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Author Topic: Fun with Koch fractals.  (Read 18532 times)
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eNZedBlue
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« Reply #15 on: January 16, 2007, 11:38:22 AM »

Stella Octangula Koch Crystal:





The Stella Octangula is made up of two intersecting tetrahedra. For each child stella octangula that comes off the parent in the above fractal solid, one of its tetrahedra lies entirely inside the parent, so it's really just made up of alternating tetrahedra after the first generation. There is another Koch solid based on the tetrahedron (see previous post) but it doesn't have the Koch Snowflake silhouette.

I think it should be possible to add some more fractal detail to the "flat" areas by adding more child stella octangula to each parent, in between the existing 8.

If you're interested in a "sacred geometry" connection to this one, it's probably the Merkabah, envisioned by some as a stella octangula on its end, with one of the tetrahedra pointing up, and the other pointing down.

Cheers,
Chris
« Last Edit: January 16, 2007, 11:40:53 AM by eNZedBlue » Logged
lycium
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« Reply #16 on: January 16, 2007, 11:49:14 AM »

The image that I saw was in some book I read on raytracing back in high school.

lucky you, we only had mandelbrot's crappy book in the computers section (most of which was for commodore 64s, cobol etc). i wasn't interested in fractals back then, but the ray tracing obsession i've had since high school too smiley
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eNZedBlue
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« Reply #17 on: January 16, 2007, 12:30:51 PM »

View of the Stella Octangula Koch fractal from the side, which shows its cubic structure more clearly:

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eNZedBlue
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« Reply #18 on: January 16, 2007, 12:34:21 PM »

The image that I saw was in some book I read on raytracing back in high school.

lucky you, we only had mandelbrot's crappy book in the computers section (most of which was for commodore 64s, cobol etc). i wasn't interested in fractals back then, but the ray tracing obsession i've had since high school too smiley
I was into fractals before raytracing, because there was no way I could code up a raytracer back then. I remember rendering my first Mandelbrot Set on an Amiga 500 using AmigaBasic. Ahh... memories. smiley

I lost interest in fractals up until about a year ago, when my interest was rekindled after reading a few wacky Terrance McKenna theories on the supposed fractal nature of reality and time.
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eNZedBlue
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« Reply #19 on: January 16, 2007, 03:41:17 PM »

Okay, here it is with each stella octangula producing 14 children at each level of recursion (8 at the vertices of the cube enclosing the stella octangula, and 6 in the middle of each face of the cube, where the two tetrahedra form a cross). A couple more well-placed tetrahedrons at each level and it should be possible to remove all the flat surfaces and make it a 100% fractally stellated solid smiley





It looks like I broke my shadowing algorithm  sad
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lycium
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« Reply #20 on: January 17, 2007, 08:11:19 PM »

It looks like I broke my shadowing algorithm  sad

if i recall correctly that stencil buffer is only 8bits, so wrap-around could cause problems at high recursion depths if you're using opengl to rasterise this. you mentioned you have a ray tracer, check out http://ompf.org/ray/sphereflake/

gtg, thunderstorm / exam studying sad
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eNZedBlue
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« Reply #21 on: January 20, 2007, 04:19:13 PM »

Fractally Stellated Stella Octangula (self-similar and non-intersecting) in perspective and isometric views:









The rules are a bit more complicated than indicated by the little graphic, because you have to remember the orientation of the tetrahedrons from one generation to the next (a bit like when using directed line segments), and you also have to completely subdivide the original tetrahedron (except for it's base) into smaller ones one third the size, with a set of orientations to make it come out "just so" in order to preserve the Koch snowflake silhouette, get complete coverage and avoid self-intersection. I'm going to do a post about it on my site, and then I'll post a link here. It should be possible to use a set of rules that work on either faces or solids. I used solids because that's what my ray-tracer deals with.

Cheers,
Chris
« Last Edit: January 20, 2007, 04:20:59 PM by eNZedBlue » Logged
Nahee_Enterprises
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« Reply #22 on: January 21, 2007, 11:17:48 AM »

Chris Hayton (eNZedBlue) wrote:
>
>    Fractally Stellated Stella Octangula (self-similar and non-intersecting)
>    in perspective and isometric views:
>        KochCrystalPerspective2.jpg

These images really came out nice.  I like the image listed just above the best of this new set.    smiley

>
>    The rules are a bit more complicated......    I'm going to do a post
>    about it on my site, and then I'll post a link here.

I look forward to viewing and reading what you will put together.

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jehovajah
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« Reply #23 on: August 29, 2008, 02:11:45 PM »

What interests me particularly about the shapes you obtained for the octahaedal shapes is the closeness in resemblance to the capsid of a particular type of virus. I wonder if you can produce a 3d image based around that form?

Of course the images are interesting and praiseworthy, but i feel i would if i had your skill attempt to explore some basic fundamental structures such as the chemical bonding arrangements say in the h2o complex, or carbon carbon bonds. I would also like some representation of M theoretic geometry in string theory. A tall order I know, but one can ask.  wink
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
lycium
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« Reply #24 on: August 30, 2008, 03:22:27 PM »

What interests me particularly about the shapes you obtained for the octahaedal shapes is the closeness in resemblance to the capsid of a particular type of virus. I wonder if you can produce a 3d image based around that form?

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cKleinhuis
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« Reply #25 on: August 31, 2008, 12:39:04 PM »

@lyc we want to see it rotateeeeeeeeee cheesy afro afro afro
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jehovajah
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« Reply #26 on: September 09, 2008, 08:53:10 PM »

Lycium you have produced an interesting form but can you explain a little bit about which viral form inspired the image you have produced. I must say it does look nasty, but most viral forms look quite beautiful.
Chris,
I am interested in the octahedral form and its relationship to the Koch snowflake. If you think about the H2O complex which forms at low temperatures the octahaedral form could represent the 2 oxygen atoms ands the 4 hydrogen atoms at the vertices:                          
                                                                      H
                                                           O      H   H        O
                                                                      H
Now could this self assemble into the octahedral forms that you have depicted? Could the larger octahedrals have the form of a 3d serpinski gasket? If you could produce a form based around this, it would be interesting to see if it follows the natural forms of snowflakes.
« Last Edit: September 12, 2008, 01:51:57 AM by jehovajah » Logged

May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
Thekikker
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« Reply #27 on: June 24, 2014, 01:06:53 PM »

Nice, I actually was introduced into fractal art by David Seied who  printed a perfect geometric version. I was so captivated with it that I had to own it. http://www.shapeways.com/model/1273033/star-tetrahedron-fractal-v2.html

This is his rendering. Thanks for sharing!
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Nicolas Douillet
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« Reply #28 on: August 15, 2017, 07:43:08 AM »

Hey, some news about 3D Koch snowflake :

I implemented it following eNZedBlue rules with a very slight variation. Here is the result at step 4 :





I also implemented the "cubic version", Koch snowflake sponge which I find beautiful since it is a sponge. My result at step 4 :



For these two fractals I also created .ply and .stl files so they are 3D printable. Actually they are both available in my online Sculpteo store at iterations 3 and 4.

[Don't trust displayed prices, they are meaningless, because tunable, and mostly function of choosen size, material, and color. I don't set them myself, it's automatically computed. There are however some minimum / limit size for the 3D objects to be robust enough regarding their details / geometry.]



« Last Edit: October 14, 2017, 10:45:13 PM by Nico » Logged
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