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3d.html
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<!doctype html>
<html lang="en">
<head>
<title>Quasicrystal 3D</title>
<meta name="viewport" content="width=device-width, initial-scale=1">
<meta name="theme-color" content="#000000" />
<link rel="stylesheet" type="text/css" href="icons/webfont.css" />
<link rel="stylesheet" type="text/css" href="umbra.css" />
<link rel="icon" sizes="400x400" href="thumb-3d.png" />
<style type="text/css">
html, body {
width: 100%; height: 100%; margin: 0px; padding: 0px;
overflow: hidden;
}
body { background: #000; }
canvas {
position: absolute;
top: 0px; left: 0px; bottom: 0px; right: 0px;
}
canvas#main { z-index: 0; }
canvas#emulation { z-index: -1; }
#config .box { text-align: center; }
.box {
background: rgba(15, 15, 15, 0.9);
box-shadow: 0em 0em 2em 0em rgba(255, 255, 255, 0.5);
}
.license { opacity: 0.75; }
h2 { text-align: center; }
#help h2, #config h2 { text-align: left; }
form ul {
list-style-type: none; text-align: left; padding-left: 0em;
margin-bottom: 0em;
}
</style>
<script type="text/javascript" src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.0/jquery.min.js"></script>
<script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<script type="text/javascript" src="hammer.min.js"></script>
<script type="text/javascript" src="umbra.min.js"></script>
<script type="text/javascript" src="anima.min.js"></script>
<script type="text/javascript" src="linear.js"></script>
<script type="text/javascript" src="latticehelper.js"></script>
<script type="text/javascript">
var awaitFlags = function (target_flags, func) {
// Run func once, when all target_flags have been hit.
// Optionally, set .done to a function that will be
// passed the return value of func.
var already_run = 0;
var flags_hit = 0;
var f = function (flag) {
if (already_run) { return; }
flags_hit |= flag;
if ((flags_hit & target_flags) == target_flags) {
already_run = 1;
return f.done(func());
}
};
f.done = function (x) { return x; };
return f;
};
var fillCircle = function (context, x, y, r) {
context.beginPath();
context.arc(x, y, r, 0, Math.PI*2, true);
context.fill();
};
var setupMovable = function () {
var m = new Anima.Movable3D();
m.uFog = [2, 1];
m.isMoving = function () {
if (!(V.isZero(this.uOriginExtra))) { return true; }
return Anima.Movable3D.prototype.isMoving.call(this);
};
m.moveReset = function () {
Anima.Movable3D.prototype.moveReset.call(this);
this.radius = this.radius || 2;
this.uDotSize = this.uDotSize || 0.2;
this.uVariance = this.uVariance || 0.4;
this.uOriginExtra = [0, 0, 0];
this.uFog[1] = 1;
this.uBTMatrix = [
[1,0,0,0],
[0,1,0,0],
[0,0,1,-this.radius],
[0,0,0,1]
];
if (window.lattice_helper) { window.lattice_helper.reset(); }
$('#config_fog').prop('checked', true);
$('#config_fp').prop('checked', false);
};
m.moveFly = function (distance) {
Anima.Movable3D.prototype.moveFly.call(this, distance);
for (var i = 0; i < 3; i++) {
window.lattice_helper.offset[i] = this.origin[i];
}
};
m.moveZoom = function (amount) {
window.lattice_helper.zoom(amount);
this.scale = window.lattice_helper.scale[0];
};
m.moveAdjust = function () {
// Viewpoint yoinging
var dist_extra = Math.sqrt(V.dot(this.uOriginExtra, this.uOriginExtra));
if ((!dist_extra) || (dist_extra <= 0.01)) {
this.uOriginExtra = [0, 0, 0];
} else if (dist_extra <= 0.1) {
this.uOriginExtra = V.scale(this.uOriginExtra, 1 - 0.005/dist_extra);
} else {
this.uOriginExtra = V.scale(this.uOriginExtra, 0.95);
}
// Rezooming
window.lattice_helper.zoomAdjust();
// Recentering
window.lattice_helper.recenter();
// Synchronize lattice_helper and movable
for (var i = 0; i < 3; i++) {
this.origin[i] = window.lattice_helper.offset[i];
}
this.scale = window.lattice_helper.scale[0];
};
m.moveReset();
m.bindKeyboard(window);
m.bindTouch($('#main'));
var h = m.hammer;
h.add(new Hammer.Tap({event: 'tripletap', taps: 3, posThreshold: 20}).recognizeWith(h.get('pan')));
h.add(new Hammer.Tap({event: 'doubletap', taps: 2, posThreshold: 20}).recognizeWith(h.get('pan')).requireFailure(h.get('tripletap')));
h.on('doubletap', function (e) {
var i = window.glc.pick(
e.center.x/$(window).width(),
e.center.y/$(window).height()
);
if (!i) { return; }
// Move origin
var new_origin = Array.prototype.slice.call(glc.glData,
(i-1)*6*11, (i-1)*6*11 + 3);
window.movable.uOriginExtra = V.add(
window.movable.uOriginExtra, V.add(
window.lattice_helper.offset,
new_origin));
for (var j = 0; j < 3; j++) {
window.lattice_helper.offset[j] = window.movable.origin[j] = -new_origin[j];
}
window.movable.velocity[4] = 0;
window.frame_manager.requestFrame();
});
h.on('tripletap', function (e) {
$('#config_fp').click();
});
return m;
};
var setupForm = function () {
$('#config_fp').on('change', function (e) {
var r = 0;
if ($(this).prop('checked')) {
r = window.movable.uBTMatrix[2][3];
window.movable.uBTMatrix[2][3] = 0;
} else {
r = window.movable.uBTMatrix[2][3] + window.movable.radius;
window.movable.uBTMatrix[2][3] = -window.movable.radius;
}
// Animate transition
window.movable.uOriginExtra = V.add(
window.movable.uOriginExtra,
M.mul(M.transpose(window.movable.rotation),
[0, 0, r]));
window.frame_manager.requestFrame();
});
$('#config_fog').on('change', function (e) {
if ($(this).prop('checked')) {
window.movable.uFog[1] = 1;
} else {
window.movable.uFog[1] = -1;
}
window.frame_manager.requestFrame();
});
$('#config_reset').on('click', function (e) {
window.movable.moveReset();
window.frame_manager.requestFrame();
});
$('#config_addpoints').on('click', function (e) {
window.lattice_worker.postMessage({type: 'addVerts', iterations: 5});
});
};
$(document).ready(function () {
// MessageBox
window.message_box = Umbra.MessageBox('messages');
// Motion
window.movable = setupMovable();
setupForm();
// Animation
var glc = new Anima.GL();
// Set up
glc.glData = new Float32Array([]);
glc.mem_limit = undefined;
glc.num_verts = 0;
glc.num_verts_avail = 0;
glc.oom = function () {
var e;
while (e = this.gl.getError()) {
if (e == this.gl.OUT_OF_MEMORY) { return true; }
}
return false;
};
glc.updateData = function (glData) {
var gl = this.gl;
var t = this;
this.num_verts = this.num_verts || 1;
function saveData(glData) {
t.glData = glData.subarray(0, t.mem_limit || glData.length);
t.num_verts_avail = t.glData.length/11;
}
saveData(glData);
if (gl) {
gl.bindBuffer(gl.ARRAY_BUFFER, this.vertex_buffer);
gl.bufferData(gl.ARRAY_BUFFER, this.glData, gl.STATIC_DRAW);
while (this.oom()) {
this.memory_full = true;
this.mem_limit = 66 * Math.ceil((this.mem_limit || glData.length)/132);
saveData(glData);
this.setupCallback();
}
}
window.frame_manager.requestFrame();
};
glc.setupCallback = function () {
var gl = this.gl;
gl.clearColor(0, 0, 0, 0);
gl.enable(gl.BLEND);
gl.disable(gl.DEPTH_TEST);
gl.blendFunc(gl.SRC_ALPHA, gl.ONE);
gl.depthFunc(gl.LEQUAL);
// Upload data
this.vertex_buffer = gl.createBuffer();
gl.bindBuffer(gl.ARRAY_BUFFER, this.vertex_buffer);
gl.bufferData(gl.ARRAY_BUFFER, this.glData, gl.STATIC_DRAW);
this.frame_buffer = gl.createFramebuffer();
gl.bindFramebuffer(gl.FRAMEBUFFER, this.frame_buffer);
var rt = gl.createTexture();
gl.bindTexture(gl.TEXTURE_2D, rt);
gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_MAG_FILTER, gl.NEAREST);
gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_MIN_FILTER, gl.NEAREST);
gl.texImage2D(gl.TEXTURE_2D, 0, gl.RGBA, 1, 1, 0, gl.RGBA, gl.UNSIGNED_BYTE, null);
var rb = gl.createRenderbuffer();
gl.bindRenderbuffer(gl.RENDERBUFFER, rb);
gl.renderbufferStorage(gl.RENDERBUFFER, gl.DEPTH_COMPONENT16, 1, 1);
gl.framebufferTexture2D(gl.FRAMEBUFFER, gl.COLOR_ATTACHMENT0, gl.TEXTURE_2D, rt, 0);
gl.framebufferRenderbuffer(gl.FRAMEBUFFER, gl.DEPTH_ATTACHMENT, gl.RENDERBUFFER, rb);
this.last_capture = new Uint8Array(4);
// Resize
this.resize($(window).innerWidth(), $(window).innerHeight());
};
glc.resizeCallback = function (w, h) {
window.movable.setScreenCenter($('#main'));
window.movable.screen_diameter = this.screen_diameter = Math.sqrt(w*w + h*h);
var ir = window.movable.units_per_px = 3/this.screen_diameter;
window.movable.touch_map.pan_x.amount = -ir;
window.movable.touch_map.pan_y.amount = -ir;
this.setProjectionMatrix(ir*w/2, ir*h/2, 0.1, 1000);
this.uPMatrix = M.fromGL4(this.projection_matrix);
window.frame_manager.requestFrame();
};
glc.prepareDrawData = function (prog) {
var gl = glc.gl;
gl.useProgram(prog);
// Enable vertex buffers
gl.bindBuffer(gl.ARRAY_BUFFER, this.vertex_buffer);
var pos = gl.getAttribLocation(prog, 'aVPosition');
gl.enableVertexAttribArray(pos);
gl.vertexAttribPointer(pos, 3, gl.FLOAT, false, 44, 0);
var clr = gl.getAttribLocation(prog, 'aVColor');
gl.enableVertexAttribArray(clr);
gl.vertexAttribPointer(clr, 3, gl.FLOAT, false, 44, 12);
var tex = gl.getAttribLocation(prog, 'aVTexCoord');
gl.enableVertexAttribArray(tex);
gl.vertexAttribPointer(tex, 2, gl.FLOAT, false, 44, 24);
var vid = gl.getAttribLocation(prog, 'aVID');
if (vid >= 0) {
gl.enableVertexAttribArray(vid);
gl.vertexAttribPointer(vid, 3, gl.FLOAT, false, 44, 32);
}
// Set uniforms
gl.uniformMatrix4fv(
gl.getUniformLocation(prog, 'uPMatrix'),
false, this.projection_matrix);
gl.uniformMatrix4fv(
gl.getUniformLocation(prog, 'uMVMatrix'),
false, M.toGL4(
M.mulMats(window.movable.uBTMatrix,
window.movable.rotation))
);
gl.uniform2f(gl.getUniformLocation(prog, 'uScale'),
window.lattice_helper.scale[0], window.lattice_helper.scale[3]);
gl.uniform2fv(gl.getUniformLocation(prog, 'uFog'),
new Float32Array(window.movable.uFog));
gl.uniform3f(gl.getUniformLocation(prog, 'uDotSize'),
window.movable.uDotSize, // straight-up dot size
-0.25 / window.movable.uVariance, // variance
3 / window.movable.uVariance); // color scaling
gl.uniform3fv(gl.getUniformLocation(prog, 'uOrigin'),
new Float32Array(V.add(
window.lattice_helper.offset,
window.movable.uOriginExtra)));
gl.uniform3fv(gl.getUniformLocation(prog, 'uOriginColor'),
new Float32Array(window.lattice_helper.offset.slice(3,6)));
};
glc.pick = function (x, y) {
var gl = glc.gl;
gl.disable(gl.BLEND);
gl.enable(gl.DEPTH_TEST);
gl.bindFramebuffer(gl.FRAMEBUFFER, glc.frame_buffer);
gl.clear(gl.COLOR_BUFFER_BIT | gl.DEPTH_BUFFER_BIT);
glc.prepareDrawData(glc.shaders.picker);
gl.uniformMatrix4fv(
gl.getUniformLocation(glc.shaders.picker, 'uPMatrix'),
false, M.toGL4(M.mulMats([
[10, 0, 0, 9-10*2*x],
[0, -10, 0, 9-10*2*y],
[0, 0, 1, 0],
[0, 0, 0, 1],
], this.uPMatrix)));
gl.drawArrays(gl.TRIANGLES, 0, Math.min(glc.num_verts, (1 << 21) * 6 - 6));
gl.readPixels(0, 0, 1, 1, gl.RGBA, gl.UNSIGNED_BYTE, glc.last_capture);
var clr = Array.prototype.slice.call(glc.last_capture);
return clr[0] + (clr[1] << 7) + (clr[2] << 14);
};
glc.draw = function (time_scheduled) {
var gl = glc.gl;
if (!gl) { return false; }
window.movable.update(this.dt);
if (!window.lattice_helper) { return; }
window.movable.moveAdjust();
gl.enable(gl.BLEND);
gl.disable(gl.DEPTH_TEST);
gl.bindFramebuffer(gl.FRAMEBUFFER, null);
gl.clear(gl.COLOR_BUFFER_BIT);
glc.prepareDrawData(glc.shaders.stars);
// Draw!
this.selectLOD('realtime');
glc.num_verts = Math.min(this.lod.num_verts, glc.num_verts_avail);
gl.drawArrays(gl.TRIANGLES, 0, glc.num_verts);
if (!V.isZero(window.movable.uOriginExtra)) {
glc.prepareDrawData(glc.shaders.locker);
gl.uniform3f(gl.getUniformLocation(glc.shaders.locker, 'uDotSize'),
2*window.movable.uDotSize, 0, 0);
gl.uniform3fv(gl.getUniformLocation(glc.shaders.locker, 'uOrigin'),
new Float32Array(window.movable.uOriginExtra));
gl.drawArrays(gl.TRIANGLES, 0, 6);
}
this.adjustLOD();
return window.movable.isMoving();
};
glc.emulatedProjection = function (f) {
return (function () {
var w = $(window).width();
var h = $(window).height();
var c2 = $('#emulation')[0].getContext('2d');
c2.globalCompositeOperation = 'lighter';
var m = M.mulMats(glc.uPMatrix,
M.mulMats(window.movable.uBTMatrix, window.movable.rotation));
var uo = V.add(window.lattice_helper.offset,
window.movable.uOriginExtra);
var usc = window.lattice_helper.scale[3];
var udv = -0.25 / window.movable.uVariance;
var udz = 3 / Math.sqrt(window.movable.uVariance);
var uds = window.movable.uDotSize * Math.sqrt(w*w + h*h) / 6;
function scaleColor(x) { return 1 / (1 + x * x * udz); }
function to255(x) { return Math.round(x * 255); }
for (var i = 0; i < glc.num_verts / 6; i++) {
var v = Array.prototype.slice.call(glc.glData, i*66, i*66 + 6);
var p = M.mul(m, V.scale(V.add(v, uo), window.lattice_helper.scale[0]).concat(1));
if (!(Math.abs(p[2]/p[3]) < 1)) { continue; }
var c = V.scale(V.add(v.slice(3,6), window.lattice_helper.offset.slice(3,6)), usc);
var clr = c.map(scaleColor);
clr = V.scale(clr, 1 / Math.max.apply(Math, clr)).map(to255);
c2.fillStyle = 'rgb(' + clr.join(',') + ')';
f(c2, p, uds * Math.exp(V.dot(c, c) * udv) / p[3], i);
}
}).bind(this);
};
glc.emulatedDraw = function (time_scheduled) {
window.movable.update(this.dt);
if (!window.lattice_helper) { return; }
window.movable.moveAdjust();
var w = $('#emulation')[0].width = $(window).width();
var h = $('#emulation')[0].height = $(window).height();
var moving = window.movable.isMoving();
var realtime = moving || this.next_id || window.movable.is_hammer_busy;
if (realtime) { this.selectLOD('realtime'); }
else { this.selectLOD('hd'); }
glc.num_verts = Math.min(this.lod.num_verts, glc.num_verts_avail);
// Draw!
glc.emulatedProjection(function (context, p, radius, i) {
var x = w * p[0]/p[3];
var y = -h * p[1]/p[3];
var diameter = 2 * radius;
if (Math.abs(x) > w + diameter || Math.abs(y) > h + diameter) {
return;
}
fillCircle(context, (w + x) / 2, (h + y) / 2, radius);
})();
this.adjustLOD();
return moving;
}
glc.emulatedPick = function (x, y) {
var last_id = -1;
var last_depth = Infinity;
var w = $(window).width();
var h = $(window).height();
glc.emulatedProjection(function (context, p, radius, i) {
var v = [
w * ((1 + p[0]/p[3]) / 2 - x),
h * ((1 - p[1]/p[3]) / 2 - y)
];
if (V.dot(v,v) < radius * radius && p[3] < last_depth) {
last_id = i;
last_depth = p[3];
}
})();
return last_id + 1;
};
window.frame_manager = new Anima.FrameManager(glc.draw);
window.frame_manager.addLOD('realtime', {
frame_min: 8, frame_max: 16, num_verts: 50,
});
window.frame_manager.addLOD('hd', {
frame_min: 250, frame_max: 500, num_verts: 60000,
});
window.frame_manager.upLOD = function () {
if (this.lod.num_verts < glc.num_verts_avail) {
this.lod.num_verts = Math.min(glc.num_verts_avail,
6 + 6 * Math.floor(this.lod.num_verts / 5));
this.requestFrame();
}
};
window.frame_manager.downLOD = function () {
this.lod.num_verts = Math.min(glc.num_verts_avail,
6 * Math.ceil(this.lod.num_verts/7));
};
var setupContinue = awaitFlags(7, function () {
try {
glc.bindCanvas($('#main')[0]);
} catch (e) {
window.message_box.post('GL: ' + e, 'error');
window.message_box.post('Falling back to emulation.', 'warning');
glc.gl = null;
glc.draw = window.frame_manager.drawCallback = glc.emulatedDraw;
glc.pick = glc.emulatedPick;
glc.resize($(window).innerWidth(), $(window).innerHeight());
}
window.movable.motionCallback = function () {
window.frame_manager.requestFrame();
};
$(window).on('resize', function (e) {
glc.resize($(window).width(), $(window).height());
});
});
glc.addProgramByURLs('stars', 'vert.c', 'frag.c', setupContinue.bind(null, 1));
glc.addProgramByURLs('picker', 'vert.c', 'fpick.c', setupContinue.bind(null, 2));
glc.addProgramByURLs('locker', 'vert.c', 'flock.c', setupContinue.bind(null, 4));
window.glc = glc;
// Init lattice
var requestMorePoints = (function () {
var i = 0;
var BASE_REQUESTS = [
{type: 'addVerts', iterations: 5},
{type: 'addVerts', iterations: 5},
{type: 'addVertsShrink'},
{type: 'addVertsGrow'},
{type: 'addVertsGrow'},
{type: 'addVertsGrow'},
];
var LIMIT_SIZE = 200;
var LIMIT_TIME = 5000;
var finalized = false;
var start_time = 0;
var progress_bar;
return function (neighbor_size) {
if (finalized) {
$('#config_addpoints').prop('disabled', false);
if (progress_bar) { progress_bar.set(1); }
return;
}
if (!progress_bar) {
progress_bar = window.message_box.makeProgress('Finding points...');
progress_bar.set(0.1);
}
if (!start_time) { start_time = (new Date()).valueOf(); }
var progress = Math.max(0,
((new Date()).valueOf() - start_time)/LIMIT_TIME,
LIMIT_SIZE/neighbor_size);
progress_bar.set(0.1 + progress * 0.8);
if (progress > 1) {
finalized = true;
window.lattice_worker.postMessage({type: 'addVertsShrink'});
window.lattice_worker.postMessage({type: 'addVertsGrow'});
window.lattice_worker.postMessage({type: 'addVerts', iterations: 5});
} else {
if (i < BASE_REQUESTS.length) {
window.lattice_worker.postMessage(BASE_REQUESTS[i]);
i++;
} else {
window.lattice_worker.postMessage({type: 'addVerts', iterations: 5});
}
}
};
})();
window.lattice_worker = new Worker('quasilattice3.js');
window.lattice_worker.onmessage = function (e) {
var data = e.data;
if (data.type == 'update') {
if (!glc.memory_full) {
glc.updateData(data.glData);
}
window.lattice_helper.translators = data.translators;
movable.uFog[0] = 0.7 * data.next_weight;
requestMorePoints((glc.screen_diameter || 512) / Math.sqrt(data.next_weight));
window.message_box.post(data.source + ' ' + data.glData.length/66 + ' ' + data.next_weight, 'debug');
} else if (data.type == 'ready') {
window.lattice_helper = new LatticeHelper(
6, 3, [], data.scale_factors);
requestMorePoints();
} else if (data.type == 'message') {
window.message_box.post(data.message, data.message_class);
} else {
window.message_box.post('Window ignored message of type ' + data.type, 'debug warning');
}
};
window.lattice_worker.postMessage({type: 'init'});
});
</script>
</head>
<body>
<canvas id="main">
<img src="thumb-3d.png" alt="Preview" />
This demo requires HTML5 Canvas support.
</canvas>
<canvas id="emulation"></canvas>
<div id="messages" class="messagebox"></div>
<div id="config" class="overlay"><div class="box">
<a class="button icon button_close button_rightmargin"
title="Cancel" href="#">×</a>
<h2>Options</h2>
<form method="post" action="#" id="form_config">
<ul>
<li><input type="checkbox" id="config_fp" /><label for="config_fp">First-person view</label></li>
<li><input type="checkbox" id="config_fog" checked="checked" /><label for="config_fog">Hide incompletely-drawn regions with fog</label></li>
<li>
<button type="button" id="config_reset"
/>Reset view</button>
<button type="button" id="config_addpoints"
disabled="disabled" />Draw more points</button>
</li>
</ul>
</form>
</div></div>
<div id="help" class="overlay">
<div class="box">
<a class="button icon button_close button_rightmargin"
title="Close" href="#">×</a>
<h2>Help</h2>
<p><a href="#help_about">What am I seeing here?</a></p>
<ul>
<li><b>Turn</b>: arrow keys and
<span class="keyboard-key"><<sub>,</sub></span>,
<span class="keyboard-key">><sub>.</sub></span>
or mouse/finger dragging.</li>
<li><b>Fly</b>: <span class="keyboard-key">spacebar</span> and
<span class="keyboard-key">Z</span> or press-and-hold.</li>
<li><b>Zoom</b>: <span class="keyboard-key">+</span> and
<span class="keyboard-key">-</span>, mouse scrolling,
or pinch zoom. (Hint: lock onto a point first.)</li>
<li><b>Lock on</b> to a point: double-tap.</li>
<li>Toggle <b>first-person</b> view: triple-tap.</li>
</ul>
<p class="license">Code written by
<a href="http://pteromys.melonisland.net/">Pteromys</a>
and released under the <a href="http://unlicense.org/">Unlicense</a>.</p>
</div>
</div>
<div id="help_about" class="overlay">
<div class="box">
<a class="button icon button_leftmargin"
title="Back" href="#help">«</a>
<a class="button icon button_close button_rightmargin"
title="Close" href="#">×</a>
<h2>About</h2>
<p>This page draws the icosahedral 3-D
<a href="http://en.wikipedia.org/wiki/Quasicrystal"
>quasicrystal</a>: a thin 3-D slice of a 6-D
<a href="http://en.wikipedia.org/wiki/Lattice_(group)"
>lattice</a> with
<a href="http://en.wikipedia.org/wiki/Icosahedral_symmetry"
>icosahedral symmetry</a>. Points are sized by
their proximity to the slice and colored
by how far they extend into the 3 unrepresented dimensions
(<span style="color: #fcc;">red</span>,
<span style="color: #cfc;">green</span>, and
<span style="color: #ccf;">blue</span>).</p>
<p>This quasicrystal has two curious properties:
<ol>
<li><a href="#help_uniqueness">It's the only 3-D lattice slice
which is truly 3-D</a>—all others with any symmetry
merely repeat a 2-D pattern.</li>
<li><a href="#help_selfsimilarity">It's self-similar</a>—a
zoomed-in view is indistinguishable from a zoomed-out view.</li>
</ol></p>
</div>
</div>
<div id="help_selfsimilarity" class="overlay">
<div class="box">
<a class="button icon button_leftmargin"
title="Back" href="#help_about">«</a>
<a class="button icon button_close button_rightmargin"
title="Close" href="#">×</a>
<h2>Self-similarity</h2>
<p>Zooming in expands space by some factor \(s\).
To compensate for the spreading out of dots,
we'll make more dots touch our slice
by scaling the unseen 3 dimensions by
\(\frac{1}{s}\) (to keep the number of dots
drawn roughly constant).</p>
<p>If \(s = 2 + \sqrt{5}\), a miracle occurs: all dots
land on other dots, and it's as if we never zoomed in.
To understand why, we need to describe the 6-D
lattice in more detail.</p>
<p>Let \(\zeta\) be the 5th
<a href="http://en.wikipedia.org/wiki/Root_of_unity"
>root of unity</a> \(e^{2\pi i/5}\).
In \(\mathbf{R} \times \mathbf{C}\), the points
\[ \pm(\sqrt{5}, 0), \pm(1, 2\zeta^k) \]
(\(k = 0\) to \(4\)) are the twelve vertices of an icosahedron,
and their sums are the dots we see.
The unseen 3 dimensions are
a second set of \(\mathbf{R} \times \mathbf{C}\)
coordinates that replace \(\zeta\) with the
<a href="http://en.wikipedia.org/wiki/Conjugate_element_(field_theory)"
>algebraically indistinguishable</a> \(\zeta^2\).
As \(\sqrt{5} = 1 + 2(\zeta + \zeta^{-1})\), this also ends up
replacing \(\sqrt{5}\) by \(-\sqrt{5}\).
You can check using
<a href="http://en.wikipedia.org/wiki/Dot_product">dot products</a>
that the twelve vertices now point in six perpendicular directions
in the 6-D combined coordinate system.</p>
<p>Scaling by \(2 + \sqrt{5}\) in the first two coordinates
amounts to replacing each vertex with the sum of
twice itself and its five neighbors—thus taking dots
to other dots.
In the last two coordinates, this becomes scaling by \(2 - \sqrt{5}\),
and the miracle stems from the fact that
\[ (2 + \sqrt{5})(2 - \sqrt{5}) = -1 . \]
That is, this scaling is what we'd obtain with a zoom factor
of \(2 + \sqrt{5}\).</p>
<p>This also occurs in
<a href="2d.html#help_zooming">the
2-D quasicrystal with 5-fold symmetry</a>,
for which a related phenomenon is illustrated by the game
<a href="http://www.math.brown.edu/~res/Java/App12/test1.html"
>Lucy and Lily</a>.</p>
</div>
</div>
<div id="help_uniqueness" class="overlay">
<div class="box">
<a class="button icon button_leftmargin"
title="Back" href="#help_about">«</a>
<a class="button icon button_close button_rightmargin"
title="Close" href="#">×</a>
<h2>Uniqueness</h2>
<p>What we're looking for is a group of symmetries \(G\)
of a higher-dimensional lattice \(\mathbf{Z}^n \subset \mathbf{R}^n\),
along with a 3-plane \(V \subset \mathbf{R}^n\) (the slice)
on which \(G\) acts irreducibly (the non-2-D condition,
which amounts to not having an axis).</p>
<p>Of the <a
href="http://en.wikipedia.org/wiki/Point_groups_in_three_dimensions#The_seven_remaining_point_groups"
>seven non-axial point groups in 3 dimensions</a>,
only the two icosahedral groups don't already
preserve an ordinary 3-D lattice. These are the
<a href="http://en.wikipedia.org/wiki/Alternating_group"
>alternating group</a> \(A_5\) and the product
\(A_5 \times \{\pm 1\}\); so it suffices to find
<a href="http://groupprops.subwiki.org/wiki/Linear_representation_theory_of_alternating_group:A5"
>a list of representations</a>
of \(A_5\) over integers and real numbers.</p>
<p>(One detail remains: \(A_5\) might only
be the quotient of \(G\) by a normal subgroup \(N\).
Fortunately, the map
\[ x \mapsto \sum_{g \in N} gx \]
is an endomorphism of \(\mathbf{Z}^n\)
on whose image \(A_5\) acts.)</p>
</div>
</div>
<a class="button icon button_root" data-key="QUESTION"
href="#help" id="button_help" title="Help">?</a>
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href="#config" id="button_config" title="Settings">⚙</a>
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