The Basics
Author: Chris Cooling, computer programmerSite manager: Emma Lucas
The familiar Mandelbrot Set with colours around the edge of the black 'Beetle'. It is considered a Wonder of the Universe. We didn't invent it. We only discovered it.
Zooming in on the edges reveals limitless coloured beauty. The patterns are made only by the behaviour of numbers.
The image is made up of any chosen coordinates on a scatter plot. The Mandelbrot formula repeatedly processes each coordinate to discover stability.
With a usual maths formula, the coordinate positions are simply predictable; the Y value depends on the X.
With the Mandelbrot formula, any chosen X and Y coordinate is repeatedly processed and the stability of that X Y combination is assessed.
Inside the 'Beetle' the coordinates are stable. Because of the enormous processing necessary, early mathematicians had no detailed knowledge of the overall 'Beetle' shape.
A BIG SHOCK. THE 'BEETLE' WAS DISCOVERED IN THE 1970'S. Benoit Mandelbrot at IBM started computer-assisted research
Take an example inside the 'Beetle'. Coordinate (-0.485 , 0.404) is moved around by repeated maths (iterations) done on each new position.
After unlimited numbers of moves, this coordinate's new positions never escape. It is stable. A black pixel is drawn, a popular colour choice.
COLOURS: On the edges of the 'Beetle', new positions can escape. For instance (-0.65 , 0.6) eventually escapes and is unstable. The number of moves gives the level of instability and colour choices
Computers assign colours representing levels of instability. The colours are an artistic choice by the computer programmers but only numbers determine the shapes.
Zooming in. Small black beetles appear again and again amongst the amazing patterns; small oases of stability where the coordinates do not escape.
Other colours are related to their level of instability. High power computers can zoom in to incredibly tiny areas and are still going into infinitely greater depths.
Youtube has stunning zoom trips into ever tinier areas of Mandelbrot.
So, to the maths. Note the X axis has usual numbers +1 , -1. However the Y axis has unusual numbers +1i , -1i. What is this letter i ?