I found a quite a popular at the time ducky like fractal from Fractint. It was called "Liar 4". What a name! There were other liars, 2, 3 but they were uninteresting. Alsou modifications called "Chaotic Dualist" and "Zeppo". Search for this formula led me to the publication on philosophy (!) and chaos theory.

1.

This formula is very mutch ducky/kaliset like but it was implemented well before colour methods which build up colour values as a summ of consequitive iterations (1/2 + 1/3 + 1/4 -> sum <1) like Lyapunov, Exponent Smoothing, (Mega) Discrete Lagrangian Descriptors, twin lamps, wave trichrome (FE orbits) working great for Ducky fractals when used as inside colouring.

2.

This formula is absolutely bailout dependant. bailout 1 is different from 3 and different from 30. But alsou escaping areas of image depends on modulus function (of Lp spaces). I used pnorm p=1 , p=2 and p=4 and it gave different patterns.

I treated Liar4 as a julia set of (0,0) thus with + C I got parameter space /mandelbrot set <-> julia set. (Zeppo had it, too) And alsou I tried number of different transformations aplied to it. Scaling by complex number before a + C gave the best results.

`float realScale = real(@scale)`

float imagScale = imag(@scale)

initial conditions:

`M set / parameter space z=0 C = pixel`

Julia set z = pixel C = julia seed

Formula (my enchanced version):

` tempZ_x = 1 - abs(Z_y - Z_x )`

tempZ_y = 1 - abs(1 - Z_x - Z_y )

; complex scaling + C

Z_x = tempZ_x*realScale - tempZ_y*imagScale + C_x

Z_y = tempZ_y*realScale + tempZ_x*imagScale + C_y

;make complex Z (for color methods)

Z = Z_x + flip(Z_y)

This could alsou be written as complex Ultra Fractal equation:

z =( 1 - abs(Z_y - Z_x ) + flip(1 - abs(1 - Z_x - Z_y ) ) )*@scale + C

bailout:

I used bailout value = 3, Fractint's version had bailout = 1. For more ducks like behaviour bailout could be increased.

Escaping areas very mutch depends on how modulus is calculated

`;p = 1, cool world of squares:`

abs(Z_x) + abs(Z_y) < @bailout

;p = 2, normal:

cabs(z) < @bailout ; or (Z_x^2 + Z_y^2)^0.5 < @bailout

;p = 4, supershapes:

(Z_x^4 + Z_y^4)^0.25 < @bailout

; or generalised for p

( abs(Z_x)^@p + abs(Z_y)^@p )^(1/@p) < @bailout

{--- CHUCK EBBERT & JON HORNER -------------------------------------------}

comment {

Chaotic Liar formulas for FRACTINT. These formulas reproduce some of the

pictures in the paper 'Pattern and Chaos: New Images in the Semantics of

Paradox' by Gary Mar and Patrick Grim of the Department of Philosophy,

SUNY at Stony Brook. "...what is being graphed within the unit square is

simply information regarding the semantic behavior for different inputs

of a pair of English sentences:"

}

Liar1 { ; by Chuck Ebbert.

; X: X is as true as Y

; Y: Y is as true as X is false

; Calculate new x and y values simultaneously.

; y(n+1)=abs((1-x(n) )-y(n) ), x(n+1)=1-abs(y(n)-x(n) )

z = pixel:

z = 1 - abs(imag(z)-real(z) ) + flip(1 - abs(1-real(z)-imag(z) ) )

|z| <= 1

}

Liar3 { ; by Chuck Ebbert.

; X: X is true to P1 times the extent that Y is true

; Y: Y is true to the extent that X is false.

; Sequential reasoning. P1 usually 0 to 1. P1=1 is Liar2 formula.

; x(n+1) = 1 - abs(p1*y(n)-x(n) );

; y(n+1) = 1 - abs((1-x(n+1) )-y(n) );

z = pixel:

x = 1 - abs(imag(z)*real(p1)-real(z) )

z = flip(1 - abs(1-real(x)-imag(z) ) ) + real(x)

|z| <= 1

}

Liar4[float=y] { ; by Chuck Ebbert.

; X: X is as true as (p1+1) times Y

; Y: Y is as true as X is false

; Calculate new x and y values simultaneously.

; Real part of p1 changes probability. Use floating point.

; y(n+1)=abs((1-x(n) )-y(n) ), x(n+1)=1-abs(y(n)-x(n) )

z = pixel, p = p1 + 1:

z = 1-abs(imag(z)*p-real(z))+flip(1-abs(1-real(z)-imag(z)))

|z| <= 1

}

F'Liar1 { ; Generalization by Jon Horner of Chuck Ebbert formula.

; X: X is as true as Y

; Y: Y is as true as X is false

; Calculate new x and y values simultaneously.

; y(n+1)=abs((1-x(n) )-y(n) ), x(n+1)=1-abs(y(n)-x(n) )

z = pixel:

z = 1 - abs(imag(z)-real(z) ) + flip(1 - abs(1-real(z)-imag(z) ) )

fn1(abs(z))<p1

}

The original source call this function Dualist. (But I didn't read it whole.)

https://www.semanticscholar.org/paper/Pattern-and-chaos%3A-New-images-in-the-semantics-of-Mar-Grim/c14bbaa0cf8a198c6069eec278eea39ca0490374or

http://www.pgrim.org/articles/patternandchaos.pdf p=2 bailout with value =1 and julia set (0,0) gives figure 29 of publication.

(3D: This image is copy url from semanticscholar)

DOI:10.2307/2215637Corpus ID: 10847722

Pattern and chaos: New images in the semantics of paradox

G. Mar, P. Grim

Philosophy

Noûs, Vol. 25, No. 5 (Dec., 1991),

The paradox of the Liar and its kin are well known as recalcitrant puzzles, perennially resistant to perennial attempts at solution. There is also a tradition, however, in which such paradoxes are treated as more than mere puzzles: a tradition running at least from Russell (1903) through Godel (1931) and Tarski (1935) to Kripke (1975), Herzberger (1982), Gupta (1982), and Barwise and Etchemendy (1987). In such approaches, patterns of paradox are respectfully treated as possible keys to a better understanding of incompleteness phenomena and semantics in general. Our work here is intended as a part of this latter tradition.

......

8. FRACTALS IN THE SEMANTICS OF PARADOX

Here we finally want to offer another way of graphing the behavior

of the Dualist functions sketched in terms of attractors immediately

above. Though we consider the results that follow to be intriguingly

beautiful, we cannot at this point claim to understand fully the semantic

lessons they may have to teach.

.....

These images clearly exhibit an intricate fractal character, or

self-similarity under magnification. 23 Nonetheless what is being

graphed within the unit square is simply information regarding the

semantic behavior for different inputs of a pair of English sentences:

X": X" is true to the extent that Y" is true

Y": Y" is true to the extent that X" is false

....

The existence of such fractal images within an infinite-valued

semantic analysis of paradoxical sentences seems to offer beauty and

an intriguing promise of some deep truths. Nonetheless at present

we cannot say precisely what they may have to tell us about truth

and paradox.

It's not a sentences in english it is a philosophical paradox.

https://en.wikipedia.org/wiki/Liar_paradox I did not expected so mutch dept from this formula with a stupid name "liar4". All the pictures below are julia sets.

Some philosopher from Crete said that all the cretans are liars. Does this mean that he laid that all cretans are liars?

2 ancient philosophers had died seeking solution for this logical paradox. One made a wow, that he will not eat untill he will found a solution. And he eventualy starwed to death. Another one had poor health and had a lot of unsleeped nights seeking the solution leading to his demise.