XaoS
Encyclopedia
XaoS is an interactive fractal
zoomer program. It allows the user to continuously zoom in or out of a fractal in real-time.
XaoS is licenced under GPL
. The program is cross-platform
, and is available for a variety of operating system
s, including GNU
/Linux
, Windows
, Mac OS X
, BeOS
and others.
XaoS can show the Mandelbrot set
(power 2, 3, 4, 5 and 6), the Octo fractal, three types of Barnsley's fractals
, the Newton fractal
(order 3 and 4), Phoenix fractal and Magnet (1 and 2). XaoS can display Julia set
s from selected fractal parts. Xaos also allows you to enter your own custom formulas.
XaoS is capable of displaying fractals as ASCII art
using AAlib
, which, in combination with being built on freely available gnu tools, allows it to run almost anywhere.
An interactive help and an animated introduction to fractals is given in this software. The introduction is in 10 chapters dealing with different formulas presented in the software and their features.
At the time fractal zoom movies were produced by completely recalculating each frame, even though they naturally had much of their area in common with each other. This made interactive zooming impossible without very powerful computers. Furthermore, unless even more processing is used in order to do antialiasing, recalculating every frame produces a 'twinkle' effect as small bright areas hit and then disappear between pixels.
Yet allowing the user to zoom, rather than jump as in Fractint
, seemed like the most natural way to interact with fractals. In order to create an interactive zoom, Hubička needed to find a way to save the calculations which were already made. It would take up too much memory to save every pixel ever calculated, so the Hubička algorithm only saves the previous frame, and rather than remembering the location of each pixel it can keep them aligned in rows and columns and remember those instead.
The most difficult part of the XaoS algorithm was choosing which saved rows and columns to draw where. Doing this wrong results in distorted images, yet it must be done quickly to be useful. After several different heuristics were tried, eventually the problem was treated as an optimization problem.
The remaining rows and columns are colored in the same as the closest row/column, and are freshly calculated as the CPU gets time to do so. This is a careful balance between keeping the zoom going and increasing the level of detail. Calculating areas where the image is being zoomed to is put at a higher priority since these will be on the screen the longest and this is likely where the user is looking anyway. Zooming out, the reverse occurs and the priority is on the edges.
The Hubička algorithm can also be applied to zooming in on other images where the pixels are calculated, and has been used in other software such as the rtzme complex function graphing program, and other fractal zoomers.
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
zoomer program. It allows the user to continuously zoom in or out of a fractal in real-time.
XaoS is licenced under GPL
GNU General Public License
The GNU General Public License is the most widely used free software license, originally written by Richard Stallman for the GNU Project....
. The program is cross-platform
Cross-platform
In computing, cross-platform, or multi-platform, is an attribute conferred to computer software or computing methods and concepts that are implemented and inter-operate on multiple computer platforms...
, and is available for a variety of operating system
Operating system
An operating system is a set of programs that manage computer hardware resources and provide common services for application software. The operating system is the most important type of system software in a computer system...
s, including GNU
GNU
GNU is a Unix-like computer operating system developed by the GNU project, ultimately aiming to be a "complete Unix-compatible software system"...
/Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...
, Windows
Microsoft Windows
Microsoft Windows is a series of operating systems produced by Microsoft.Microsoft introduced an operating environment named Windows on November 20, 1985 as an add-on to MS-DOS in response to the growing interest in graphical user interfaces . Microsoft Windows came to dominate the world's personal...
, Mac OS X
Mac OS X
Mac OS X is a series of Unix-based operating systems and graphical user interfaces developed, marketed, and sold by Apple Inc. Since 2002, has been included with all new Macintosh computer systems...
, BeOS
BeOS
BeOS is an operating system for personal computers which began development by Be Inc. in 1991. It was first written to run on BeBox hardware. BeOS was optimized for digital media work and was written to take advantage of modern hardware facilities such as symmetric multiprocessing by utilizing...
and others.
XaoS can show the Mandelbrot set
Mandelbrot set
The Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognisable two-dimensional fractal shape...
(power 2, 3, 4, 5 and 6), the Octo fractal, three types of Barnsley's fractals
Michael Barnsley
Michael Fielding Barnsley is a British mathematician, researcher and an entrepreneur who has worked on fractal compression; he holds several patents on the technology. He received his Ph.D in Theoretical Chemistry from University of Wisconsin–Madison in 1972...
, the Newton fractal
Newton's method
In numerical analysis, Newton's method , named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley's method...
(order 3 and 4), Phoenix fractal and Magnet (1 and 2). XaoS can display Julia set
Julia set
In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function...
s from selected fractal parts. Xaos also allows you to enter your own custom formulas.
XaoS is capable of displaying fractals as ASCII art
ASCII art
ASCII art is a graphic design technique that uses computers for presentation and consists of pictures pieced together from the 95 printable characters defined by the ASCII Standard from 1963 and ASCII compliant character sets with proprietary extended characters...
using AAlib
AAlib
AAlib is a software library which allows applications to automatically convert still and moving images into ASCII art.It has been used in a wide variety of programs, including a conversion of Quake II which allows the 3D game to be played on dumb terminals, and to let media players output in text...
, which, in combination with being built on freely available gnu tools, allows it to run almost anywhere.
An interactive help and an animated introduction to fractals is given in this software. The introduction is in 10 chapters dealing with different formulas presented in the software and their features.
Hubička algorithm
XaoS was originally just a "poorly written" Mandelbrot viewer until Jan Hubička added efficient zooming, using a technique sometimes called the XaoS algorithm or Hubička algorithm.At the time fractal zoom movies were produced by completely recalculating each frame, even though they naturally had much of their area in common with each other. This made interactive zooming impossible without very powerful computers. Furthermore, unless even more processing is used in order to do antialiasing, recalculating every frame produces a 'twinkle' effect as small bright areas hit and then disappear between pixels.
Yet allowing the user to zoom, rather than jump as in Fractint
Fractint
FractInt is a freeware program that can render and display many kinds of fractals.-Name:Its name comes from the words fractal and integer, since the first versions of it computed fractals by using only integer arithmetic , which led to much faster rendering on x86 computers without math coprocessors...
, seemed like the most natural way to interact with fractals. In order to create an interactive zoom, Hubička needed to find a way to save the calculations which were already made. It would take up too much memory to save every pixel ever calculated, so the Hubička algorithm only saves the previous frame, and rather than remembering the location of each pixel it can keep them aligned in rows and columns and remember those instead.
The most difficult part of the XaoS algorithm was choosing which saved rows and columns to draw where. Doing this wrong results in distorted images, yet it must be done quickly to be useful. After several different heuristics were tried, eventually the problem was treated as an optimization problem.
The remaining rows and columns are colored in the same as the closest row/column, and are freshly calculated as the CPU gets time to do so. This is a careful balance between keeping the zoom going and increasing the level of detail. Calculating areas where the image is being zoomed to is put at a higher priority since these will be on the screen the longest and this is likely where the user is looking anyway. Zooming out, the reverse occurs and the priority is on the edges.
The Hubička algorithm can also be applied to zooming in on other images where the pixels are calculated, and has been used in other software such as the rtzme complex function graphing program, and other fractal zoomers.