I like Nautilus shells. They have the shape of a logarithmic spiral. This served me as a new inspiration for creating new pictures.
First I found an anamorphosis that maps the logarithmic spiral onto a square lattice. I use polar coordinates. Making one turn, the distance r from the center increases by a factor a and we get exponential growth. Taking the natural logarithm of r, I get a Archimedean spiral. Then log(r)=log(a)*φ/2π, where the angle φ changes by 2π if one makes one complete rotation. The Archimedean spiral is simply a strip of constant width wound around the center. It can be easily unwound. The new coordinate perpendicular to the strip is y=(log(r)-log(a)*φ)/2π. Imposing y=0 I get the spiral growth in r for increasing angles φ. Now I want that the anamorphosis leaves right angles intact and I note that the radius r and the angle φ are perpendicular directions. Thus x=m*(φ+log(a)*log(r))/2π is orthogonal to y. Here m is an integer number. It determines how many times an image is repeated upon making one turn.
Now we make a periodic picture from an input image. It has periodic length equal to one in x- and y-directions and is mirror symmetric in y-direction to get a seamless image. Then we get a self-similar picture made of tiles of only one shape and decor, bur of varying size.
Here I am using additional mirror-symmetries as in the square kaleidoscope. The results depend strongly on the input image. I can get something resembling a petrified Nautilus:
or entirely different images: