Welcome to Fractal Forums

EDIT: I posted more similar stuff here: www.fractalforums.com/new-theories-and-research/extending-the-definition-of-a-burning-ship
The Burning Ship formula is very similar to the Mandelbrot formula, but at the beginning of every iteration, the real and imaginary parts of z are set to their absolute values. In terms of transformations, this can be thought of as folding the complex plane in half like a piece of paper twice. The Perpendicular Burning Ship is the same, but only the imaginary coordinates are set to their absolute values. This is like only folding the piece of paper once (along the real axis)
The Burning Ship (left) and the Perpendicular Burning Ship (right)
(http://i.imgur.com/A20Uq40.png)(http://i.imgur.com/NbAgLPC.png)

Cool, but what happens when you fold the complex plane diagonally? What if you fold it at an arbitrary angle? I wanted to know, so I quickly hacked together some code to make some animations. The way to fold the complex plane at an arbitrary angle is to first rotate the plane, fold it on one of the axis with absolute value then rotate it back. The effect is the same as folding the plane on an axis with opposite the angle you rotated by. In the following animations, every frame is a Burning Ship, but with a different folding angle

Two folds like the regular b-ship (left) and one fold like the pb-ship (right)
The Celtic fish shape at 45 degrees, and a crown shape at negative 45 degrees (left)!
A heart shape at 45 90 degrees and an anteater (???) at negative 45 90 degrees(right)!
(http://i.imgur.com/RCJ9NvN.gif)(http://i.imgur.com/ZQRWtTt.gif)

Amazing! But there's more! What if the folds aren't at 90 degrees? What if I make 4 folds at 45 degrees? What about 2 folds at 45 degrees? What do those look like when you rotate the folding angles?

Four folds at 45 degrees from each other (left) and two folds at 45 degrees from each other (left).
I see a buffalo followed by a porcupine (right)!
Looks kinda like a possum to me  ;D  (left)
(http://i.imgur.com/KeRnmPR.gif)(http://i.imgur.com/DmH0zKR.gif)

Since different angles produce different Burning Ships, all of the Burning Ships can be thought of as a circle. Any point on the circle represents a unique burning ship. (Infinite burning ship possibilities!)

EDIT: Bonus Images
(http://i.imgur.com/TyEwfq6.png)
(http://i.imgur.com/PRRMjWi.png)
(http://i.imgur.com/V6honJq.png)
(http://i.imgur.com/mM83v3g.png)
(http://nocache-nocookies.digitalgott.com/gallery/20/14759_16_03_17_10_06_32.png)

 O0

 :ok: :educated:

fascinating, I didn't know the close connection between these fractals.
thanks for sharing!

Glad you like them :) I didn't know they were connected either before I made the animations. I started by rendered an image of the BS rotated at 45 degrees and just got the Celtic. I was shocked!

I just now logged on to post some zoomed in images of the half celtic/half burning ship, so here's those: (EDIT: Moved all the images to the OP)

Very cool animations!
 :thumbsup1:

Really very cool explorations! I never knew... :)  O0

Very insightful. Thanks

Nice! How about folds that don't go through the center? (I.e. additionally to rotating, also shift it forth and back before/after the fold)

I'm on it! I've got more to share tonight or tomorrow. I'll make sure to try moving the folds ;)

By the way, presumably that's obvious to you, but just in case (it only just occurred to me), you don't actually have to shift stuff in all 2D directions. Only shifts perpenticular to the fold line will actually matter. So the process would be:

• Rotate such that the desired fold line goes along, say, the real axis
• Shift along (in that case) the imaginary axis (by adding an arbitrary value from the imaginary part)
• fold (take the absolute value of the imaginary part)
• shift back
• rotate back
• (optional) repeat the same with more cuts

This will be sufficient to generate all possible such fold variations. Shifts along the fold line (the real axis in this case) will not be affected by the fold and thus transformed exactly to where they came from with the unshift.

And by stacking them, you get a 3D version.

Burning Ship Bulb ???

I like in the animations how you get little glimpses of Mandelbrot set popping in and out.