Chaotic Maps in Processing 3, The Knot Map

The pattern is generated with the following code:

//Initial values
float x = 0.125;
float y = - x;
void setup() {
  frameRate(60);
  size(800,600);
  colorMode(RGB, 255, 255, 255);
  background(0);
  stroke(255, 255, 255);
  strokeWeight(0.125);
}
void draw() {
  float dx = - y;
  float dy = pow(x, 3) - x - y;
  x = dx;
  y = dy;
  translate(width / 2, height / 2);
  point(x * 128, y * 128);
}

It creates the following image:

 

Chaotic Maps in Processing 3, The Sinai Map

The pattern is generated with the following code:

//Attractors
float a = 2.25;
//Initial values
float x = 1;
float y = 1;
void setup() {
  frameRate(60);
  size(800,600);
  colorMode(RGB, 255, 255, 255);
  background(0);
  stroke(255, 255, 255);
  strokeWeight(0.125);
}
void draw() {
  float dx = (x + y + a * cos(TAU * y)) % 1;
  float dy = (x + 2 * y) % 1;
  x = dx;
  y = dy;
  translate(width / 2, height / 2);
  point(x * 96, y * 96);
}

It creates the following image:

 

Chaotic Maps in Processing 3, The Gingerbread Man Map

Just in time for the season! The pattern is generated with the following code:

//Attractors
float a = 1;
float b = a;
//Initial values
float x = 0.125;
float y = - x;
void setup() {
  frameRate(60);
  size(800,600);
  colorMode(RGB, 255, 255, 255);
  background(0);
  stroke(255, 255, 255);
  strokeWeight(0.125);
}
void draw() {
  float dx = 1 - a * y + b * abs(x);
  float dy = x;
  x = dx;
  y = dy;
  translate(width / 2, height / 2);
  point((x - 1) * 32, (y - 1) * 32);
}

It creates the following image:

 

Chaotic Maps in Processing 3, The Business Cycle Map

The pattern is generated with the following code:

//Attractors
float a = 1;
float b = a;
//Initial values
float x = 0.125;
float y = x;
void setup() {
  frameRate(60);
  size(800,600);
  colorMode(RGB, 255, 255, 255);
  background(0);
  stroke(255, 255, 255);
  strokeWeight(0/125);
}
void draw() {
  float dx = x + y;
  float dy = (a * y) - (a + 1) * pow(y, 3) - (b * x);
  x = dx;
  y = dy;
  translate(width / 2, height / 2);
  point(x * 80, y * 80);
}

It creates the following image:

 

Chaotic Maps in Processing 3, The Sine-Sine Map

Short post. I'll make some more when I render more. I should really be updating this blog more too, I have been doing a lot in chaos-theory-related areas of math.

The pattern is generated with the following code:

//Attractors
float a = 2.75;
//Initial values
float x = 0.125;
float y = x;
void setup() {
  frameRate(60);
  size(800,600);
  colorMode(RGB, 255, 255, 255);
  background(0);
  stroke(255, 255, 255);
  strokeWeight(0.125);
}
void draw() {
  float dx = sin(x) - sin(a * y);
  float dy = x;
  x = dx;
  y = dy;
  translate(width / 2, height / 2);
  point(x * 80, y * 80);
}

It creates the following image:

The Weird World of Cryptographic Programming Languages

In this post I will be showing programming languages that rely on self-encryption and what not to make programming in it as hard as possible.

First one: SHAFuck (Gregor Richards).

SHAFuck is a BrainFuck derivative that relies on the SHA1 hash of each 1024-bit chunk of the program data and executes the hash as code (I assume it runs the binary data of the hash through BinaryFuck)

Here's a program that prints "Hello, World!" (compressed through bzip2):
QlpoOTFBWSZTWf3IZ1IACGmBgL/22fAAODABAlQhUobUADTQE1SqDTNT9UGj9SeoUGjRoMgNPyIjh86tRVE0ISxDFElTh00WhUpAp5iRimWiIgkIqCpmbRUEcgAiMjACI5aeb4L9DKqvuppTJcPZTYbqOE1LhqptNNkr5JXrI/vEuU3QFFFFJRSNAxC8nZHYF5bU8qOkdcT8qbDo7ucysqzFYVVPBelE5gDu/XUARH+LuSKcKEh+5DOpAA==

Second one: Malbolge (Ben Olmstead)

This one uses functions that ARX (add, rotate, XOR) ciphers use (such as Speck and Simon or hash functions such as Blake). In particular, it relies on the tritwise "crazy" operator instead of the bitwise XOR operator and also relies on tritwise rotations (yes, trit sounds wrong but just ask a French speaker what they think about the term "bits" for binary digits). Get ready, this language is named after the eighth circle of hell in Dante's Inferno. It took years before the first program was found by a computer program designed by Andrew Cooke. Nowadays programs generators exist (even online).

So how does it work? This was confusing to me at first but I think I understand it now.
Malbolge has 3 registers (a, c ,d). A is the accumulator, it is set to the value written by all write operations on memory and used for input/output. C is the instruction pointer. D is the data pointer.

Malbolge takes the code instruction [c], adds c to it and divides it by 94, the remainder is what is interpreted as an instruction. The rest are the instructions:

If the remainder is:
4 : Jump to instruction [d]. It copies the value [d] to c (note that c increments every instruction just like every programming language that goes left to right).

5 : Print the ascii value of a.

23 : Input a single ASCII character and store it in a.

39 : Tritwise rotates the value [d] by one trit, stores it in [d] and a.

40 : Copies the value [d] into d.

62 : Does the "crazy" operator with the value of [d] and the value of a. Stores at both [d] and a.

68 : Nop (no-operation).

81 : Halts the VM.

Any other value is also a no-op like 68.

----------------------------------------------------------------------------------------------------------------------

The crazy operation is a logic gate: For example let me show you the logic gate XOR

      0  1
      _  _
0   |0  1
1   |1  0

Now for crazy:


      0  1  2
      _  _  _
0   |1  0  0
1   |1  0  2
2   |2  2  1

It encrypts the trits of the 2 ascii values. After every instruction, the value at [c] is replaced with itself modulo 94 and then encrypted as follows:

ORIGINAL:    !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~
TRANSLATED:  5z]&gqtyfr$(we4{WP)H-Zn,[%\3dL+Q;>U!pJS72FhOA1CB6v^=I_0/8|jsb9m<.TVac`uY*MK'X~xDl}REokN:#?G"i@

Finally, after that is performed, c and d are incremented modulo 59049 and the execution cycle repeats.

Third one: Cryptoleq.

Subleq is a turing-complete function that runs as follows:
It's name is a shortened version of "SUBtract and branch if Less than or EQual to zero". This language has only one instruction making it an OISC. This is similar but it is cryptographic. Unlike other ones I discussed here, this one is quite fast (assuming you have the right hardware to execute this code) and generally subleq can run quite fast despite how hard it can be to code in at times. IBM made the IBM 1620, A great computer at the time (this was 1960, computers looked completely different, see the PDP-7 (what Unix was built on!). This was nicknamed CADET for "Computer with ADvanced Economic Technology" (although it jokingly can also be named "Can't Add, Doesn't Even Try"). Back on topic to Cryptoleq, it is a close relative to Subleq, where Subleq programs can run on Cryptoleq, making it backwards-compatible and also Turing-complete. I still haven't wrapped my head around how it works but there are a few resources on it: Esolang Wiki, the paper on it, Wikipedia, Github.

Fourth one: ROTFuck.

Back to BrainFuck derivatives, this one has the instructions rotated by one each time an instruction is called. It's more of a rotation cipher, ROT13-esque basic cipher you may have made then you were really young and just learned the Caesar Cipher, but still fun as it's a good challenge and also because of the fact that ARX architecture exists as I mentioned on the Malbolge paragraph. the instructions go as follows:
><+-.,[]
]><+-.,[
[]><+-.,
,[]><+-.
.,[]><+-
-.,[]><+
+-.,[]><
<+-.,[]>

Since the "Hello, World!" program was never completed, I will now complete it.
<.>]-,]>]>,][,.->].,[,+-+<..[[<>]><.-[,+<>->-<+<>]<[,.,-+<>][[+<++,.-+-.
Don't interpret this as BrainFuck. It will give you a lot of errors.

Cellular automata (and other chaotic systems) gallery, March 2020.

Combinations: A glider destroying what pattern was once there

Strange Attractors: Fractal Dream 3 (evolution of)

Stochastic Cellular Automata: Modelling Virus Spread (inspired by this video)

Langton's Ant: A colorful version of rule LLRR (visit here for the live version)

Hodgepodge Machine: Algae