Some observations about the classic Mandelbrot and Julia sets
by Kai Krause
Part 1: Spirals
There are some fundamental shapes in nature and one of them is the spiral - from the DNA helix to galactic nebulae they can be found in sheer endless variety and across all size scales.
One might expect that the abstract world of mathematics is equally full of spirals and within that the classic fractals of the Mandelbrot and Julia sets also contain many different forms and across all scales as well.
But there are some surprises in how this all happens - and while you may well have seen fractal imagery for years, it is the small details that are worth examining.
This little essay is trying to bring some of these unusual visual forms to light, show examples of the variety and oddity as well - and and muse about their implications a bit…
The images are all generated with Frax on an iPhone and iPad, see the website for more information.
So to jump right in, let us begin with the basic starting point: the Mandelbrot set, here in plain monochrome:
We only need to zoom in at the thin spikes that separate the circular shapes from one another and, already within just a few zoom stages, smaller features will come into view and almost instantly there they are: spirals!
And the first thing to realize is: there is never an actual singular object that would be ‘turning in a spiralling way’, but what is perceived as a spiral is composed of multitudes of repeating smaller shapes that are ‘forming a composite spiral path’ - quite a different concept.
One way to think of it is that for example bighorn sheep: they have horns formed in a physical spiral shape, whereas a galaxy may look like a spiral, but is composed of myriads of individual stars, organized in that shape.
The Mandelbrot and Julia sets are really not objects as such either, they are mathematical virtual constructs which represent boundaries of a very simple equation.
As you evaluate these limits you are computing the coordinates of a point in a continuous curve, just like the formula for a sine wave will draw that familiar shape.
Here is a region with such points drawn in simple 1 bit “on or off” black and white:
It is the special nature of Frax however, that it will not merely draw those lines as show above here… but it is then filling the spaces between them. This is actually the entire definition and character of the Frax images, that they are using those spaces and applying 3D height to raise them up, then adding complex color gradients and textures and then 3D lighting with gloss and reflections: The result are depthy forms as if made from plastic or metal or glass, which kind of “live within” the boundaries given by the Mandelbrot and Julia sets.
Here is an example of what that looks like for the area shown above:
The result is a visualization of the M-Set, but almost in reverse, showing the negative spaces instead!
Note how that “big grey spiral” fits exactly into the white space of that flat drawing above…they complement each other perfectly - and not only in a 2D image, but in 3D as you continue to zoom in! It is important to understand that every picture you see here is merely a 2D slice of a 3D zoomable construct.
To continue further there are many surprises waiting for us. If you look closer at this area you will find that there are two kinds of spiralling behaviours to be found:
The first type is individual small elements that seem to be attracted by a certain point in their center and will look like a path that orbits around that, getting ever closer to it.
The second is also a point in a center, one that attracts the shapes in a radial fashion, seen above in the left half…
It can be likened to a rocket orbiting the earth at a fast speed and slowly getting closer, versus one that has no radial velocity of its own and is falling straight towards the center of gravity.
Both these forms are related and can often be found in may surprising combinations: just look at this region… :
An odd double cross of each spiral type… :
This is really just four points that are attracting the shapes like gravity - but how elegant that turns out to be!
You can click here to download that as a QuadHD 4K image ( 3840 x 2160 ), look at it at 1:1 pixels, zoomed in!
Another unexpected finding as you look around other areas… look at this spiral and see if you spot the oddity:
Yes indeed: what looked like a spiral at first glance is actually TWO identical spirals intertwined perfectly to form one large one! Look at the large grey area in the middle of the left half and follow it towards the center and you soon notice that it is only half of the shape, the rest coming in from the right….
Here is another such thing, easier to see… :
It looks so simple that one may not even realize that this is not at all a straightforward obvious thing to do.
Consider this: if I asked you to take a pencil and draw three spirals, interlocking, that perfectly form a large spiral in the center, always balancing the distances between them…. you would see that you end up having quite a hard time to get that done: things bump into one another very easily and it turns out dilettante and ugly as heck ;)
And yet the fractal boundaries solve this task effortlessly and perfectly - and remember this all is a 3D zoomable depthy shape as well ! You could enlarge that center there and find that it is just as perfect with both arms flowing around it in perfect harmony as you go deeper and deeper…
But can it do three spirals ?? Indeed it can… Here is one example…
…and you are also beginning to see another miraculous property: this entire region here is filled with more copies of these triple spirals…! You can find them on the left and right, at all the edges… and the more you look the more you realize: the entire spiral itself is composed of triple spirals that are feeding into it from all sides! And it is not “a few hundred” or “a few thousand”, it is really infinitely many spirals continuing in-depth as well, each edge composed of further edges with further spirals!
Here you can download the small inset region in the upper right at full Quad HD 4K size ( 3840 x 2160 )
We will come back to his concept of self-similarity, but let us just finish up on the previous notion: It may look so effortlessly easy here, but that trivializes the task: filling a plane - or worse: 3D space with depth! - with balanced interlocking multiple spirals is not at all obvious, let alone simple.
Soon one begins to find more and more examples in different regions for just about every number of spiral arms (and each of the infinite number of copies along the edges and in depth are all using seven as well…)
If that is not awe-inspiring…:
Lest you just figured that “there is a pattern with odd numbers only”, here is an example with four contributories:
Again: it strikes me as an extremely complex problem to solve, to come up with any of these designs if you were given the task just to make a single one on a sheet of paper - but this is not one, not thousands but zillions of them, layered in depth!
At this point it seems appropriate to mention people like M.C. Escher, who has done SO much for the general appreciation of the link between mathematics and aesthetics: showing the sheer beauty in such mundane topics as tessellation ( filling a plane gaplessly ). His images are still timeless and almost unprecedented, a darn shame that he got run over by the zeitgeist, how it could ever be true that an MC Hammer became better known I shall never understand, but such is the ‘wisdom’ of the crowd.
Someone who invested thousands of hours in meticulously executed manual drafting of recursive graphics like Maurits… would probably self-combust instantaneously at the sight of the details of fractals… I will devote another essay some time to exploring similarities in his drawings to things I found in Frax, some eerie parallels.
In no way am I trying to say the Mandelbrot set is “art”, on the contrary: the creative effort and ingenious solutions Escher came up with are truly the pinnacle of real art… but what I am wondering really is: how much a mind like his would be inspired by the shapes and forms of fractals, and how puzzled he would be, to find them hardwired into such a tiny formula.
But before we get deeper let us first look at more examples of the unexpected oddities hiding deep inside the Mandelbrot and Julia sets… and focussing on spirals still!
So we step back and examine the nature of the spirals some more: the images show lots of the little swirlies at all the edges - infinitely many of them. Btw: in Frax itself there is a depth limit as a trade-off to remain fully realtime immersive. In the pure mathematics of fractals however, the zoom to the edges continues to infinity. The computation does get iteratively worse as well, of course - but the self-similarity makes it a diminishing return to spend the time to go there.
Here is an example again: the myriad of little spirals showing up near all edges of this shape:
That looks like a real solid object with carved ornamentals along the edges - but… there are also entirely different kinds of spirals that are most mysterious!
Look at this innocent bit of turquoise spirality here: at first glance not all that different really…
But let us zoom in just a little bit… you notice the gap there in the center? They are clearly not touching there…:
And if you look a little closer, you see that there are several such groups, with gaps in between.. right? Right… but… it gets worse: even the areas that DO look connected… aren’t ! Lets zoom into the center bottom area:
Even if you zoom in some 10,000% deeper, you still get these kind of “cursive P ”-like shapes, which are composed of spirals with spirals:
and then after a while you do realize: NONE of them are EVER connected to anything at all! The whole thing is a construct of a certain “look” that consists of the same kind of lego blocks, all of which have that same look again, and are themselves built up of further lego blocks…
There are two rather profound statements here: one that there are shapes that preserve perfect self-similarity without any limit to scale changes and second that it contains shapes from entirely unconnected building blocks. Seeing that in a graphical depiction in front of you, and letting you zoom in directly in realtime continuously is just a truly deep concept if you think about it a little…
But let us continue with more examples: here is another look at infinite spirals built from spirals of spirals, this time a Julia set in its entirety:
You can download the 4K size here and the extreme bottom tip in the center again as a 4K zoom image here…
But there are still further levels of complexity to be found… Here is another one of those challenges to try:
Drawing a point that attracts spirals radially is already quite a feat. It can look something like this:
But now the surprise part: as you zoom out of this area a bit, you find that that entire massive attractor radial spiral is replicated zillions of times and arranged in yet another super-spiral!
Have a look at this gigantic field of nested mega structures, the image above is a tiny spot in the center here now:
And now we have hit upon a rich vein: examples of dense fields of shapes…!
Here are a few such snapshots, starting with a Julia (often recognizable with the center symmetry)
and here a fun skewed Mini-Mandelbrot Set, red in the lower left corner. Notice that they continue following inside the spiral path towards the center, surrounded by the spirals on the arms, which themselves are all spiralling towards their own centers, wonderfully dense.
This one is available here as a Quad HD 4K image as well, to be viewed zoomed in at 1:1 pixels 100% scale…
Btw, the Mini-Mandels are a topic for a post by themselves: Frax has a built-in database of the largest 10,000 sets, and a little hidden easter-egg: if one shuffles locations with a Mini in view, one lands on another random Mini… Thus one can quickly examine the fields around Minis and they can be extremely dense: here is an example - notice the spiral types everywhere: they are not “regular” in any way, and yet still “evenly balanced” wherever you look and always remember that this is also organized in the z axis, in depth: as you zoom in, the same evenly balanced density continues perfectly… There is no precedence for this, nor an equivalence in nature - it is the pure perfection of the principles in mathematics. ( This image is particularly fun in QuadHD 4K size, download here )
A little earlier we had the case of ‘spirals made up of building blocks that actually never touched’, a myriad of smaller elements,ever receding further towards smaller self-similar ones at the next scale down. But it is not as simple as saying “that is how fractals are made” - here is a region of …… the exact opposite!
In this area everything you see is all connected up as one gigantic structure! If you follow the lacquered black shapes ( all with little spirals at the edges ) you will find out that no matter where you start, it will connect back up and as you can see on the far left edge, there are yet larger “stalks” on the perimeter that are the common link even for those that seem to be on separate spiral arms. (See it better up close in 4K resolution here )
There is a wonderfully perfect balance here between “figure and ground” - which is a theme of Escher in many paintings and drawings, several even called that - and it also touches upon other very central notions such as YinYang, being and nothingness. There it is, interleaved in 2D space and also in 3D depth!
One can create such a thing with spheres in a lattice grid - but… this is achieving it without regular simplistic rules, it looks organic and has wildly differing local phenomena, changing continuously without ever losing that balance. The more you look, the more you can appreciate the impossible challenge of that - and yet it is all contained in that tiny formula.
Now to switch gears, other spiral images that can be simply inspiring. Starting with a little blue thing:
Lovely shapes, wonderful especially the interplay of sharp and soft, the colors condensing.
And yet another very large topic opens up now: as mentioned before, Frax is adding a complex texture engine to the classic Mandelbrots and Julias - and not just simple flat images like wood or marble or metal, but algorithmic textures that scale along with the zooming into depth! They must maintain their clean features across gigantic orders of magnitude. And while they are not part of the pure definition of the Mandelbrot, they illustrate another principle: one can create absolutely beautiful images by using the fractal boundaries as a kind of ‘guard rail’, akin to a ‘playing field’ defined by the M and J Sets, in which textures and color gradients can sprawl.
Here is such an example exploding into irregularities, non-continuities, gigantic spaces of diversity in form, color, detail - pure inspiration for any visual artist. What would Maurits have felt, flying around continuously in this:
(Much more interesting in the larger 4K size here)
There will be purists arguing this is an ‘arbitrary’ image, but it IS related very intimately to the Mandelbrot set: that is what provides the shape in which the texture can live - as if you were to pour a liquid into a plastic spiral. This is a kind of ‘plasticine ferrofluid’ which visualizes the underlying field lines as dictated by the classic M Set - the combination is a wonderful explosion in possibilities: to escape the self-similarity and yet retain the rigorous perfection, even in 3D depth: this “object” is fully zoomable !
There are endless ways to continue there, embellishing the pure points and lines with more complex patterns and forms. The general geometry here is all defined by the Mandelbrot and Julia sets, no objects are modelled here. Just the style, color and lighting are custom features of Frax - a nice symbiosis.
And another one here with lovely “wobbly” shapes adding a further organic feel to the ‘often all too perfect’ lines: Also one for the full 4K resolution here
This example is another supercluster of spiral centers being whirled around in a larger spiral. It has a delicate ornamental “white lace” quality, especially in the full size 4K, download here
In a way we are just getting started with the example of complex textures cast into the spiral shapes, but it is probably enough for starters ;) If you have not seen it, there is a showcase of large files and galleries.
As a closing thought: the self-similarity in the Mandelbrot and Julia sets is often compared to ever smaller scales in nature and you may well know the kinds of “ferns and broccoli and river deltas” that are often used as examples. But there is always a limit, beyond which the broccoli is not looking like the molecules it is composed of, and once you get down to the atom level, they also bear no resemblance to the shapes they build - let alone quarks, where we are at a loss to describe their true nature, grasping at straws of ‘11 dimensional strings vibrating’ and such. At best it works for a few orders of magnitude, but then you hit the Planck length as a brick wall.
In the mathematical version however, there are infinite magnitudes with no end in sight and no limits to express! If you are willing to type in a million digits you can zoom that much deeper - but we have no reason to believe that you will not find the same shapes repeated further down. In fact, some areas have been examined at scarily gigantic scales (easily found in the many articles everywhere).
To me, exploring these fine details and then realizing the extent of the ungraspable is bringing me back to a quotation that influenced me greatly…
The deepest, most sublime emotion we are capable of
is the experience of the mystical and the miraculous.
From that alone true science can emerge.
He to whom this emotion is foreign,
who can no longer wonder and lose himself in awe,
is already spiritually dead.