Burning Ship Fractal and Friends 3

Hello, this is 108Hassium.

I wrote an article like this before.

In the first article, I introduced famous fractal figures that are the same as the Burning Ship fractal, which are "fractal figures created by combining z^2+c with absolute value functions," and in the second article, I introduced lesser-known ones.

And recently, I discovered a new group of Burning Ship fractal-related figures, so in this article, I would like to introduce them all.

Rare

The group I want to introduce this time is further divided into three subgroups internally, so I have given each group a name for the time being.

Note that I have not given names to the individual fractal figures within those groups.

(x+iy)(x+i|y|)+c

What I will introduce in this article are those obtained by applying the following transformations to (α+iβ)(α+iβ)+c.

• α→x

• α→-x

• α→|x|

• α→-|x|

• β→y

• β→-y

• β→|y|

• β→-|y|

However, the number of times an absolute value function transformation is used shall be from one to three times.

There are 4^4-2^4×2=224 possible expressions that can be generated with this rule, but when identifying those that can be shown to be the same function through simple algebraic manipulation or those that generate the same Mandelbrot set shape, it came down to 20 types.

(x-iy)(x+i|y|)+c

The Mandelbrot set for the function $${(-x-i|y|)(-x+iy)+c}$$ has the exact same shape as this, but this is because

$${(-x-i|y|)(-x+iy)\\=(-x+iy)(-x-i|y|)\\=-(x-iy)(-x-i|y|)\\=(x-iy)(x+i|y|)}$$

...is a possible algebraic transformation, and the reason is that $${(-x+iy)(-x-i|y|)+c}$$ and $${(x-iy)(x+i|y|)+c}$$ are the same function.

To avoid such "functions that look different but are essentially the same," in the preliminary research for this article (intended to identify all fractal figures included in the group introduced this time), I established rules such as "place absolute value functions as far back in the expression as possible" and "fix the sign of the first $${x}$$ in the expression as positive" to narrow down the scope of the investigation.

(x+iy)(|x|+iy)+c

(x+iy)(-|x|+iy)+c

(x-iy)(|x|+iy)+c

(x-iy)(-|x|+iy)+c

Medium

The classification of small groups is based on the number of times the absolute value function is used, and the group names are Rare, Medium, and Well-done in order starting from those with one absolute value. (I chose the Burning Ship fractal because it has four absolute value functions, for the reasons that it is "related to fire" and "there is an order.")

(x+iy)(|x|+i|y|)+c

(x+iy)(-|x|+i|y|)+c

(x-iy)(|x|+i|y|)+c

(x-iy)(-|x|+i|y|)+c

Among the medium group, it seems that all those with absolute value functions concentrated in the latter half result in point-symmetric Mandelbrot sets.

Conversely, among those introduced this time, only these four in this group were point-symmetric.

(x+i|y|)(x+i|y|)+c

This is the "Perpendicular Burning Ship" introduced in a previous article.

(x+i|y|)(x-i|y|)+c

This figure is a bit special; it has the same shape as $${(|x|+iy)(|x|-iy)+c}$$, etc., where the position of the absolute value function is different.

(x+i|y|)(|x|+iy)+c

(x+i|y|)(-|x|+iy)+c

(|x|+iy)(|x|+iy)+c

This is the "Heart Mandelbrot" introduced in a previous article.

(|x|-iy)(|x|-iy)+c

This is the "Perpendicular Mandelbrot" introduced in a previous article.

Well done

(x+i|y|)(|x|+i|y|)+c

(x+i|y|)(-|x|+i|y|)+c

(|x|+iy)(|x|+i|y|)+c

(|x|+iy)(-|x|+i|y|)+c

Bonus

This is the lineage chart for each group.

The 0 and 1 parts represent the positional relationship of the absolute value functions, the subsequent number is the number of generatable expressions, and the final number is the actual count of Mandelbrot sets obtained.

This is the true identity of $${(x+i|y|)(x-i|y|)+c}$$, which I described as "a bit special".

When you expand $${(x+i|y|)(x-i|y|)}$$ and $${(|x|+iy)(|x|-iy)}$$, they become $${x^2+|y|^2}$$ and $${|x|^2+y^2}$$ respectively, and since $${|r|^2=r^2}$$, they both result in the same Mandelbrot set.