[HN Gopher] A Closer Look at Fractals
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A Closer Look at Fractals
Author : signa11
Score : 70 points
Date : 2021-07-05 13:22 UTC (9 hours ago)
(HTM) web link (blog.fract.al)
(TXT) w3m dump (blog.fract.al)
| anjel wrote:
| Alt title: whatever became of Kai Krause?
| [deleted]
| beefman wrote:
| [2013]
| [deleted]
| jancsika wrote:
| Just looks the same as a broad overview to me.
|
| Yuk yuk yuk
| warent wrote:
| 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.
| Sharlin wrote:
| 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).
| scotty79 wrote:
| 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.
| pmoriarty wrote:
| 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.
| scotty79 wrote:
| 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.
| sysadm1n wrote:
| 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.
| danbruc wrote:
| 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
| crazygringo wrote:
| 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...
| aardvark179 wrote:
| 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 a
| 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.
| rssoconnor wrote:
| 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.
| wombatmobile wrote:
| "Bottomless wonders spring from simple rules, which are repeated
| without end."
|
| -- Benoit Mandelbrot
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