NextFractal

The beauty of chaos

NextFractal is an application for creating amazing fractal images. The fractal images are generated from simple scripts written in M language. The M language is a Domain Specific Language designed for creating Mandelbrot and Julia sets. NextFractal provides also tools for zooming fractals, browsing images, and exporting images as PNG files.

How to install

NextFractal is free software, available for Mac, Windows and Linux. NextFractal requires Java SDK 8 or later, a computer with multi-core CPU, a modern graphics card, and 1Gb of RAM. Download the software archive for your platform and decompress the archive in your preferred folder. The latest release and the source code are available on GitHub.

DOWNLOAD

The M language

The M language is a Domain Specific Language designed to create Mandelbrot and Julia sets. You can define the orbit algorithm and the color algorithm to use for generating the fractal image. You can define multiple color palettes and multiple orbit traps. You can use expressions to activate rules and compute colors, and you can use opacity to compose rules.

Grammar Specification

Please read the grammar specification of the M language or download the grammar in PDF format. The specification provides the exact definition of all the lexical tokens and syntactic rules you can use in any M script.

GRAMMAR

Grammar Tutorial

Please see the grammar tutorial to know more about the M language. The tutorial shows how to use the M language to generate fractals and implement various techniques for computing colors.

TUTORIAL

The Mandelbrot set

The Mandelbrot set is one of the most fascinating objects that we can encounter in geometry. The discovery of this object in 1979 is attributed to BenĂ´it Mandelbrot, who also founded the new branch of Fractal Geometry. Mandelbrot introduced the word Fractal from latin Fractus, which means irregular, to describe a new class of objects which have similar properties like self-similarity and fractal dimension.

What is a Fractal?

A Fractal is an object with an irregular geometric shape each part of which is statistically similar to the whole. In other words a Fractal contains copies of the whole object at different scale levels.

Definition of Mandelbrot set

The Mandelbrot set is defined as the set of points of the complex plane which generate orbits of a not linear system that don't converge to any attractor for that system. The simplest Mandelbrot set you may create is the set of equation x ^ 2 + w which represents the not linear system x(n + 1) = x(n) * x(n) + w, where x and w are complex numbers, n is a positive integer, and the initial state is x(0) = 0. Even a simple equation like x ^ 2 + w generates a complex object with unexpected properties. One of the most surprising property is that the Mandelberot set is totally connected, which means that given two points of the set you can always connect them with a line contained in the set.

Computer algorithm

In order to represent the Mandelbrot set of equation x ^ 2 + w on the screen of your computer, you have to map the pixels of the screen to the points of the complex plane. For each point p = (px,py) set w = px + py i, and iterate the system equation from 0 to n <= N, where |x(n)| > 2 or n = N, and N is the maximun number of iterations. You will generate a sequence x(0), x(1), x(2), ..., x(n) for state variable x, which represents the orbit of point p. Finally color the pixel with a color derived as function of the state values and/or iteration n.

M script for Mandelbrot set

The M script for generating the Mandelbrot set of equation x ^ 2 + w is very simple. You can easily identify the region [<-3.0,-1.5>,<0.0,1.5>], the iteration interval [0,200], the loop condition |x| > 2, the system equation x = x * x + w, the background color (1,0,0,0), and the color rule which sets the color to element n - 1 of linear gradient when n > 0:

fractal {
	orbit [<-3.0,-1.5>,<0.0,1.5>] [x,n] {
		loop [0, 200] (|x| > 2) {
			x = x * x + w;
		}
	}
	color [(1,0,0,0)] {
		palette gradient {
			[#FFFF0000 > #FFFFFFFF, 10];
			[#FFFFFFFF > #FF000000, 190];
		}
		rule (n > 0) [1.0] {
			gradient[n - 1]
		}
	}
}
				

The script above generates the image below when executed in NextFractal. You can see that the Mandelbrot set has a irregular frontier with many tentacles which are departing from the central bulb. You can't see it but the tentacles are very thin and actually they spread all around.

You may change the region and zoom close to the frontier of the Mandelbrot set to see more details. The more you zoom, the more you find new details and you may find copies of the whole Mandelbrot set. The Mandelbrot set has infinite levels of details which make it the most complex object in mathematics.

NextFractal provides tools for zooming, moving and rotating the images to discover new details. Unfortunately a computer has a finite resolution of decimal numbers, so you cannot continue to zoom the Mandelbrot set over a certain limit.

Julia and Fatou sets

The Julia sets and the Fatou sets are fractal objects derived from the Mandelbrot set. They are generated from the same equation of the Mandelbrot set but with a different initial state. Actually the Mandelbrot set is a map of all possible Julia and Fatou sets. Each point inside the Mandelbrot set generates a Julia set, and each point outside the Mandelbrot set generates a Fatou set. All the Julia sets are totally connected as the Mandelbrot set, and all the Fatou sets are not connected (for that reason they are also called fractal dust).

Computer algorithm

In order to represent a Julia or a Fatou set on the screen of your computer, you have to map the pixels of the screen to the points of the complex plane. Given a constant w = wx + wy i, for each point p = (px,py) set x = px + py i and iterate the system equation from 0 to n <= N, where |x(n)| > 2 or n = N, and N is the maximun number of iterations. You will generate a sequence of state values like x(0), x(1), x(2), ..., x(n), which represents the orbit of point p. Finally color the pixel with a color derived as function of the state values and/or iteration n.

M script for Julia and Fatou sets

The M script for generating a Julia or a Fatou set is very similar to the script for generating the Mandelbrot set. They differ for the initial values of w and x. For Julia and Fatou sets w is a constant and x is a variable. The constant w determines the shape of the fractal:

fractal {
	orbit [<-1.5,-1.5>,<1.5,1.5>] [x,n] {
		begin {
			x = w;
			w = <0.26,0.52>;
		}
		loop [0, 200] (|x| > 2) {
			x = x * x + w;
		}
	}
	color [(1,0,0,0)] {
		palette gradient {
			[#FFFF0000 > #FFFFFFFF, 10];
			[#FFFFFFFF > #FF000000, 190];
		}
		rule (n > 0) [1.0] {
			gradient[n - 1]
		}
	}
}
				

The script above generates the image below when executed in NextFractal. NextFractal provides tools for generating a Julia or Fatou set for a specific constant w from the Mandelbrot set. You don't need to provide the value of w in most of the cases and you don't need to insert the begin statement in your script unless you want to force a specific value as in the example above.

Remember that Mandelbrot set and Julia and Fatou sets are deeply related by the system equation. You can observe that the shape of the Julia or Fatou set for constant w = wx + wy i reflects the shape of Mandelbrot set around the same point w of complex plane.

NextFractal provides a tool for generating a preview of Juila or Fatou set for any point of Mandelberot set. You can use the Mandelbrot set to predict the shape of the Julia or Fatou set.