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Author Topic: Fun with Koch fractals.  (Read 18692 times)
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eNZedBlue
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« on: January 10, 2007, 05:18:25 PM »

Here are some Koch fractals I have made recently. I'll keep the explanations minimal, since the rules are mostly self-evident from looking at them.

I started in 2-dimensions, beginning with the classic Koch Snowflake fractal and recursively subdividing it into 6 smaller snowflakes and one larger (but still smaller than the original) snowflake in the centre, like so:



I colourised the fractal by giving each layer of snowflakes a contribution factor, which influenced how much the colouring affected the pixel. Each time I subdivided and recursed, the contribution factor for snowflakes at that level was halved. Since each pixel is in a different snowflake depending on the recursion level, I summed the colourisation at each level of recursion (multiplied by the contribution factor for that recursion level) over all the levels of recursion (I went about 6 levels deep). Up-and-down pointing snowflakes were coloured black, side-to-side pointing snowflakes were coloured white. After adding over all the recursion layers the result was various shades of grey.

I made a little GIF animation showing continuous zooming into the centre of the fractal (I just cross fade while zooming, so the colourisation adapts as it goes, as if the contrast knob is gradually being cranked up):

Koch Snowflake zoom animation

Next, I tried using a 2D circle-inversion transform to distort the fractal. I colourised it using GIMP, and added an "all-seeing eye" symbol to make it more interesting:



Looking at the top image, an idea suggested itself: the roughly hexagonal meandering outline of the Koch Snowflake suggests the outline of a cube viewed at a 45 degree angle, looking diagonally at one of the vertices. I played around with this idea and figured out that you can make all the "dents" in the outline of the snowflake by chopping away at this cube - the trianglular insets of the 2D Koch Snowflake corresponding to insets in the cube's surface.

The rule to create this cube-based 3D fractal is simple: Start off with a cube of width n, and then attach 8 smaller cubes of width n / 3, centred on the 8 vertices of the original cube. Apply this recursively and voila!



When rendered in an isometric projection, the outline of this object forms the Koch Snowflake:



This means that if you project an infinite, directional lightsource at the object from a diagonal 45 degree angle, a perfect outline of the Koch Snowflake will be cast onto any flat surface behind it.

Hope you like the images  smiley

Cheers,
Chris Hayton
« Last Edit: January 11, 2007, 12:53:31 AM by eNZedBlue » Logged
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« Reply #1 on: January 11, 2007, 12:37:35 PM »

Chris Hayton (eNZedBlue) wrote:
>
>    Here are some Koch fractals I have made recently.
>        ........
>    Hope you like the images   smiley

Some interesting explorations into the Koch, especially the 3-D cube views and their shadows.

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eNZedBlue
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« Reply #2 on: January 11, 2007, 03:05:33 PM »

Some interesting explorations into the Koch, especially the 3-D cube views and their shadows.
Thanks  smiley

The Koch Snowflake overlaid two different ways with Metatron's Cube:





 and the Flower of Life pattern:



"Sacred Geometry" symbols are a good source of inspiration and ideas for fractals in my opinion.
« Last Edit: January 11, 2007, 03:35:11 PM by eNZedBlue » Logged
eNZedBlue
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« Reply #3 on: January 11, 2007, 03:45:08 PM »

It just occurred to me that the same trick with the cube fractal might also work with octahedrons (the polyhedral double of the cube), if you oriented the camera so that it was looking face-on to one of the triangular surfaces. Start off with an octahedron and then add smaller octahedrons at each of the six vertices. Wash, rinse, repeat etc.
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lycium
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« Reply #4 on: January 12, 2007, 06:17:59 AM »

those designs are awesome, there's an excellent matchup between them and the fractals!

about the rendering though: blurring is not the same as antialiasing wink a simple way to draw antialiased circles is by a distance check; you do many of these within the "area" of a pixel (pixels don't have area, but that needs more explanation) and average the results - the result can be made reasonably smooth. btw, this handles the intersection of your circles perfectly, whereas opengl's line rendering can't. yesyes it's a lot slower, but you can also do it in an opengl pixelshader if you like, and the results are definitely worth it smiley

oh and your koch zooming animation ruuuules!!
« Last Edit: January 12, 2007, 06:21:16 AM by lycium » Logged

eNZedBlue
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« Reply #5 on: January 12, 2007, 06:59:40 AM »

those designs are awesome, there's an excellent matchup between them and the fractals!

about the rendering though: blurring is not the same as antialiasing wink a simple way to draw antialiased circles is by a distance check; you do many of these within the "area" of a pixel (pixels don't have area, but that needs more explanation) and average the results - the result can be made reasonably smooth. btw, this handles the intersection of your circles perfectly, whereas opengl's line rendering can't. yesyes it's a lot slower, but you can also do it in an opengl pixelshader if you like, and the results are definitely worth it smiley

oh and your koch zooming animation ruuuules!!
Thanks for your comments, and for the tips on circle rendering. I actually overlaid the designs manually in GIMP using some images of Metatron's Cube and the FoL I found on the net, but I think it's worthwhile doing it properly sometime using the antialiased circle technique you describe.
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eNZedBlue
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« Reply #6 on: January 14, 2007, 05:35:14 PM »

Same thing with octahedrons:





It should be possible to get a Koch Snowflake silhouette by placing the smaller octahedrons on the vertices of the larger one, instead of on the faces (as above). I think it will work several other ways with cubes as well (such as placing the smaller cubes on the lines connecting the vertices instead of one the vertices themselves). I'm going to give tetrahedrons and spheres a go tomorrow night after work smiley
« Last Edit: January 15, 2007, 12:55:52 PM by eNZedBlue » Logged
GFWorld
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« Reply #7 on: January 14, 2007, 05:47:53 PM »

My compliments Chris - I have always a deep respect for really computer / fractal specialists !  smiley
Thanks for the links here to Wikipedia here too  smiley
Margit
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« Reply #8 on: January 14, 2007, 05:50:33 PM »

Wow.

Just... Wow.
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« Reply #9 on: January 14, 2007, 05:54:39 PM »

Sockratease - thats again a wonderful meeting! cheesy
And, I like your Chaoscope basic here too  smiley
Margit
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eNZedBlue
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« Reply #10 on: January 15, 2007, 01:09:30 PM »

Thanks for the compliments Margit and Sockratease smiley

Here's a couple more renders, showing another octahedron-based variation of the solid with the smaller octahedrons positioned on the vertices instead floating off the faces of the larger octahedron.

Perspective:



Isometric:

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eNZedBlue
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« Reply #11 on: January 15, 2007, 02:24:02 PM »

Sorry to flog a dead horse, but here it is with spheres:









I'm pretty sure I've seen the first one (with 8 smaller spheres attached to the larger one) in some raytracing book ages ago.
« Last Edit: January 15, 2007, 02:40:32 PM by eNZedBlue » Logged
lycium
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« Reply #12 on: January 15, 2007, 03:52:06 PM »

I'm pretty sure I've seen the first one (with 8 smaller spheres attached to the larger one) in some raytracing book ages ago.

pretty recently actually: http://ompf.org/forum/viewtopic.php?t=336
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eNZedBlue
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« Reply #13 on: January 15, 2007, 04:47:01 PM »

I'm pretty sure I've seen the first one (with 8 smaller spheres attached to the larger one) in some raytracing book ages ago.

pretty recently actually: http://ompf.org/forum/viewtopic.php?t=336

Nice renders. I see I've got a lot of work to do on my raytracer  smiley

The image that I saw was in some book I read on raytracing back in high school. It was from Pixar or "Palo Alto Research Laboratories" or somewhere like that. I wish I could remember the book's name.  huh?

Is there some kind of established theory on which polyhedra can be used to make "looks like a Koch Snowflake in silhouette" solids? I think I can do it with stella octangula, truncated tetrahedra and cuboctahedrons (that'll be fun, lol) but that might be the limit. Tetrahedra on their own wont work, you'll end up with the regular 3D Koch Snowflake:



Of course you can do it with any roughly spherical shape (e.g. dodecahedrons) but you won't get a perfect Koch Snowflake outline with no holes in the middle... I think.
« Last Edit: January 15, 2007, 04:49:35 PM by eNZedBlue » Logged
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« Reply #14 on: January 16, 2007, 11:36:30 AM »

This is inspirational!

I will be experimenting with programming my own fractal stuff once I learn how these new-fangled computers think!!

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