A Closer Look at Fractals | Hacker News Reader

A Closer Look at Fractals

blog.fract.al

97 pointssigna1123 comments

I don't think the author understands fractals well. I just skimmed the first part of the article and there are at least two really wrong ideas.

The Mandelbrot set is a set but the images are not generated by evaluating some function like a sine function that traces out the border of that set and then coloring the exterior with nice colors. Instead the colors represent the escape times of points and the trajectories of those obtained by iterated evaluation of the defining function are not actually visible. Buddhabrot [1] visualizes the trajectories to some extend but more for artistic beauty than clear visualization.

Secondly there is no third dimension that you access by zooming such that each image is just a slice through some three dimensional object. Yes, one can do this and look at the Mandelbrot set as a slice through a higher dimensional object but zooming into the Mandelbrot set in your standard fractal drawing app really just magnifies parts of it. Any three dimensional appearance is just an [intended] artifact of the color palette used.

EDIT: Maybe the app used treats the escape time as height coordinate turning the Mandelbrot set into a three dimensional landscape and that is what he is trying to say. While a common idea for visualizations it is not an intrinsic aspect of the Mandelbrot set.

EDIT: I just realized that the author also wrote the fractal visualizer used to make those images. Maybe I am just misunderstanding his words or he is not great at explaining what he means or he wrote the app without a good understanding what it actually does...I don't know. Especially given that he has seemingly worked in computer graphics for a long time according to Wikipedia [2].

[1] https://en.wikipedia.org/wiki/Buddhabrot

[2] https://en.wikipedia.org/wiki/Kai_Krause

Believe me, he definitely understands fractals well. :)

But I was also thrown off by the sentence about a "2D slice of a 3D zoomable construct". I think it's just sloppy writing, and indeed he's describing the escape times as generating a 3D surface that you can zoom "deeper" into with more iterations. Still not really a 2D slice... but given his creations I'll just assume maybe he hadn't had his morning cup of coffee yet...

I remember using Kai Power Tools back in the 90's! Certainly one of the most creative interfaces at the time as well...

There are quite a few ways to map aspects of points outside the set to colours or other variables. It’s been a long time since I looked at Frax but I seem to remember it maps iterations, the direction in which points are escaping, and a few other things to a height, colour index, speculative index, etc., and then renders the set by rendering that surface.

It’s not as fun as the four dimensional construct that is all the Julia sets, but probably easier to explain to people.

Not three dimensions, but fractional and fractal dimensions : https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_...

This are different things. The Mandelbrot set iterates z <- z² + c usually with z initialized to zero and c varying across the complex plane with each value corresponding to a pixel in the image. But you can also vary the initial value of z and then the standard Mandelbrot set is a two dimensional slice through a four dimensional generalized Mandelbrot set. You can add another degree of freedom by varying the exponent. The Hausdorff dimension on the other hand characterizes the fractal structure itself, not the number of degrees of freedom the fractal is defined over.

  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 horn is composed of myriads of individual particles organized in that shape. The illusion is stronger but it's no more physical than a galaxy. Then the fractal which is composed of myriads of points is the same.

I don't get what the author is saying, but I think they're just adding confusing noise to their message by mixing some incomplete philosophy around the nature of objects.

I think he's saying "Wow, those fractals! How profound!"

For me the amazing thing about fractals is that they come often as a result of very simple equations.

They show that simplicity and complexity go hand in hand.

You can get complexity from a simple pseudo-random number generator.

Fractals are more interesting than PRNG's to me because they not only result in complexity from simplicity, but because of their self-similarity and beauty.

Pseudo-random generators also have this self-similarity and beauty if you dig into them deep enough.

And if you take an arbitrary thin slice through a fractal it looks pseudo-random.

Consider equation like x <- kx(1-x) that turns chaotic for some values of k.

When you plot its behavior for various values of k around that point you'll see weird self-similarities like in fractals on the border of apparent chaos.

But fractals have self-similarity on many (infinite) scales. They don't simply repeat. Do PRNG's also have this quality?

Check out this video about the chaotic equation I mentioned: https://m.youtube.com/watch?v=ovJcsL7vyrk

To be fair, there is sort of a qualitative difference between the two: the spirals of spiral galaxies are density waves moving independently from the orbiting stars themselves.

When it comes to fractals, there is a much deeper and fundamendal distinction that can be made, though: the difference between connected and disconnected sets. The Mandelbrot set is connected (and indeed simply connected), no matter how much some of its regions might look like made of disjoint components. There can always be found "tendrils" connecting everything to the main bulb. On the other hand, Julia sets are either connected (precisely those that correspond to points in the Mandelbrot set) or disconnected (in which case they are "dust" made of uncountably infinite disjoint points).

I was, similarly to some things Kai says here very inspired to find ever more 'dense' areas in fractals, full of detail everywhere.

I found two basic ways to find fractals that appear to be 'detailed everywhere' rather than only near the inside/outside edge:

One way is to use a negative non-integer exponent (say, -1.528) with the regular generalised Mandelbrot formula I call it 'negmandel'. This makes areas with loads of sharp 'cuts' all compounding with each other. The cuts are actually a digital 'artifact' caused by the way complex logarithm is (most often) evaluated on a computer. I'd be very interested to explore what happens if you try to do the logarithm in such a way that the cuts don't happen, but the maths is beyond me for the moment.

A second way is to use a formula such as 'Nova' (in UltraFractal which is related to Newton's method for root-finding I believe) and turn up the 'relaxation' parameter. On the 'inside' (in the 'lake' if you like) some extremely detailed and very beautiful structures can be found.

Visual examples on my blog:

https://ultraiterator.blogspot.com

Here is are two very clear explanations of fractal dimension by the man himself, Grant from 3b1b.

[1] Fractals are typically not self-symmetric: https://www.youtube.com/watch?v=gB9n2gHsHN4

[2] Fractal charm: Space filling curves: https://www.youtube.com/watch?v=RU0wScIj36o

Alt title: whatever became of Kai Krause?

This is a great writeup. Since humans can't really grasp the concept of infinity, it would be worth considering that the Universe could be only a small fraction of an even bigger Universe that exists alongside the one we inhabit, and that there could be infact multiple Universes. A sort of fractal Universe that never ends and has all possible realities.

“Bottomless wonders spring from simple rules, which are repeated without end.”

-- Benoit Mandelbrot

[2013]

Fun fact: the Mandelbrot set (including its interior) is not known to be computable. A computable (compact) set is one where a program can compute arbitrarily close approximations (in Hausdorff distance) by using a finite set of (rational) points.

Makes you wonder a little bit about the drawings you see rendered.

It's not a practical problem. Renderings are all based on fixed resolution, so the approximation doesn't have to be arbitrarily close. Note this doesn't mean most of the renderers aren't wrong in subtle and not so subtle ways ways, but that you can do something reasonably careful with it and not need unbounded computation.

i tried looking closer but it mostly looked the same. what am i missing?

Great title.

Just looks the same as a broad overview to me.

Yuk yuk yuk

Reply on news.ycombinator.com