# Knitting and crocheting the Mandelbrot set

## Knitting and crocheting the Mandelbrot set

A babushka is knitting the Mandelbrot set converted into a knitting pattern. She is bounding the void.

*“Boundaries of the void are defined through the process of knitting. The void, in this case, is that which is outside the boundaries of the knitted fabric. Thus, is the knitter herself “suspended” in the void? Or perhaps, as long as a thread connects her to the knitted set, she is part of the non-void – and that’s what makes the knitting so important? “** Marina Sokolovskaya*

*” **The Mandelbrot set can be called the boundary of escaping to infinity. One takes a point not far from zero and then substitutes its two coordinates, x and y, into two simple expressions – resulting in another two numbers, the coordinates of a new point, which are then substituted into the same expression, and so on. So if the initial point is lucky (or “lucky”?) to be within the Mandelbrot set, then, passing through the equation, all the subsequent points will stay close to the origin. But if the initial point is even a little bit beyond the set’s boundary, then its descendants will not stay put – they will lose touch with the origin and fly to infinity. The coordinates of the iterated points will only grow and will never return to the vicinity of zero, where their ancestors dwell. The boundary of the Mandelbrot set cannot be described by even the most complex of equations. It is always generated by trial and error: one takes a point, performs repeated calculations, and sees whether the results remain bounded. It is impossible to check each and every point there is, as their number is even greater than the standard (countable) infinity that, in our childhood, used to begin somewhere beyond a million or a billion. Therefore, the generated boundary is always approximate: this million of points is definitely inside the set, that million of points is definitely outside it, and the boundary is somewhere between these two. A program can zoom in on any section of the known boundary: take new points between the already checked ones and determine which ones are inside the boundary and which ones are outside. The points within the set are traditionally painted black. Any fragment of the boundary, no matter how small, looks similar to the entire boundary. That’s why it is called a self-similar shape, or a fractal. It is impossible to generate and draw it without a computer. Many structures similar to the Mandelbrot set are found in nature: for example, blood vasculature, coastlines, etc.What we are interested in here is the attitude towards the indeterminable boundary: is it better to stay inside and be painted the traditional black, or to stay outside knowing for sure that one will have to fly to infinity, or to exist on the boundary and become infinitely self-similar?”*

*Evgeniy Prigorodov*

**Description of the process:**

Let **С** denote a single crochet (), then **С**² – a double crochet (), – chain stitch () – transition to the next point

Let the power of the number **С** denote the multiplicity of crochet, so that **С****⁵**– is a five double crochet, **С****⁶**– a six double crochet, etc. Thus, we convert mathematical notation into a crochet pattern that any crocheter can read and follow.

After the fourth point, the graphic crochet pattern becomes too unwieldy, and so the babushka crocheting the set naturally switches to the mathematical description of a point: because by this time it is easier and more intelligible for her. For example, if the crocheter needs to represent **42****С****⁶**, she makes **42** six double crochets **470****С****⁹**, – means **470** nine double crochets. Thus it is easier for the crocheter to use the mathematical notation rather than the graphic one. The crocheters are currently beginning work on the seventh point (its representation is given on the picture above) and, since they are dealing with large numbers they have to constantly count and write down their actions.

## Knitting and crocheting the Mandelbrot set

2007-2017, work in progress

Performance:

Threads for knitting, text documentation of the process, knitting hook

Exhibitions:

**2017** / Tyumen / Special project of the 4th Ural Industrial Biennale of Contemporary Art , exhibition “The work never ends”

**2014**** ****/ ****Perm / PERMM, solo-exhibition “Registry”**

**2012** / London / Calvert gallery 22 “The russian art show”

**2011 **/ Moscow / NCCA / exhibition of nominees for the “Innovation” award

**2011 **/ Yekaterinburg / UF NCCA / Special program of the 4th Moscow Biennale of Contemporary Art / exhibition “City-port: there is no sea here”

**2007** / Aizpute (Latvia) / residence “SERDE” / The Second Summer Art Camp “Myths and Technologies”