Spiral designs are attractive and we can easily get them by transforming periodic designs. An example is the “Iris Spiral” created by Frank Farris. In my earlier post “Nautilus” I tried to explain the method and showed some results. Unfortunately, that post is too confusing. Everything becomes easier using complex functions.
The position (x,y) of a pixel of the output image is the same as a complex number z=x+iy. The mapping to pixels of the input image takes several steps. First we use polar coordinates z=r exp(iφ) and the complex logarithm ln(z)=ln(r)+iφ. This maps the plane to an infinite horizontal strip of width 2π. We get a smooth output image if the image in this plane is periodic in y-direction with a period length of 2π. This corresponds to a translation vector (0,2π). We then can use a translation and scaling as a second mapping to a periodic transformation of the input image. This second mapping matches the translation vector to multiples of the periods of this image.
In the most simple case we get the periodic image from the input image with a mapping using packages of waves with square symmetry. If the components of the wave vectors are integers, then the basic periods are 2π in x- and y-direction and we have to transform the translation vector (0,2π) to a vector of the form 2π(m,n), where m and n are integers. If n is much larger than m then we see m different spiral arms. Each arm repeats n times a distorted primitive cell of the periodic image for one turn around the center. Other spirals will appear depending on the structure of the input image.
If h is the greatest common divisor of n and m then we get an h-fold rotational symmetry around the center of the spiral. As an example suppose that n is a multiple of m. Then we get an n-fold rotational symmetry.
We can similarly use periodic intermediate images of 3- and 6-fold rotational symmetry. Instead of a Cartesian lattice of integer choices (n,m) we then have a lattice of Eisenstein integers.
Here is a result for n=7 and m=2 with an intermediate periodic image of square symmetry. There is no particular rotational symmetry around the center because 2 and 7 are relatively prime. Strong diagonal accents of the periodic image give rise to additional spirals: