It’s can be obvious that the outer bands around the Mandelbrot set form complete loops around the Mandelbrot set. Look at this image. It is the Mandelbrot set with just two iterations calculated.
You can see that the band representing two iterations, travels smoothly around the outer edges and then connects back up to itself. There are no other points that have an iteration count of two except on this band, and all the points on this band are connected by other points with an iteration count of two. This is less obvious but equally true for all other bands. Take a look at this image of the same Mandelbrot set below. It has ten iterations calculated.
You can travel all the way around the Mandelbrot set, following that band, and return to where you started. You could try it for the band representing 100 or 1,000 iterations, but it would take a very long time.
This little idea, of all the bands being single bands going all the way around, doesn’t seem too amazing when you look at the outside of the Mandelbrot set. But when you’ve magnified it 60 times as in the image below
so you are looking at this complex shape surrounded by spirals, it’s quite amazing to think that each band you see somehow works it way around each of the individual nodes in the structure, into and out of all the arms in the spiral and then onwards around the Mandelbrot set, without ever crossing another band or ever quite disappearing.
The Mandelbrot set, the internal black (by default) area, itself is not excluded from this rule. Whenever you come across a miniature copy of the Mandelbrot set (and there are an infinite number of them within the original), you can be sure that this tiny copy is connected to the main Mandelbrot set by one, and only one, infinitely thin filament that you will never see, but whose presence can be detected by seeing the constantly thinning bands of colour squeezing in on it from both sides. as seen below.
Mesmerizing stuff on this blog. Fantastic colors…A bit of a puzzle… A ver, as the Spanish say which means that I shall see and will be educated about all this… Eventually…
Glad you like the content! I add posts and images every week.
Terrific work. Can only suggest more YouTube posts to capture the imaginations of more people. I didn’t know about the connectedness, although it’s probably be derivable from just looking. BTW, I generated the ‘entire’ Mandelbrot Set on a Commodore 64 back in the eighties – wrote the program in C64-BASIC, ran it, and went out for the rest of the day. Came home and there it was on my screen, in monochrome. But I didn’t know how to save the image, nor could I have – I had no non-volatile storage at the time, so I had to turn the machine off and lose it all. But I did it. That is the only program I ever wrote that had non-textual output. Thanks to Douglas Hofstadter for the original interest. BTW2, my first 2 computer programs written on a IBM PC were also inspired by D.H. and Martin Gardener respectively. The first I don’t know how to describe anymore, the second was a John Conway Game of Life cellular automata that output 48×48 square grid and could have wrap turned on or off as well as having its rules altered. It used ASCII graphic chrs for output. I learned about the chaotic nature of society just by looking at the evolution of initial random patterns and imagining the squares were busy people.
Now I’m REALLY impressed! How did you know I wear glasses and have a hexagonal head?
Thanks Tim, I’m glad you like this article. I try to explain fractals in a way that people who are not hardcore maths fans will find easy to read and understand – although I think that maybe you are not like that! I remember when I first came across the Mandelbrot set and though I had the pleasure of using a pc to create the image, it still took a full 20 mins to draw the whole set and I could only magnify it around 10 times before the computer choked and crashed – nether the less I would reboot and try again and again searching around the set and wondering at the beauty of infinity played out on the complex plane, oh how things have moved on! of course with the set being infinitely deep I will never be satisfied as no matter how powerful the computer in the end I always hit a wall in the computing power as I plunge ever deeper into the fractal – It is nice to be able to save these journeys though and I love to share them – on youtube – vimeo and where ever else I can. Indeed the chaotic nature of society can be seen in these patterns and I appreciate your interest.
as for your hexagonal head – check me out – I think nothing beats looking like a tic-tac!!!!
I don’t get this AT ALL! But I kinda like it. Next dog i get will be named mandelbrot, fer shur.
Nice explanation! Connectedness. I think an animated gif with one LONE iteration per frame may help show the exponentially increasing complexity without the previous iterations distracting.
Interesting how each fold of each iteration is a copy and yet unique… kinda like… everything
good idea. I will make an iteration movie and add it to the post soon