Title: Why are there minibrots? Post by: fractal_dust on November 17, 2010, 09:10:08 PM 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! Title: Re: Why are there minibrots? Post by: mrob on November 18, 2010, 09:58:15 PM 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". Hypercalc is distributed under the terms and conditions of the GNU General Public License, version 2, June 1991. Type 'help gpl' at the Hypercalc prompt for details. Go ahead -- just TRY to make me overflow! _ _ |_| . . ._ _ ._ _ ._ | _ | | | | | ) (-` | ( ,-| | ( ~ ~ 7 |~ ~' ~ ~ `~` ~ ~ -' mrob.com/hypercalc Enter expressions, or type 'help' for help. 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 Title: Re: Why are there minibrots? Post by: miner49er on November 19, 2010, 12:04:55 AM 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 Title: Re: Why are there minibrots? Post by: fractal_dust on November 19, 2010, 01:26:07 PM 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!! Title: Re: Why are there minibrots? Post by: ker2x on November 19, 2010, 01:47:52 PM 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 :embarrass: |