Why are there minibrots?

[fractal_dust]

I am looking for a mathematical understanding of why there are minibrots in the mandelbrot set.

Can you give me some direction toward this please!

[mrob]

It looks like a tough thing to prove, but the problem is made much simpler by noticing that there are minibrots on the real axis.

That means we can deal with "z -> z^2+c" using normal math (not complex numbers), if we can explain the minibrots on the real axis, a similar explanation must apply to the whole complex plane.

The largest minibrot is on the real axis, located just to the left of -1.75. It has period 3. If you define C to be number like -1.76 and apply the calculation: z -> z^2+c (starting with z=0) you get a period 3 cycle. The existence of the cycle means it's in the Mandelbrot set. Other points like -1.7 or -1.6 do not cycle. So we have an "island".

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C1 = old iter-r2.1-3s     

C1 = list
10  c = -1.76
20  z = 0
30  for i=0 to 15;
40    z = z^2 + c
50  next i
60  end

C1 = run

R1 = c: -1.76
R2 = z: 0
R3 = z: -1.76
R4 = z: 1.3376
R5 = z: 0.02917376
R6 = z: -1.7591488917274
R7 = z: 1.3346048232659
R8 = z: 0.0211700342847
R9 = z: -1.7595518296483
R10 = z: 1.3360226412189
R11 = z: 0.0249564978497
R12 = z: -1.759377173215
R13 = z: 1.3354080376302
R14 = z: 0.0233146269675
R15 = z: -1.7594564281693
R16 = z: 1.3356869226265
R17 = z: 0.0240595552754
R18 = z: -1.7594211377999


As for why the island is shaped like a Mandelbrot set, that's similar to the overall question of why does the Mandelbrot set have a self-similar fractal shape. It relates to the "period doubling bifurcation" in 1-dimensional case (look it up on Wikipedia). There is a more complex (literally and figuratively  ) type of period bifurcation in the general case that results in the shape of the Mandelbrot set.

- Robert Munafo

[miner49er]

Very interesting but you've taken all the mysticism out of the Mandelbrot set! You're like one of those horrible scientists or athiests that explains the world around us without invoking God! Pah, I'm not listening - it's magic I tell you or the signature of God or something!

/joke

[fractal_dust]

Thank you Robert Munafo! I'll start trying to understand the axis-aligned minibrots then.

Also I am a fan of your site, I used to spend hours and hours on your brilliant encyclopedia when I was a child!!

[ker2x]

Very interesting but you've taken all the mysticism out of the Mandelbrot set! You're like one of those horrible scientists or athiests that explains the world around us without invoking God! Pah, I'm not listening - it's magic I tell you or the signature of God or something!

/joke

And i laughed