Variations on a star

I still have a lot to do on tilings of 12-fold rotational symmetry – but summer is too hot to do serious work. Meanwhile I amuse myself with iterative decorations.

A star made of two large triangles or six smaller ones.

Replace each triangle by a star to get six smaller stars.

Two triangles superimposed form a six-pointed star. This is a well-known old design. Alternatively we can think that there are six smaller triangles put together in a circle. They are smaller by a factor of three compared to the large triangles. We can replace each of these triangles by a star. This defines an iterative procedure which results in a fractal shape if repeated.

Fractal shape resulting from iteration.

I think that I have seen similar designs before. It is interesting that the outline of the shape (a line going around between the blue and the yellow colors) becomes infinitely long. Some holes might be filled out to get a more space-filling image. However, this design is too regular for my taste.

To get something more interesting I transformed the image. I change the distance of the points from the center and keep the direction the same. In particular I used the inversion at a circle of radius R. Then, the new distance of a point from the center is d_new=R/d, where d is the distance before. This is strongly nonlinear. All points outside the circle will go inside it and the points inside will be placed outside. The center of the image will be infinitely far away. This is not correctly done in the coloring. In addition I used factors larger than 1/3 in the iteration. A typical result:

Result after changing the radial position of the points and slightly bending the iteration rules.

Then I used the inversion at a circle mapping also in the iteration steps. This again strongly changes the image:

Further bending the iteration rules …

This shows that a somewhat plain design can be easily made up into a more exciting image.

Notes added later:

This method might also be used to transform images resulting from other iteration methods. Instead of quasiperiodic tilings one gets less scientific but maybe more decorative designs.

Inversion at a circle is an important method in geometry, see Lines become circles at inversion. Thus the straight sides of the triangles actually become bent arcs and should be drawn as such. In this sense, the pictures presented here are not correct.

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