More results from the iteration of rhombs

Iterative methods can give many different results with only some small changes. An example is the iteration of rhombs to create a tiling with eight-fold rotational symmetry, see my post “Iteration of rhombs”. Here we can use different angles for the rhomb than 45 degrees. Its angle simply has to divide 360 degrees (the full circle) by an even number to get a good tiling and not overlapping rhombs. To get a two-color rotational symmetry the angle has to divide 360 degrees by a multiple of 4.

First I show my results for an angle of 36 degrees and thus a ten-fold rotational symmetry. I am showing only the outlines of the rhombs because there is no two-color symmetry. Note that this is not a Penrose tiling.

Tiling with 10-fold rotational symmetry created by iteration of rhombs.

Subdivision of a green rhomb into smaller rhombs.

Subdivision of an orange rhomb.

For an angle of 30 degrees we get a 12-fold rotational symmetry with a two-fold color symmetry. Here we can use a special trick because the gap between the rhombs is a rhomb with an angle of 60 degrees. It can be divided into two equilateral triangles and we can put an additional rhomb between these triangles. This rhomb then has the same size as the rhombs resulting from a further iteration of the other rhombs. Thus we have rhombs of just two different sizes for any number of iterations.

Tiling with twelve fold two-color rotational symmetry resulting from the iteration of rhombs.

To me these tilings have both the character of fractal patterns and of quasiperiodic tilings. As an example, the stars of 12 rhombs in the last result show a part of the quasiperiodic tiling discovered a long time ago by Stampfli. Programming is really fun, because you just have to change a few lines of code to get quite different results.

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2 Responses to More results from the iteration of rhombs

  1. Jim Millar says:

    I was playing with something like this a while ago, and haven’t seen anyone else do this. See http://home.comcast.net/~patternblock/infoct.htm and http://home.comcast.net/~patternblock/infdodec.htm if you are interested (though comcast is taking my pages away in a month or so and need to find a new site. 😦 ) If anyone is interested then I’ll post an updated link when I have a new home. I just found your site and am making my way through it. Very cool!

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