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Author Topic: Burning Ship fractal  (Read 28526 times)
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youhn
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Shapes only exists in our heads.


« Reply #30 on: August 04, 2014, 02:02:06 PM »

When searching the web, not so many burning ship fractal galleries are found. The few pictures that do exists seem to be focused on the (mini)ship(s). In order to share the beauty of the variety of shapes in this set, it feels good to post some more zoom pictures here in the general discussion:

  1.


  2.


  3.


  4.


  5.


  6.


  7.


  8.


  9.


  10.


  11.


  12.


  13.


  14.


  15.


  16.


  17.


  18.


  19.


  20.


  21.


  22.


  23.


  24.


  25.


  26.


  27.


  28.


If anyone is interested, I can attach all location in a zip file. Please let me know if you want the Kalles Fraktaler location files.

Link to gallery on Imgur: http://imgur.com/a/gkCBz/
« Last Edit: August 04, 2014, 06:10:26 PM by youhn, Reason: Added numbers to the images as reference points » Logged
laser blaster
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« Reply #31 on: August 04, 2014, 05:16:01 PM »

Wow, nice pictures, youhn!

I've heard a couple people mention that they couldn't find miniships south of the western antenna. Well, I don't know if this counts, but if you look near the lower-rightmost roundish bulb on the bottom of a miniship on the antenna, then look at the whiskers underneath of the antenna tucked right in the cusp above this roundish bulb, you will find miniships.

Interestingly, as you move away from the bulb on the miniship, the miniships will disappear, first by becoming distorted, then collapsing into dust, then turning into an empty void. This could have something to do with the iteration function being locally very non-conformal near those points, whereas above the antenna, the function is locally almost completely conformal, except at the folding lines. That's just my speculation.
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Kalles Fraktaler
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« Reply #32 on: August 04, 2014, 06:04:44 PM »

Wow, that's really diverse patterns in your set of images!
Very beautiful!
I really like the ones with thick lines in them, image 10, 17 and 25, but all are awesome and while I was scrolling down it got better and better  shocked
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Kalles Fraktaler
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« Reply #33 on: August 05, 2014, 11:40:40 PM »

Btw yes I would like to have the locations in a zip smiley
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youhn
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Shapes only exists in our heads.


« Reply #34 on: August 09, 2014, 05:04:49 PM »

Some are missing, but most locations of the 28 images are zipped and attached to this post. Anyone without Kalles Fraktaler could download it at http://www.chillheimer.de/kallesfraktaler/ or just open the kfr file, since these are regular text files.

Too bad imgur randomized the image file names, but the location should be more or less in the same order as the images.

* burning-ship-locations.zip (103.69 KB - downloaded 262 times.)
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M Benesi
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« Reply #35 on: August 10, 2014, 05:56:39 AM »

  Funny, while I worked with the formula that did the 3d burning ship, this other formula I worked with (original mag vs. xyz) seems to follow the pattern of the above image (whoops, page before... ehhh  Youhn's post with the symmetry).  Wondering about the more basic formula now...  Gotta look at that.


  
« Last Edit: August 10, 2014, 06:45:16 AM by M Benesi » Logged

stardust4ever
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« Reply #36 on: September 16, 2014, 12:36:50 AM »

Yeah, I'm familar with the burning ship fractal. I always thought it looked a bit like a mistake. I wonder if this formula can be 'made 3D'?
I think it would be possible.

The current implementation of the 2D Mandelbulb has the Perpendicular Mandelbrot on the XZ plane with the Mandelbrot on the XY plane. Although an original triplex trig system was created to generate the fractal, a polynomial version was subsequently created. I've forgotten the polynomial Mandelbulb formula, but setting all the Ys to zero gives the Perpendicular Mandelbrot set on the XZ plane.

The formula for the Perpendicular Mandelbrot set is:

Zr1 = Zr^2 + Zi^2 + Cr
Zi1 = |Zr| * Zi * -2 + Ci

The formula for the Mandelbrot set is:
Zr1 = Zr^2 - Zi^2 +Cr
Zi1 = Zr * Zi * 2 + Ci


Because the Burning ship fractal is asymmetric, if doing a mandelbulb burning ship, it may help to make the fractal represent the burning ship along the vertical axis, and use a symmetric fractal along the horizontal axis. The Burning ship fractal is similar to the Mandelbrot with an abs() command applied to the the Zi portion of the equation:

The Burning Ship fractal:
Zr1 = Zr^2 - Zi^2 +Cr
Zi1 = |Zr * Zi| * 2 + Ci

Likewise, applying an abs command to the Zr portion instead will yield the sister fractal "Celtic Mandelbrot":

The Celtic Mandelbrot fractal:
Zr1 = |Zr^2 - Zi^2| +Cr
Zi1 = Zr * Zi * 2 + Ci

The Celtic Mandelbrot, along with Celtic Mandelbar,
Zr1 = |Zr^2 - Zi^2| +Cr
Zi1 = Zr * Zi * -2 + Ci

are sister fractals to the Burning ship, all with similar types of designs withing them. The Burning ship can have minis shaped like Celtics as well as Burning ships and other shapes in between, likewise for the Celtic varieties. (remember, the burning ship is chiral so it along with it's mirror image form half of the group. I have an theory, but not rigid proof, that the orientation of the mini in part determines it's shape)

Anyway, I believe if a triplex system of algebra could be formulated such that the burning ship is on one axis with one of the two Celtic varieties on the other, that beautiful patterns could exist in such a fractal.

Also in common with the Mandelbrot set:

<Quoted Image Removed>

Looks pretty symmetric around the most vertical axis.


EDIT: I've been out of the loop too long on this one, looks like it's already been made. The screenshot definitely looks Celtic to me, if it is in fact a slice of the BS Mandelbulb... tongue stuck out
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youhn
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« Reply #37 on: September 16, 2014, 03:32:32 PM »

That image was created with Kalles Fraktaler, from the standard 2D burning ship. Params:

Re: 0.642889267562326942967270123357015
Im: -1.22729902944794937886379577301386
Zoom: 1.0E10

I've been hacking the source code of Mandelbulber 1.12-1 a little bit. While understanding just a little about the whole 3D fractal math thing, I came up with this change:

Code: (original)
double x2 = z.x * z.x;
double y2 = z.y * z.y;
double z2 = z.z * z.z;
double temp = 1.0 - z2 / (x2 + y2);
double newx = (x2 - y2) * temp;
double newy = 2.0 * z.x * z.y * temp;
double newz = -2.0 * z.z * sqrt(x2 + y2);
z.x = newx + constant.x;
z.y = newy + constant.y;
z.z = newz + constant.z;
r = z.Length();

Code: (hack)
double x2 = z.x * z.x;
double y2 = z.y * z.y;
double z2 = z.z * z.z;
double temp = 1.0 - z2 / (x2 + y2);
double newx = (x2 - y2) * temp;
double newy = 2.0 * z.x * z.y * temp;
double newz = -2.0 * z.z * sqrt(x2 + y2);
z.x = fabs(newx) + constant.x;
z.y = fabs(newy) + constant.y;
z.z = fabs(newz) + constant.z;
r = z.Length();

This results in images like:





Actually I wanted to use the formula for the 3D burning ship in Mandelbulber, but I don't know how ...  sad
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TheRedshiftRider
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« Reply #38 on: September 17, 2014, 09:25:14 AM »

I found some combined burning ships in the cubic burning ship:





Are these common? Are they special? I've never seen any combined mini-set in a fractal this way.
« Last Edit: September 17, 2014, 09:27:27 AM by Toofgib » Logged

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stardust4ever
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« Reply #39 on: September 17, 2014, 10:21:59 AM »

I found some combined burning ships in the cubic burning ship:


<Quoted Image Removed>


Are these common? Are they special? I've never seen any combined mini-set in a fractal this way.
Yeah, with the abs() functions, the sets become mirrored on itself. You can often find Minis in close proximity to each other at moderate zoom levels. When Panzerboy released his absmandvars plugin for Fractal Extreme, one of my first zooms into the needle of the 2nd order Buffalo produced this little formation, at a "moderate" zoom depth of 2^179!

http://www.fractalforums.com/fractal-exteme/fractal-extreme/msg42568/

Despite having needles which is very uncharacteristic of odd-order fractals, the 3rd order BS is still very much third order. It contains rotational symmetries of 2 and 6 sided shapes when you zoom in enough, often producing ornate snowflake like patterns, and during deep zoom sequences, minis appear 50% deeper than your last detour rather than twice as deep like with all 2nd order fractals. At shallow zoom levels this causes the minis to appear larger than in the 2nd order abs() fractals, so much so that they often collide with each other. In the cubic Buffalo, as well as Burning Ship, the minis will often get stacked together at shallow depths.

I took some measurements of the set. The three needles of the Cubic Burning ship terminate exactly √2 (square root) distance from the origin, with the diagonal having absolute coordinates of ±1 on both axes. Even Mandelbrots always have needles of legnth (n-1) root of 2. For instance the westward needle of the 2nd order Mandelbrot is the first root of 2, which is located at (duh) -2. 4th order mandelbrot needle terminates at the negative cube root of 2, so the math checks out. Again despite being an odd powered fractal, the presence of abs commands in the formula parameters will create needles due to the even symmetries the reflections create.

The buffalo fractal (2nd or 3rd order) has two abs() functions applied to the entire formula. This confines the resultant exponent to the upper right quadrant always, so reflections occur across both axes. If one of these reflection planes happens to pass in close proximity of a mini, the mini will be reflected as well. The Burning Ship and Celtic fractals have one plane of symmetry in the 2nd order sets (For clarity, the reflections I'm referring to occur in the orbitals, not the shape of the resultant fractal itself. During Zoom sequences, the Burning Ship reflects objects across the X axis while Celtic reflects objects across the Y axis, ultimately deleting the other half of the pattern. This property can be used constructively to customize the levels of complexity during a zoom sequence, much like repeating patterns can be generated in the base 2 Mandelbrot by selective zooming), but I'm still figuring out the symmetries of the Cubic Burning Ship.

Specifically with the Burning Ship fractal, the abs() command is applied to both the real and imaginary components prior to the exponent. The dual abs() commands confine the initial complex value to the upper right quadrant. In the 2nd order Burning Ship, this results in complex coordinates within the upper half of the complex plane after squaring. As a result, you won't find minis in this area (remember the BS fractal is normally rendered flipped vertically). In the 3rd order Burning Ship, the coordinates can exist in one of three quadrants, lower right being the exception. As a result, minis can exist anywhere in the fractal except the upper left quadrant of the fractal behind the twin masts (assuming the fractal is not rendered flipped or rotated).

The reflections in the The Cubic Burning ship appears to have some sort of quasi 3/4 reflective symmetries which ultimately get divided by symmetry lines into forths and twelths due to 3rd order periodicity. Somehow this 270 degree symmetry also generates a third needle equidistant and diagonally offest by 135 degrees to the other two. Being 3rd order, this ultimately creates rotation symmetry of modulo 2, 6, 18, and so on during a zoom sequence. The masts in the 3rd order Burning ship show similarities to the 2nd order BS being assymetrical along the orthagonal needles, and also shares similar properties to the 2nd order Celtic, being perfectly symmetric along the diagonal needle. As an added bonus, the third order Burning Ship also has less "rats nest" material outside of the mast areas compared to the 2nd order version, with even more options ripe for exploration.

Sadly (and this is only my humble opinion) the 2nd and 3rd order Burning Ship fractals are where the amazing symmetry ends. Starting with 4th order and up, the Burning Ship fractals begin to get chaotic and disorganized, with niether the beautifully ornate masts of the 2nd order nor the amazing symmetries contained within the 3rd order. Also like with any progressively higher Mandelbrot powers, fractals eventually tend to get "blobby". That's why I'm not insisting on arbitrary powers. The attempt to expand into a generic arbitrary powers formulation with Kalles Fraktaler might have been why the previous attempt failed. If you can just get the 3rd order BS working alongside the 2nd order, I'll be a very happy camper! cheesy

EDIT: Man, I'm becoming like a spokesperson for this formula, or something...  tongue stuck out
« Last Edit: September 17, 2014, 11:44:18 AM by stardust4ever, Reason: Derp. » Logged
Tame
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« Reply #40 on: October 17, 2014, 06:37:56 PM »

Hi!

I made some burning ship images and I guess this is a good place to post. I haven't seen too many renders with interior, but I think the interior is interesting.

Here's a part of the bottom showing all kinds of interesting patterns: http://tam3n.deviantart.com/art/DIST-BS-150-halfsize-489019896
I cut the noisy part away to save on the file size.

Here are some glory renders of the mini ship, though you might already be bored with it...
Ver1: http://tam3n.deviantart.com/art/BurningShip-TEST-halfsize-489014844
Ver2: http://tam3n.deviantart.com/art/BurningShip-TEST-ver2-big-489016121
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stardust4ever
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« Reply #41 on: October 17, 2014, 10:56:44 PM »

Hi!

I made some burning ship images and I guess this is a good place to post. I haven't seen too many renders with interior, but I think the interior is interesting.

Here's a part of the bottom showing all kinds of interesting patterns: http://tam3n.deviantart.com/art/DIST-BS-150-halfsize-489019896
I cut the noisy part away to save on the file size.

Here are some glory renders of the mini ship, though you might already be bored with it...
Ver1: http://tam3n.deviantart.com/art/BurningShip-TEST-halfsize-489014844
Ver2: http://tam3n.deviantart.com/art/BurningShip-TEST-ver2-big-489016121
I don't know too much about internal rendering methods. I guess anything is possible, but typically, the glorious fractal detail is within the iteration bands, not the main fractal or minis. I'm not saying that an interior couldn't be made into deep zoom material, but typically the entirety of the video is spent starting with the main fractal and ending with the mini. Don't get me wrong, I think internal render methods are great for stills, but I don't feel they are much use for deep zoom videos because in most deep zoom movies, the vast majority of the video contains only iteration bands.
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laser blaster
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« Reply #42 on: October 18, 2014, 10:15:33 PM »

I think the those interior shots look great! They really show the strong connection this fractal has with Kalisets (both use cabs() for their folding operation, but use different stretch operations). I also like how the burning ship has actual minisets in its interior. Interior zoom videos present a challenge, but maybe if you keyframed the iterations to increase with the zoom, you could get something nice.
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laser blaster
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« Reply #43 on: October 18, 2014, 10:41:06 PM »

Nice pictures, youhn, although your 3D fractal has a Buffalo cross-section along the xy plane (and no BS cross-section), so you've found the 3D Buffalo fractal, not the burning ship. This is because you take the absolute values after the power operation- if you take instead take the abs values at the start of each iteration, you'll get a nice Burning Ship cross-section along the xy-plane.

Here's the formula I'm using, in glsl. It's equivalent to yours except with the abs() in a different spot and without the negated z-component (which doesn't seem to affect the shape in any way...).

Code:
vec3 iterate(vec3 z) {
z = abs(z); //component-wise absolute value

float r = length(z.xy);
vec2 newXY =  complexSqr(z.xy/r);
vec2 newRZ = complexSqr(vec2(r,z.z));

newXY = newXY * newRZ.x;

z = vec3(newXY.x, newXY.y, newRZ.y);
        z += c; // c is the initial position as a vec3
        return z;
}
Here are some pictures. The first is a head-on view of the front. The second one is rotating over the top to see part of the the backside. The third one is a different view of the back.


* Pic1.jpg (14.69 KB, 453x320 - viewed 4692 times.)

* Pic2.jpg (23.14 KB, 449x320 - viewed 2960 times.)

* Pic3.jpg (21.03 KB, 428x320 - viewed 2822 times.)
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laser blaster
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« Reply #44 on: October 18, 2014, 10:43:07 PM »

This next one is a view of the main miniship. You can't see it well from here, but it has a long "tail" sticking out behind it one the z-axis. The last one is a closeup view of the tail, with some interesting lattice patterns.


* Pic4.jpg (27.12 KB, 451x320 - viewed 3709 times.)

* Pic5.jpg (61.91 KB, 680x480 - viewed 691 times.)
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