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

The Mandelbrot set can be called a boundary of escaping to infinity. One chooses a point not far from zero and then inputs two coordinates —x and y —, into two simple expressions. A resulting two numbers, the coordinates of a new point, are substituted into the same expression, and so on. So if the initial point is lucky (or not so “lucky”) to be within the Mandelbrot set, then, passing through the equation, all the subsequent points stay close to the origin. If the initial point is even a little bit beyond the set's boundary, then its descendants will not hold a position; they will lose touch with the origin point 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 point, as their number is even greater than the standard (countable) infinity, which in our childhood, used to begin somewhere beyond a million or a billion. The generated boundary is always approximate: one million of points is definitely inside the set, another million of points is definitely outside it, and the boundary is somewhere in-between. Any fragment of the boundary, even smallest one, 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.

There are many structures similar to the Mandelbrot set that exist in nature: blood vasculature, coastlines, etc. Our interest here is the attitude towards the indeterminable boundary. What is better: to stay inside and be marked black (the points within the set are traditionally marked black), or to stay outside knowing that one will have to fly to infinity anyway, or to exist on the boundary and infinitely self-similar?

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