Multibrot set

Click to play video of multibrot set with d changing from 0 to 8

In mathematics, a multibrot set is the set of values in the complex plane whose absolute value remains below some finite value throughout iterations by a member of the general monic univariate polynomial family of recursions.^[1][2][3] The name is a portmanteau of multiple and Mandelbrot set.

$z \mapsto z^d + c . \,$

where d ≥ 2. The exponent d may be further generalized to negative and fractional values.^[4]

Several graphics are available but, as far as can be verified, none of these have been taken a step further to display a 3-D stack of the various stages so that the evolution of the general shape can be seen from other than vertically above.

Examples^[5]

The case of

$d = 2\,$

is the classic Mandelbrot set from which the name is derived.

The sets for other values of d also show fractal images^[6] when they are plotted on the complex plane.

Each of the examples of various powers d shown below is plotted to the same scale. Values of c belonging to the set are black. Values of c that have unbounded value under recursion, and thus do not belong in the set, are plotted in different colours, that show as contours, depending on the number of recursions that caused a value to exceed a fixed magnitude in the Escape Time algorithm.

Positive powers

The example d = 2 is the original Mandelbrot set. The examples for d > 2 are often called Multibrot sets. These sets include the origin and have fractal perimeters, with (d − 1)-fold rotational symmetry.

z ↦ z^2 + c

z ↦ z^3 + c

z ↦ z^4 + c

z ↦ z^5 + c

z ↦ z^6 + c

z ↦ z^96 + c

z ↦ z^96 + c detail x40

Negative powers

When d is negative the set surrounds but does not include the origin. There is interesting complex behaviour in the contours between the set and the origin, in a star-shaped area with (1 − d)-fold rotational symmetry. The sets appear to have a circular perimeter, however this is just an artifact of the fixed maximum radius allowed by the Escape Time algorithm, and is not a limit of the sets that actually extend in all directions to infinity.

z ↦ z^−2 + c

z ↦ z^−3 + c

z ↦ z^−4 + c

z ↦ z^−5 + c

z ↦ z^−6 + c

Fractional powers

Rendering images

All the above images are rendered using an Escape Time algorithm that identifies points outside the set in a simple way. Much greater fractal detail is revealed by plotting the Lyapunov exponent,^[7] as shown by the example below. The Lyapunov exponent is the error growth-rate of a given sequence. First calculate the iteration sequence with N iterations, then calculate the exponent as

$\lambda = \frac{1}{N} \ln | \mathbf{z}|$

and if the exponent is negative the sequence is stable. The white pixels in the picture are the parameters c for which the exponent is positive aka unstable. The colours show the periods of the cycles which the orbits are attracted to. All points colored dark-blue (outside) are attracted by a fixed point, all points in the middle (lighter blue) are attracted by a period 2 cycle and so on.

Enlarged first quadrant of the multibrot set for the iteration z ↦ z⁻² + c rendered with the Escape Time algorithm.

Enlarged first quadrant of the multibrot set for the iteration z ↦ z⁻² + c rendered using the Lyapunov exponent of the sequence as a stability criterion rather than using the Escape Time algorithm. Periodicity checking was used to colour the set according to the period of the cycles of the orbits.

Pseudocode

ESCAPE TIME ALGORITHM
=====================
For each pixel on the screen do:
{
  x = x0 = x co-ordinate of pixel
  y = y0 = y co-ordinate of pixel

  iteration = 0
  max_iteration = 1000

  while ( x*x + y*y <= (2*2) AND iteration < max_iteration )
  {
    /* INSERT CODE(S)FOR Z^d FROM TABLE BELOW */

    iteration = iteration + 1
  }

  if ( iteration == max_iteration )
  then
    colour = black
  else
    colour = iteration

  plot(x0,y0,colour)
}

The complex value z has coordinates (x,y) on the complex plane and is raised to various powers inside the iteration loop by codes shown in this table. Powers not shown in the table can be obtained by concatenating the codes shown.

z^−1	z^2 (for Mandelbrot set)	z^3	z^5	z^n
d=x^2+y^2 if d=0 then ESCAPE x = x/d + a y= -y/d + b	xtmp=x^2-y^2 + a y=2*x*y + b x=xtmp	xtmp=x^3-3*x*y^2 + a y=3*x^2*y-y^3 + b x=xtmp	xtmp=x^5-10*x^3*y^2+5*x*y^4 + a y=5*x^4*y-10*x^2*y^3+y^5 + b x=xtmp	xtmp=(x*x+y*y)^(n/2)*cos(n*atan2(y,x)) + a y=(x*x+y*y)^(n/2)*sin(n*atan2(y,x)) + b x=xtmp

^[8]

References

1. "Definition of multibrots". Retrieved 2008-09-28.
2. "Multibrots". Retrieved 2008-09-28.
3. "Homeomorphisms on Edges of the Mandelbrot Set, Wolf Jung, p.23 "The Multibrot set Md is the connectedness locus of the family of unicritical polynomials z^d+c, d ≥ 2"" (PDF).
4. "WolframAlpha Computation Knowledge Engine".
5. "http://www.nihilogic.dk/labs/javascript_canvas_fractals/ Javascript application "Multibrots, variations on the Mandelbrot theme created by raising z to higher powers, rather than the standard z² of the Mandelbrot.". External link in |title= (help)
6. "Animated morph of multibrots d = −7 to 7". Retrieved 2008-09-28.
7. Ken Shirriff (Sep 1993). "An Investigation of Fractals Generated by z → 1/zⁿ + c". Computers & Graphics 17 (5): 603–607. doi:10.1016/0097-8493(93)90012-x. Retrieved 2008-09-28.
8. P.-O. Parisé & D. Rochon. A study of dynamics of the tricomplex polynomial ηp+c, Non Lin. Din., (2015), http://link.springer.com/article/10.1007/s11071-015-2146-6