Galois Orbits

Translated by Martin Schneider

Based on: Untitled 1 by Vladimir Bonacic
Category: experimental
Move mouse over the canvas to redraw.
        
/*
Part of the ReCode Project (http://recodeproject.com)
Based on "Galois Orbits" by Vladimir Bonacic
Originally published in "Computer Graphics and Art" vol2 no3, 1977
Copyright (c) 2012 Martin Schneider - OSI/MIT license (http://recodeproject/license).
*/

////////////////////////////////////////////////////////////////////////
//                                                                    //
//           "Orbits of Galois Fields" by Vladimir Bonačić            //
//                                                                    //
//                       ( interactive version )                      //
//                                                                    //
////////////////////////////////////////////////////////////////////////

// These patterns are displayed using 32 x 32 dots.
// They visualize objects from  abstract algebra,
// which are known as orbits of galois fields.

// Source:
// "Research and Teaching in Art and Science" by Vladimir Bonačić
// Computer Graphics and Art Vol. 2, No. 3 pp. 4-8

// See also:
// "Kinetic Art: Application of Abstract Algebra to Objects with Computer-
// Controlled Flashing Lights and Sound Combinations" by Vladimir Bonačić
// Leonardo, Vol. 7, No. 3
// Note: This paper is quite mathematical at times.
// The implementation however is actually really simple.


///////////////////////// INTERACTIVE VERSION  /////////////////////////

// - Use space to flip through the presets
// - Use '+' and '-' to generate a different galois field
// - Use 'b' to toggle background color
// - Use the mouse to explore the orbits


////////////////////////////////////////////////////////////////////////

let n = 5;           // bits per dimension
let d = 1 << n;      // cells per dimension
let led = 10;        // size of the dot
let w = led * d + 1; // screen size

let preset = [1087, 1157];
let pick = 0;
let p = preset[pick];

let debug = true;
let i0, bg = 0;


function setup() {
  createCanvas(w, w);
  ellipseMode(CORNER);
  noStroke();
}


function draw() {
  // adding some afterglow
  fill(bg, 30);
  rect(0, 0, w, w);
  fill(255 - bg);

  // use mouse coordinates to get initial cell
  let x = floor(mouseX / led) & (d - 1);
  let y = floor(mouseY / led) & (d - 1);
  let i = y * d + x;

  // create empty field
  let field = new Array(d * d).fill(false);

  // find all cells in the orbit
  i0 = d * d;
  while (!field[i]) {
    i0 = min(i, i0);
    field[i] = true;
    i *= 2;
    if (i >= d * d) i ^= p;
  }

  // draw display
  for (i = 0; i < d * d; i++) {
    if (field[i]) {
      ellipse(led * (i % d), led * floor(i / d), led, led);
    }
  }
}


function keyPressed() {
  switch (key) {
    // switch between presets
    case ' ':
      pick = (pick + 1) % preset.length;
      p = preset[pick];
      break;
    // next pattern
    case '+':
      p = (p + 1) | d;
      break;
    // previous pattern
    case '-':
      p = (p - 1) | d;
      break;
    // toggle debugging
    case 'd':
      debug = !debug;
      break;
    // switch background color
    case 'b':
      bg = 255 - bg;
      break;
    default:
      return;
  }
}


function mouseClicked() {
  info();
}


// show generator polynomial + orbit entry point
function info() {
  if (debug) console.log("p = [" + p.toString(2).padStart(d, '0') + "] = " + p + "; i0 = " + i0);
}