During my summer holidays I got an inspiration from the work of the french military architect Vauban. To be able to defend the walls of a city he built fortifications at its corners. Then, to defend the walls of the fortifications he built towers at their corners. This suggests an iterative scheme.

We start with a square. We replace it by five smaller squares. The length of their sides is half the length of the original square. One of them has its center at the center of the original square and the other four have their centers at its corners, see the first figure at the left. Then we can iterate. If two squares have a corner in common they will have a common square after the next iteration step, see the second figure at left. This defines a quasiperiodic tiling of four-fold rotational symmetry. In the figure below you see the result of four iterations. The color of a square shows the number of other squares it is connected to. Black is for 4, dark grey for 2 and light grey for one connection.

This image is much more ordered than the usual quasiperiodic tilings, yet it has many of their properties. We easily see the lines of connected squares with various length. A part of the tiling repeats a certain distance away. This distance increases roughly proportional to its size.

We now just look at the sides of the squares and replace right angles by quarter circles. This gives us a space-filling and self-similar curve. Something like a Sierpinski-curve. You can find an interesting discussion at http://www.cut-the-knot.org/Curriculum/Geometry/SierpinskiCurve.shtml. In the image below I have colored the region inside the curve.

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