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A representation of a ‘space-filling curve’.

There are several curves similar to this. Technically, this is a Sierpinski Closed Peano Curve. Giuseppe Peano was the first to describe this type of curve. Waclaw Sierpinski used his ideas to generate a curve with no beginning and no end.

Theoretically these curves cover every point on the surface until nothing is left. This can be imagined in mathematical theory but does not work in real life.
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Square Snowflake



The background is worked as a filet crochet grid. It is important that the squares should be square. The pattern gives various ways to achieve that.

The curve is attached to the grid by surface crochet. It is an easy technique but needs a lot of concentration.

The filet crochet background is a 64 x 64 grid. It is best worked in a smooth yarn.

The curve can be added with any thicker yarn.

This is a very light and airy afghan, which could also be doubled over and used as a shawl.
Spacecraft is on a 64 x 64 grid. The examples given here are on a 16 x 16 grid to show how the principle works.
Stage 1

Imagine the grid divided into four squares. The curve takes the same shape through each of those squares.
Stage 2

Each of the four  large squares can be divided into four more squares. The curve passes through each of these squares in the same way.
Stage 3

The squares are subdivided into four again and the curve follows the same path around these.

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larger pictures