bits that stick out. It follows only makes sense that socks are also complex. To create these little forms of feet clothing, the crafter must consider the length of the foot, the circumference of the foot at multiple points, and the features of the foot arch. In addition, there are elements of design, such as how high up the leg the socks will go, that must be considered before the cast on is initialized.  

These simple socks use a regular cast-on in the round, as opposed to a toe-up cast-on.

Though we will continue to explore yarnbombing, Angela and I have taken on the challenge of knitting socks - a fitting project for these cold-weather months. We had an idea that the process of making would be tricky, but it was not until we began to leaf through books on sock-making that we began to gain awareness of the depth of design that goes into the creation of socks. 

In the beginning, I was excited by all of the sock patterns (what better way to have cozy toes?), but soon, with some exasperation, I realized that the book I was working with was much too advanced for me. Even the cast on was complicated, let alone the sheer magic that is Judy's Magic Cast-On. Just wait until we write about Judy's Magic Loop! 

I crafted this sock specifically for my body dimensions. Each crafter can personalize the sock pattern for herself. 

We were not trying to find the mathematics of feet and socks, but it was impossible for us to start these knitting projects without seeing the three-dimensional features and possible variations. It was as if the sock maths found us. 

Many socks are made using a toe-up approach, meaning that the cast-on starts at the toe and the knitter knits from toe up to the ankle. To simplify the crafting practice for myself, I chose to start with a pattern that worked opposite in direction, so the cast on was more similar to a hat, a technique I was already familiar with. Angela is currently knitting toe-up socks. 

Both of us are still engaged in the processes of making for our sock projects. 

The heel for this sock pattern is created with a special heel flap, which is knitted with a series of slipped stitches.

The beginning of the socks were fairly simple, except for the series of slipped stitches and yarn-overs that led to the rugged stitch pattern along the ankle. Now, I am at the heel flap, which will eventually reconnect with the body of the heel and remainder of the sock. 

I have had to consider the length of my ankle, the desired length of the sock, and the dimensions of my heel. We have both learned that socks are highly personalizable. 

(Note: In retrospect, I do wish that I had started two socks in the beginning, instead of one, as now I will need to complete a full second sock once the first is finished before I can wear either or both.)

 

How to hold a 2D plane

People used to talk about a flat world. Two dimensions. Length and width. 

We learned about this concept of the world in elementary school. It sounded strange then and still sounds strange today. Why? How are we even able to imagine a flat world? 


When we think about a flat world we initially hold with us the image of a three-dimensional world - a sphere. Then, somehow, we unfold this shape through our mind's eye in order to examine an imagined rectangle with bits of Pacific Ocean on both sides. We move with fluidity between 2D and 3D spaces. It's fascinating that we are able to do this, and even further, it's incredibly useful. 


Angela and I recently challenged ourselves with different knitting routes that lead to the creation of 3D objects - both in the round and on flat planes. We engaged in this exercise as part of a multi-part yarnbomb installation, which we will both expound on in posts to come. One component of this yarnbomb was the installation various 3D shapes, including spheres and pyramids. 


Here, I will describe the process of knitting that emerged as I learned to knit a sphere flatly, without joining in the round. The initial attempts were oddly shaped, but over time I gained a better understanding of what happens when a 2D shape is wrapped and seamed in order to form a 3D object. 

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To begin, the knitter must cast on only a few stitches, X. Larger cast ons will yield only slightly spherical objects with teardropped ends. Then the knitter should make one stitch, or increase, every X number of stitches when knitting on the right side (RS). The wrong side (WS) should be knitted or purled straight across, depending on the intended design (either garter stitch with all rows knitted or stockinette with alternations between knitted and purled rows). 


Continue to increase by X number of stitches until the piece is about twice the length across as you would like for the sphere to be. While knitting, try to engage with the knitted piece imaginatively by keeping in mind that the length of the longest row will ultimately be the circumference of the sphere at the largest (widest) point of the sphere. Once you have reached the desired length across your piece, begin to decrease by knitting two stitches together (k2tog) every X number of stitches. Make sure to decrease evenly, just as you increased, so that the sphere is symmetrical across the widest point. 


To end, bind off just as you would with a scarf, but leave a long tail to seam with. Line up the edges of the piece and use the seam to join the edges. The finished shape will need to be stuffed with something in order to look fully spherical. As we will show in forthcoming posts, Angela and I used styrofoam packaging peanuts to stuff the 3D objects.


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There are many knit and purl processes that will lead to 3D objects. Angela experimented with other forms, in the round and piecewise, which she will write about later. At this stage in our yarn bomb exploration, we aim to incorporate many techniques and shapes so that we can create interesting, and potentially interactive, installations in our community. More to follow.