Further Moves

The second move: Square (0.08+0.56i) giving

-0.3072 + 0.0896i.

Added to the FIRST (0.2 + 0.4i) gives

-0.1072 + 0.4896i

giving the MOVE coordinate (-0.1072 , 0.4896i)

Further moves

To decide if the FIRST coordinate (0.2 , 0.4i) is stable we have to repeat the moves many times. After 7 moves or infinite moves, they stay in the 'Beetle'.

No matter how many more iterations, this FIRST coordinate MOVES remain within the 'Beetle'. It is stable. The common choice black has been decided here for (0.2 , 0.4i).

NOW, COLOURS ON THE EDGES - HERE IS WHERE THE FANTASTIC INFINITE PATTERNS ARE FOUND

On the edges of the 'Beetle', new positions can escape. Our example choice (-0.65 , 0.6) eventually escapes and is unstable. The number of moves gives the level of instability and an associated colour choice.

Try this iteration for yourself.

Convert the coordinate (-0.65 , 0.6) to a complex number and add it to its square

Follow the process described at the top of this page. After the moves shown on the diagram there is an escape.

Mathematicians use an iteration to describe this process. The FIRST coordinate complex number is called C such as 0.2 + 0.4i. The MOVE coordinate complex numbers are called Z such as -0.31 + 0.09i. Each MOVE iteration is written:

Zn+1 = Zn2 + C

That is, the current move is the previous move squared plus the original FIRST complex number.

For simplicity our examples have omitted the first iteration Zn+1 = (0 + 0i)2 + C assigning C to the first Z.

Our example, first 2 iterations:

Zn+1 = (0 + 0i)2 + (0.2 + 0.4i) Omitted

Zn+1 = (0.2 + 0.4i)2 + (0.2 + 0.4i ) Our start

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