These unique variations on the Julia set theme, presented in The Science of Fractal Images, challenge us to expand our pre-conceived notions of how fractals should be iterated. We start with a very basic Julia formula:
z(n+1) = z(n)^2 + (0, 0i)
The standard algorithm would test each iterated point to see if it "escapes to infinity". If its size or "modulus" (its distance from the origin) exceeds a preselected Bailout Test value, it is outside the Julia set, and it is banished to the world of multicolored level sets which color-cycle spectacularly. But another way of describing an escaped point is to say that it is "attracted" to infinity. We make this decision by calculating whether the point falls within the "target set" of all points closer to infinity than the boundary created by the bailout value. In this way, the "disk around infinity" is conceptually no different from the disks around Finite Attractors such as those used for Newton fractals.
In the above formula, with c = (0, 0i), this standard algorithm yields a rather unexciting circle. But along comes Peitgen to tell us that "since T [the target set] can essentially be anything, this method has tremendous artistic potential. For example, T could be a so-called p- norm disk ... or a scaled filled-in Julia set or something designed by hand. This method opens a simple [beware when he uses words like that] and systematic approach to Escher-like tilings."
So, what we do is iterate the above formula, scale each iteration, and plug it into a second Julia formula. This formula has a value of c selected by the user. If the point converges to this non-circular target set:
T = [ z: | (z * 15.0)^2 + c | < BAILOUT ]
we color it in proportion to the overall iteration count. If not, it will be attracted to infinity and can be colored with the usual outside coloring options. This formula uses a new Fractint programming feature which allows the use of a customized coloring option for the points which converge to the target Julia set, yet allows the other points to be handled by the standard fractal engine with all of its options.
With the proper palette and parameters for c, and using the Inversion option and a solid outside color from the Color Parameters you can create a solar eclipse, with the corona composed of Julia-shaped flames radiating from the sun's surface.
If you question the relevance of these images to Escher, check out his Circle Limit series (especially III and IV). In his own words: "It is to be doubted whether there exist today many ... artists of any kind, to whom the desire has come to penetrate to the depths of infinity.... There is only one possible way of ... obtaining an "infinity" entirely enclosed within a logical boundary line.... The largest ... shapes are now found in the center and the limit of infinite number and infinite smallness is reached at the circumference.... Not one single component ever reaches the edge. For beyond that there is "absolute nothingness." And yet this round world cannot exist without the emptiness around it, not simply because "within" presupposes "without", but also because it is out there in the "nothingness" that the center points of the arcs that go to build up the framework are fixed with such geometric exactitude."
References:
Ernst, B. The Magic Mirror of M. C. Escher, Barnes & Noble, 1994, pp.102-11. Peitgen, H.-O. and Saupe, D. The Science of Fractal Images, Springer-Verlag, 1988; pp. 185, 187.