ChaosGame/docs/examples-in-julia.html at main · BetaAndBit/ChaosGame · GitHub

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
<!DOCTYPE html>
<html lang="" xml:lang="">
<head>

  <meta charset="utf-8" />
  <meta http-equiv="X-UA-Compatible" content="IE=edge" />
  <title>1.6 Examples in Julia | Chaos Game</title>
  <meta name="description" content="1.6 Examples in Julia | Chaos Game" />
  <meta name="generator" content="bookdown 0.32 and GitBook 2.6.7" />

  <meta property="og:title" content="1.6 Examples in Julia | Chaos Game" />
  <meta property="og:type" content="book" />
  <meta property="og:image" content="/images/foldout_00.png" />


  <meta name="twitter:card" content="summary" />
  <meta name="twitter:title" content="1.6 Examples in Julia | Chaos Game" />


  <meta name="twitter:image" content="/images/foldout_00.png" />

<meta name="author" content="" />


<meta name="date" content="2023-05-15" />

  <meta name="viewport" content="width=device-width, initial-scale=1" />
  <meta name="apple-mobile-web-app-capable" content="yes" />
  <meta name="apple-mobile-web-app-status-bar-style" content="black" />


<link rel="prev" href="examples-in-r.html"/>
<link rel="next" href="fractal-that-is-fixed-point.html"/>
<script src="libs/jquery/jquery-3.6.0.min.js"></script>
<script src="https://cdn.jsdelivr.net/npm/fuse.js@6.4.6/dist/fuse.min.js"></script>
<link href="libs/gitbook/css/style.css" rel="stylesheet" />
<link href="libs/gitbook/css/plugin-table.css" rel="stylesheet" />
<link href="libs/gitbook/css/plugin-bookdown.css" rel="stylesheet" />
<link href="libs/gitbook/css/plugin-highlight.css" rel="stylesheet" />
<link href="libs/gitbook/css/plugin-search.css" rel="stylesheet" />
<link href="libs/gitbook/css/plugin-fontsettings.css" rel="stylesheet" />
<link href="libs/gitbook/css/plugin-clipboard.css" rel="stylesheet" />


<link href="libs/anchor-sections/anchor-sections.css" rel="stylesheet" />
<link href="libs/anchor-sections/anchor-sections-hash.css" rel="stylesheet" />
<script src="libs/anchor-sections/anchor-sections.js"></script>
<!-- Global site tag (gtag.js) - Google Analytics -->
<script async src="https://www.googletagmanager.com/gtag/js?id=UA-5650686-12"></script>
<script>
  window.dataLayer = window.dataLayer || [];
  function gtag(){dataLayer.push(arguments);}
  gtag('js', new Date());

  gtag('config', 'UA-5650686-12');
</script>


<style type="text/css">
pre > code.sourceCode { white-space: pre; position: relative; }
pre > code.sourceCode > span { display: inline-block; line-height: 1.25; }
pre > code.sourceCode > span:empty { height: 1.2em; }
.sourceCode { overflow: visible; }
code.sourceCode > span { color: inherit; text-decoration: inherit; }
pre.sourceCode { margin: 0; }
@media screen {
div.sourceCode { overflow: auto; }
}
@media print {
pre > code.sourceCode { white-space: pre-wrap; }
pre > code.sourceCode > span { text-indent: -5em; padding-left: 5em; }
}
pre.numberSource code
  { counter-reset: source-line 0; }
pre.numberSource code > span
  { position: relative; left: -4em; counter-increment: source-line; }
pre.numberSource code > span > a:first-child::before
  { content: counter(source-line);
    position: relative; left: -1em; text-align: right; vertical-align: baseline;
    border: none; display: inline-block;
    -webkit-touch-callout: none; -webkit-user-select: none;
    -khtml-user-select: none; -moz-user-select: none;
    -ms-user-select: none; user-select: none;
    padding: 0 4px; width: 4em;
    color: #aaaaaa;
  }
pre.numberSource { margin-left: 3em; border-left: 1px solid #aaaaaa;  padding-left: 4px; }
div.sourceCode
  {   }
@media screen {
pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
}
code span.al { color: #ff0000; font-weight: bold; } /* Alert */
code span.an { color: #60a0b0; font-weight: bold; font-style: italic; } /* Annotation */
code span.at { color: #7d9029; } /* Attribute */
code span.bn { color: #40a070; } /* BaseN */
code span.bu { color: #008000; } /* BuiltIn */
code span.cf { color: #007020; font-weight: bold; } /* ControlFlow */
code span.ch { color: #4070a0; } /* Char */
code span.cn { color: #880000; } /* Constant */
code span.co { color: #60a0b0; font-style: italic; } /* Comment */
code span.cv { color: #60a0b0; font-weight: bold; font-style: italic; } /* CommentVar */
code span.do { color: #ba2121; font-style: italic; } /* Documentation */
code span.dt { color: #902000; } /* DataType */
code span.dv { color: #40a070; } /* DecVal */
code span.er { color: #ff0000; font-weight: bold; } /* Error */
code span.ex { } /* Extension */
code span.fl { color: #40a070; } /* Float */
code span.fu { color: #06287e; } /* Function */
code span.im { color: #008000; font-weight: bold; } /* Import */
code span.in { color: #60a0b0; font-weight: bold; font-style: italic; } /* Information */
code span.kw { color: #007020; font-weight: bold; } /* Keyword */
code span.op { color: #666666; } /* Operator */
code span.ot { color: #007020; } /* Other */
code span.pp { color: #bc7a00; } /* Preprocessor */
code span.sc { color: #4070a0; } /* SpecialChar */
code span.ss { color: #bb6688; } /* SpecialString */
code span.st { color: #4070a0; } /* String */
code span.va { color: #19177c; } /* Variable */
code span.vs { color: #4070a0; } /* VerbatimString */
code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warning */
</style>

<style type="text/css">
/* Used with Pandoc 2.11+ new --citeproc when CSL is used */
div.csl-bib-body { }
div.csl-entry {
  clear: both;
}
.hanging div.csl-entry {
  margin-left:2em;
  text-indent:-2em;
}
div.csl-left-margin {
  min-width:2em;
  float:left;
}
div.csl-right-inline {
  margin-left:2em;
  padding-left:1em;
}
div.csl-indent {
  margin-left: 2em;
}
</style>

<link rel="stylesheet" href="css/style.css" type="text/css" />
</head>

<body>


  <div class="book without-animation with-summary font-size-2 font-family-1" data-basepath=".">

    <div class="book-summary">
      <nav role="navigation">

<ul class="summary">
<li><a href="./"><h3>Chaos Game</h3></a></li>

<li class="divider"></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i>Cover</a></li>
<li class="chapter" data-level="" data-path="how-to-read-this-book.html"><a href="how-to-read-this-book.html"><i class="fa fa-check"></i>How to read this book?</a></li>
<li class="chapter" data-level="1" data-path="hello-fractals.html"><a href="hello-fractals.html"><i class="fa fa-check"></i><b>1</b> Hello, fractals!</a>
<ul>
<li class="chapter" data-level="1.1" data-path="wacław-sierpinski.html"><a href="wacław-sierpinski.html"><i class="fa fa-check"></i><b>1.1</b> Wacław Sierpinski</a></li>
<li class="chapter" data-level="1.2" data-path="hello-fractals-1.html"><a href="hello-fractals-1.html"><i class="fa fa-check"></i><b>1.2</b> Hello, fractals!</a>
<ul>
<li class="chapter" data-level="1.2.1" data-path="hello-fractals-1.html"><a href="hello-fractals-1.html#cantor-dust"><i class="fa fa-check"></i><b>1.2.1</b> Cantor dust</a></li>
<li class="chapter" data-level="1.2.2" data-path="hello-fractals-1.html"><a href="hello-fractals-1.html#sierpinski-triangle"><i class="fa fa-check"></i><b>1.2.2</b> Sierpinski triangle</a></li>
</ul></li>
<li class="chapter" data-level="1.3" data-path="but-why-does-it-work.html"><a href="but-why-does-it-work.html"><i class="fa fa-check"></i><b>1.3</b> But why does it work?</a>
<ul>
<li class="chapter" data-level="1.3.1" data-path="but-why-does-it-work.html"><a href="but-why-does-it-work.html#topological-dimension"><i class="fa fa-check"></i><b>1.3.1</b> Topological dimension</a></li>
<li class="chapter" data-level="1.3.2" data-path="but-why-does-it-work.html"><a href="but-why-does-it-work.html#minkowski-dimension-or-fractal-box-counting-dimension"><i class="fa fa-check"></i><b>1.3.2</b> Minkowski dimension, or fractal box-counting dimension</a></li>
<li class="chapter" data-level="1.3.3" data-path="but-why-does-it-work.html"><a href="but-why-does-it-work.html#sierpinski-carpet"><i class="fa fa-check"></i><b>1.3.3</b> Sierpinski carpet</a></li>
</ul></li>
<li class="chapter" data-level="1.4" data-path="examples-in-python.html"><a href="examples-in-python.html"><i class="fa fa-check"></i><b>1.4</b> Examples in Python</a>
<ul>
<li class="chapter" data-level="1.4.1" data-path="examples-in-python.html"><a href="examples-in-python.html#cantor-dust-1"><i class="fa fa-check"></i><b>1.4.1</b> Cantor dust</a></li>
<li class="chapter" data-level="1.4.2" data-path="examples-in-python.html"><a href="examples-in-python.html#sierpinski-triangle-1"><i class="fa fa-check"></i><b>1.4.2</b> Sierpinski triangle</a></li>
<li class="chapter" data-level="1.4.3" data-path="examples-in-python.html"><a href="examples-in-python.html#sierpinski-carpet-1"><i class="fa fa-check"></i><b>1.4.3</b> Sierpinski carpet</a></li>
</ul></li>
<li class="chapter" data-level="1.5" data-path="examples-in-r.html"><a href="examples-in-r.html"><i class="fa fa-check"></i><b>1.5</b> Examples in R</a>
<ul>
<li class="chapter" data-level="1.5.1" data-path="examples-in-r.html"><a href="examples-in-r.html#cantor-dust-2"><i class="fa fa-check"></i><b>1.5.1</b> Cantor dust</a></li>
<li class="chapter" data-level="1.5.2" data-path="examples-in-r.html"><a href="examples-in-r.html#sierpinski-triangle-2"><i class="fa fa-check"></i><b>1.5.2</b> Sierpinski triangle</a></li>
<li class="chapter" data-level="1.5.3" data-path="examples-in-r.html"><a href="examples-in-r.html#sierpinski-carpet-2"><i class="fa fa-check"></i><b>1.5.3</b> Sierpinski carpet</a></li>
</ul></li>
<li class="chapter" data-level="1.6" data-path="examples-in-julia.html"><a href="examples-in-julia.html"><i class="fa fa-check"></i><b>1.6</b> Examples in Julia</a>
<ul>
<li class="chapter" data-level="1.6.1" data-path="examples-in-julia.html"><a href="examples-in-julia.html#cantor-dust-3"><i class="fa fa-check"></i><b>1.6.1</b> Cantor dust</a></li>
<li class="chapter" data-level="1.6.2" data-path="examples-in-julia.html"><a href="examples-in-julia.html#sierpinski-triangle-3"><i class="fa fa-check"></i><b>1.6.2</b> Sierpinski triangle</a></li>
<li class="chapter" data-level="1.6.3" data-path="examples-in-julia.html"><a href="examples-in-julia.html#sierpinski-carpet-3"><i class="fa fa-check"></i><b>1.6.3</b> Sierpinski carpet</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="2" data-path="fractal-that-is-fixed-point.html"><a href="fractal-that-is-fixed-point.html"><i class="fa fa-check"></i><b>2</b> Fractal, that is fixed point</a>
<ul>
<li class="chapter" data-level="2.1" data-path="stefan-banach.html"><a href="stefan-banach.html"><i class="fa fa-check"></i><b>2.1</b> Stefan Banach</a></li>
<li class="chapter" data-level="2.2" data-path="fractal-that-is-fixed-point-1.html"><a href="fractal-that-is-fixed-point-1.html"><i class="fa fa-check"></i><b>2.2</b> Fractal, that is fixed point</a>
<ul>
<li class="chapter" data-level="2.2.1" data-path="fractal-that-is-fixed-point-1.html"><a href="fractal-that-is-fixed-point-1.html#dragons-and-ferns"><i class="fa fa-check"></i><b>2.2.1</b> Dragons and ferns</a></li>
<li class="chapter" data-level="2.2.2" data-path="fractal-that-is-fixed-point-1.html"><a href="fractal-that-is-fixed-point-1.html#sierpinski-pentagon"><i class="fa fa-check"></i><b>2.2.2</b> Sierpinski pentagon</a></li>
<li class="chapter" data-level="2.2.3" data-path="fractal-that-is-fixed-point-1.html"><a href="fractal-that-is-fixed-point-1.html#pythagoras-tree"><i class="fa fa-check"></i><b>2.2.3</b> Pythagoras tree</a></li>
<li class="chapter" data-level="2.2.4" data-path="fractal-that-is-fixed-point-1.html"><a href="fractal-that-is-fixed-point-1.html#heighway-dragon"><i class="fa fa-check"></i><b>2.2.4</b> Heighway Dragon</a></li>
<li class="chapter" data-level="2.2.5" data-path="fractal-that-is-fixed-point-1.html"><a href="fractal-that-is-fixed-point-1.html#barnsley-fern"><i class="fa fa-check"></i><b>2.2.5</b> Barnsley fern</a></li>
</ul></li>
<li class="chapter" data-level="2.3" data-path="but-why-does-it-work-1.html"><a href="but-why-does-it-work-1.html"><i class="fa fa-check"></i><b>2.3</b> But why does it work?</a>
<ul>
<li class="chapter" data-level="2.3.1" data-path="but-why-does-it-work-1.html"><a href="but-why-does-it-work-1.html#metric-space"><i class="fa fa-check"></i><b>2.3.1</b> Metric space</a></li>
<li class="chapter" data-level="2.3.2" data-path="but-why-does-it-work-1.html"><a href="but-why-does-it-work-1.html#hausdorff-distance"><i class="fa fa-check"></i><b>2.3.2</b> Hausdorff distance</a></li>
<li class="chapter" data-level="2.3.3" data-path="but-why-does-it-work-1.html"><a href="but-why-does-it-work-1.html#cauchy-sequence"><i class="fa fa-check"></i><b>2.3.3</b> Cauchy sequence</a></li>
<li class="chapter" data-level="2.3.4" data-path="but-why-does-it-work-1.html"><a href="but-why-does-it-work-1.html#complete-space"><i class="fa fa-check"></i><b>2.3.4</b> Complete space</a></li>
<li class="chapter" data-level="2.3.5" data-path="but-why-does-it-work-1.html"><a href="but-why-does-it-work-1.html#contraction"><i class="fa fa-check"></i><b>2.3.5</b> Contraction</a></li>
<li class="chapter" data-level="2.3.6" data-path="but-why-does-it-work-1.html"><a href="but-why-does-it-work-1.html#affine-transformation"><i class="fa fa-check"></i><b>2.3.6</b> Affine transformation</a></li>
<li class="chapter" data-level="2.3.7" data-path="but-why-does-it-work-1.html"><a href="but-why-does-it-work-1.html#hutchinsons-theorem"><i class="fa fa-check"></i><b>2.3.7</b> Hutchinson’s theorem</a></li>
<li class="chapter" data-level="2.3.8" data-path="but-why-does-it-work-1.html"><a href="but-why-does-it-work-1.html#banachs-fixed-point-theorem"><i class="fa fa-check"></i><b>2.3.8</b> Banach’s fixed point theorem</a></li>
</ul></li>
<li class="chapter" data-level="2.4" data-path="examples-in-python-1.html"><a href="examples-in-python-1.html"><i class="fa fa-check"></i><b>2.4</b> Examples in Python</a>
<ul>
<li class="chapter" data-level="2.4.1" data-path="examples-in-python-1.html"><a href="examples-in-python-1.html#sierpinski-triangle-4"><i class="fa fa-check"></i><b>2.4.1</b> Sierpinski triangle</a></li>
<li class="chapter" data-level="2.4.2" data-path="examples-in-python-1.html"><a href="examples-in-python-1.html#sierpinski-pentagon-1"><i class="fa fa-check"></i><b>2.4.2</b> Sierpinski pentagon</a></li>
<li class="chapter" data-level="2.4.3" data-path="examples-in-python-1.html"><a href="examples-in-python-1.html#heighways-dragon"><i class="fa fa-check"></i><b>2.4.3</b> Heighway’s Dragon</a></li>
<li class="chapter" data-level="2.4.4" data-path="examples-in-python-1.html"><a href="examples-in-python-1.html#symmetric-binary-tree-pythagoras-tree"><i class="fa fa-check"></i><b>2.4.4</b> Symmetric binary tree / Pythagoras’ tree</a></li>
</ul></li>
<li class="chapter" data-level="2.5" data-path="examples-in-r-1.html"><a href="examples-in-r-1.html"><i class="fa fa-check"></i><b>2.5</b> Examples in R</a>
<ul>
<li class="chapter" data-level="2.5.1" data-path="examples-in-r-1.html"><a href="examples-in-r-1.html#sierpiński-triangle"><i class="fa fa-check"></i><b>2.5.1</b> Sierpiński triangle</a></li>
<li class="chapter" data-level="2.5.2" data-path="examples-in-r-1.html"><a href="examples-in-r-1.html#sierpiński-pentagon"><i class="fa fa-check"></i><b>2.5.2</b> Sierpiński pentagon</a></li>
<li class="chapter" data-level="2.5.3" data-path="examples-in-r-1.html"><a href="examples-in-r-1.html#heighways-dragon-1"><i class="fa fa-check"></i><b>2.5.3</b> Heighway’s Dragon</a></li>
<li class="chapter" data-level="2.5.4" data-path="examples-in-r-1.html"><a href="examples-in-r-1.html#symmetric-binary-tree-pythagoras-tree-1"><i class="fa fa-check"></i><b>2.5.4</b> Symmetric binary tree / Pythagoras’ tree</a></li>
</ul></li>
<li class="chapter" data-level="2.6" data-path="examples-in-julia-1.html"><a href="examples-in-julia-1.html"><i class="fa fa-check"></i><b>2.6</b> Examples in Julia</a>
<ul>
<li class="chapter" data-level="2.6.1" data-path="examples-in-julia-1.html"><a href="examples-in-julia-1.html#sierpinski-triangle-5"><i class="fa fa-check"></i><b>2.6.1</b> Sierpinski triangle</a></li>
<li class="chapter" data-level="2.6.2" data-path="examples-in-julia-1.html"><a href="examples-in-julia-1.html#sierpinski-pentagon-2"><i class="fa fa-check"></i><b>2.6.2</b> Sierpinski pentagon</a></li>
<li class="chapter" data-level="2.6.3" data-path="examples-in-julia-1.html"><a href="examples-in-julia-1.html#heighways-dragon-2"><i class="fa fa-check"></i><b>2.6.3</b> Heighway’s Dragon</a></li>
<li class="chapter" data-level="2.6.4" data-path="examples-in-julia-1.html"><a href="examples-in-julia-1.html#symmetric-binary-tree-pythagoras-tree-2"><i class="fa fa-check"></i><b>2.6.4</b> Symmetric binary tree / Pythagoras’ tree</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="3" data-path="chaos-game.html"><a href="chaos-game.html"><i class="fa fa-check"></i><b>3</b> Chaos Game</a>
<ul>
<li class="chapter" data-level="3.1" data-path="hugo-steinhaus.html"><a href="hugo-steinhaus.html"><i class="fa fa-check"></i><b>3.1</b> Hugo Steinhaus</a></li>
<li class="chapter" data-level="3.2" data-path="chaos-game-1.html"><a href="chaos-game-1.html"><i class="fa fa-check"></i><b>3.2</b> Chaos Game</a>
<ul>
<li class="chapter" data-level="3.2.1" data-path="chaos-game-1.html"><a href="chaos-game-1.html#chaos-game-and-triangles"><i class="fa fa-check"></i><b>3.2.1</b> Chaos game and triangles</a></li>
<li class="chapter" data-level="3.2.2" data-path="chaos-game-1.html"><a href="chaos-game-1.html#whats-going-on-here"><i class="fa fa-check"></i><b>3.2.2</b> What’s going on here?</a></li>
<li class="chapter" data-level="3.2.3" data-path="chaos-game-1.html"><a href="chaos-game-1.html#chaos-game-and-fractals"><i class="fa fa-check"></i><b>3.2.3</b> Chaos game and fractals</a></li>
<li class="chapter" data-level="3.2.4" data-path="chaos-game-1.html"><a href="chaos-game-1.html#affine-transformations"><i class="fa fa-check"></i><b>3.2.4</b> Affine transformations</a></li>
<li class="chapter" data-level="3.2.5" data-path="chaos-game-1.html"><a href="chaos-game-1.html#ferns-and-chaos"><i class="fa fa-check"></i><b>3.2.5</b> Ferns and chaos</a></li>
<li class="chapter" data-level="3.2.6" data-path="chaos-game-1.html"><a href="chaos-game-1.html#more-formally"><i class="fa fa-check"></i><b>3.2.6</b> More formally</a></li>
</ul></li>
<li class="chapter" data-level="3.3" data-path="applications.html"><a href="applications.html"><i class="fa fa-check"></i><b>3.3</b> Applications</a>
<ul>
<li class="chapter" data-level="3.3.1" data-path="applications.html"><a href="applications.html#chaos-game-and-genetics"><i class="fa fa-check"></i><b>3.3.1</b> Chaos game and genetics</a></li>
<li class="chapter" data-level="3.3.2" data-path="applications.html"><a href="applications.html#chaos-game-and-music"><i class="fa fa-check"></i><b>3.3.2</b> Chaos game and music</a></li>
<li class="chapter" data-level="3.3.3" data-path="applications.html"><a href="applications.html#the-fractal-dimension-of-the-brain"><i class="fa fa-check"></i><b>3.3.3</b> The fractal dimension of the brain</a></li>
</ul></li>
<li class="chapter" data-level="3.4" data-path="examples-in-python-2.html"><a href="examples-in-python-2.html"><i class="fa fa-check"></i><b>3.4</b> Examples in Python</a>
<ul>
<li class="chapter" data-level="3.4.1" data-path="examples-in-python-2.html"><a href="examples-in-python-2.html#the-sierpinski-triangle-once-again"><i class="fa fa-check"></i><b>3.4.1</b> The Sierpinski triangle once again</a></li>
<li class="chapter" data-level="3.4.2" data-path="examples-in-python-2.html"><a href="examples-in-python-2.html#barnsley-fern-1"><i class="fa fa-check"></i><b>3.4.2</b> Barnsley fern</a></li>
<li class="chapter" data-level="3.4.3" data-path="examples-in-python-2.html"><a href="examples-in-python-2.html#maple-leaf"><i class="fa fa-check"></i><b>3.4.3</b> Maple leaf</a></li>
<li class="chapter" data-level="3.4.4" data-path="examples-in-python-2.html"><a href="examples-in-python-2.html#spiral"><i class="fa fa-check"></i><b>3.4.4</b> Spiral</a></li>
</ul></li>
<li class="chapter" data-level="3.5" data-path="examples-in-r-2.html"><a href="examples-in-r-2.html"><i class="fa fa-check"></i><b>3.5</b> Examples in R</a>
<ul>
<li class="chapter" data-level="3.5.1" data-path="examples-in-r-2.html"><a href="examples-in-r-2.html#the-sierpinski-triangle-once-again-1"><i class="fa fa-check"></i><b>3.5.1</b> The Sierpinski triangle once again</a></li>
<li class="chapter" data-level="3.5.2" data-path="examples-in-r-2.html"><a href="examples-in-r-2.html#barnsley-fern-2"><i class="fa fa-check"></i><b>3.5.2</b> Barnsley fern</a></li>
<li class="chapter" data-level="3.5.3" data-path="examples-in-r-2.html"><a href="examples-in-r-2.html#maple-leaf-1"><i class="fa fa-check"></i><b>3.5.3</b> Maple leaf</a></li>
<li class="chapter" data-level="3.5.4" data-path="examples-in-r-2.html"><a href="examples-in-r-2.html#spiral-1"><i class="fa fa-check"></i><b>3.5.4</b> Spiral</a></li>
</ul></li>
<li class="chapter" data-level="3.6" data-path="examples-in-julia-2.html"><a href="examples-in-julia-2.html"><i class="fa fa-check"></i><b>3.6</b> Examples in Julia</a>
<ul>
<li class="chapter" data-level="3.6.1" data-path="examples-in-julia-2.html"><a href="examples-in-julia-2.html#the-sierpinski-triangle-once-again-2"><i class="fa fa-check"></i><b>3.6.1</b> The Sierpinski triangle once again</a></li>
<li class="chapter" data-level="3.6.2" data-path="examples-in-julia-2.html"><a href="examples-in-julia-2.html#barnsley-fern-3"><i class="fa fa-check"></i><b>3.6.2</b> Barnsley fern</a></li>
<li class="chapter" data-level="3.6.3" data-path="examples-in-julia-2.html"><a href="examples-in-julia-2.html#maple-leaf-2"><i class="fa fa-check"></i><b>3.6.3</b> Maple leaf</a></li>
<li class="chapter" data-level="3.6.4" data-path="examples-in-julia-2.html"><a href="examples-in-julia-2.html#spiral-2"><i class="fa fa-check"></i><b>3.6.4</b> Spiral</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="4" data-path="pocket-atlas-of-fractals.html"><a href="pocket-atlas-of-fractals.html"><i class="fa fa-check"></i><b>4</b> Pocket atlas of fractals</a>
<ul>
<li class="chapter" data-level="4.1" data-path="barnsley-fern-4.html"><a href="barnsley-fern-4.html"><i class="fa fa-check"></i><b>4.1</b> Barnsley fern</a></li>
<li class="chapter" data-level="4.2" data-path="maple-leaf-3.html"><a href="maple-leaf-3.html"><i class="fa fa-check"></i><b>4.2</b> Maple leaf</a></li>
<li class="chapter" data-level="4.3" data-path="mcworters-pentigree.html"><a href="mcworters-pentigree.html"><i class="fa fa-check"></i><b>4.3</b> McWorter’s pentigree</a></li>
<li class="chapter" data-level="4.4" data-path="trees.html"><a href="trees.html"><i class="fa fa-check"></i><b>4.4</b> Trees</a></li>
<li class="chapter" data-level="4.5" data-path="spiral-3.html"><a href="spiral-3.html"><i class="fa fa-check"></i><b>4.5</b> Spiral</a></li>
<li class="chapter" data-level="4.6" data-path="sierpinski-polygons.html"><a href="sierpinski-polygons.html"><i class="fa fa-check"></i><b>4.6</b> Sierpinski polygons</a></li>
<li class="chapter" data-level="4.7" data-path="dragons.html"><a href="dragons.html"><i class="fa fa-check"></i><b>4.7</b> Dragons</a></li>
<li class="chapter" data-level="4.8" data-path="sky-is-the-limit.html"><a href="sky-is-the-limit.html"><i class="fa fa-check"></i><b>4.8</b> Sky is the limit</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="bibliography.html"><a href="bibliography.html"><i class="fa fa-check"></i>Bibliography</a></li>
<li class="divider"></li>
<li><a href="https://www.mi2.ai/beta-bit.html" target="_blank">Other books in the Beta and Bit series</a></li>

</ul>

      </nav>
    </div>

    <div class="book-body">
      <div class="body-inner">
        <div class="book-header" role="navigation">
          <h1>
            <i class="fa fa-circle-o-notch fa-spin"></i><a href="./">Chaos Game</a>
          </h1>
        </div>

        <div class="page-wrapper" tabindex="-1" role="main">
          <div class="page-inner">

            <section class="normal" id="section-">
<div id="examples-in-julia" class="section level2 hasAnchor" number="1.6">
<h2><span class="header-section-number">1.6</span> Examples in Julia<a href="examples-in-julia.html#examples-in-julia" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>When creating fractals, certain operations must be repeated infinitely many times, or at least for a very long time. Efficient, flexible, and very very fast programming languages are great for this. Julia is such a language. It can be downloaded from <a href="https://julialang.org/">https://julialang.org/</a>.</p>
<p>Here is some information to help you understand the following examples.</p>
<ol style="list-style-type: decimal">
<li>The following examples use the <code>Plots</code> library, a basic library for the Julia language. The <code>plot()</code> function from this library creates an empty graph. You can add more figures to it with the <code>plot!()</code> function. In the examples below, we will create additional filled polygons, triangles, and squares in this way.</li>
<li>Julia is a language very friendly to mathematical operations, you can use mathematical functions such as <code>sqrt()</code> without additional modules. Arithmetic operators such as <code>+</code> or <code>/</code> work for both numbers and vectors.</li>
<li>To make the code more readable, we use recursion, that is, a construction in which a function calls itself. New functions are defined using the word <code>function</code>. You can use a simplified definition with the <code>=</code> operator for shorter functions. We will use the shorter way when defining functions <code>line</code>, <code>triangle</code>, <code>square</code>.</li>
</ol>
<div id="cantor-dust-3" class="section level3 hasAnchor" number="1.6.1">
<h3><span class="header-section-number">1.6.1</span> Cantor dust<a href="examples-in-julia.html#cantor-dust-3" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Let’s start drawing fractals with Cantor dust. For this, we will use the recursive function <code>dust</code>, which takes three arguments: <code>x</code> – the beginning of the fractal, <code>scale -- the size of the fractal, and</code>depth` – the current nesting level of the fractal.</p>
<div class="sourceCode" id="cb7"><pre class="sourceCode julia"><code class="sourceCode julia"><span id="cb7-1"><a href="examples-in-julia.html#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Package with graphics functions.</span></span>
<span id="cb7-2"><a href="examples-in-julia.html#cb7-2" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">Plots </span></span>
<span id="cb7-3"><a href="examples-in-julia.html#cb7-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-4"><a href="examples-in-julia.html#cb7-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Function that defines a line segment.</span></span>
<span id="cb7-5"><a href="examples-in-julia.html#cb7-5" aria-hidden="true" tabindex="-1"></a><span class="fu">line</span>(x, scale) <span class="op">=</span> <span class="fu">Shape</span>([x, x<span class="op">+</span>scale], [<span class="fl">0</span>, <span class="fl">0</span>]) </span>
<span id="cb7-6"><a href="examples-in-julia.html#cb7-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-7"><a href="examples-in-julia.html#cb7-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Rekurencyjna funkcja do rysowania kurzu. </span></span>
<span id="cb7-8"><a href="examples-in-julia.html#cb7-8" aria-hidden="true" tabindex="-1"></a><span class="co"># Recursive function to draw dust. </span></span>
<span id="cb7-9"><a href="examples-in-julia.html#cb7-9" aria-hidden="true" tabindex="-1"></a><span class="co"># If depth=1, it draws a line segment, </span></span>
<span id="cb7-10"><a href="examples-in-julia.html#cb7-10" aria-hidden="true" tabindex="-1"></a><span class="co"># if depth&gt;1 it draws two copies of Cantor dust side by side.</span></span>
<span id="cb7-11"><a href="examples-in-julia.html#cb7-11" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">dust</span>(x, scale, depth<span class="op">=</span><span class="fl">1</span>) </span>
<span id="cb7-12"><a href="examples-in-julia.html#cb7-12" aria-hidden="true" tabindex="-1"></a>    <span class="cf">if</span> depth <span class="op">==</span> <span class="fl">0</span> </span>
<span id="cb7-13"><a href="examples-in-julia.html#cb7-13" aria-hidden="true" tabindex="-1"></a>        <span class="fu">plot!</span>(<span class="fu">line</span>(x, scale), color<span class="op">=:</span>black, legend<span class="op">=:</span><span class="cn">false</span>)</span>
<span id="cb7-14"><a href="examples-in-julia.html#cb7-14" aria-hidden="true" tabindex="-1"></a>    <span class="cf">else</span> </span>
<span id="cb7-15"><a href="examples-in-julia.html#cb7-15" aria-hidden="true" tabindex="-1"></a>      <span class="fu">dust</span>(x, scale<span class="op">/</span><span class="fl">3</span>, depth <span class="op">-</span> <span class="fl">1</span>)</span>
<span id="cb7-16"><a href="examples-in-julia.html#cb7-16" aria-hidden="true" tabindex="-1"></a>      <span class="fu">dust</span>(x <span class="op">+</span> scale<span class="op">*</span><span class="fl">2</span><span class="op">/</span><span class="fl">3</span>, scale<span class="op">/</span><span class="fl">3</span>, depth<span class="op">-</span><span class="fl">1</span>)</span>
<span id="cb7-17"><a href="examples-in-julia.html#cb7-17" aria-hidden="true" tabindex="-1"></a>    <span class="cf">end</span></span>
<span id="cb7-18"><a href="examples-in-julia.html#cb7-18" aria-hidden="true" tabindex="-1"></a><span class="kw">end</span></span>
<span id="cb7-19"><a href="examples-in-julia.html#cb7-19" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-20"><a href="examples-in-julia.html#cb7-20" aria-hidden="true" tabindex="-1"></a><span class="co"># Clear the screen and start drawing dust.</span></span>
<span id="cb7-21"><a href="examples-in-julia.html#cb7-21" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(<span class="fl">0</span>, xlim<span class="op">=</span>(<span class="op">-</span><span class="fl">0.1</span>,<span class="fl">1.1</span>), ylim<span class="op">=</span>(<span class="op">-</span><span class="fl">0.1</span>,<span class="fl">0.1</span>), axis<span class="op">=</span><span class="cn">nothing</span>) </span>
<span id="cb7-22"><a href="examples-in-julia.html#cb7-22" aria-hidden="true" tabindex="-1"></a><span class="fu">dust</span>(<span class="fl">0.0</span>, <span class="fl">1.0</span>, <span class="fl">4</span>)</span></code></pre></div>
<div class="figure">
<img src="images/julia_20_01.png" alt="" />
<p class="caption">The result of executing the following instructions</p>
</div>
</div>
<div id="sierpinski-triangle-3" class="section level3 hasAnchor" number="1.6.2">
<h3><span class="header-section-number">1.6.2</span> Sierpinski triangle<a href="examples-in-julia.html#sierpinski-triangle-3" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>It’s time for the iconic Sierpiński triangle. To draw it, we will use the recursive function <code>sierpinski()</code>, which takes four arguments: <code>x</code>, <code>y</code> – the location from which to draw the fractal, <code>scale</code> – the size of the fractal, and depth <code>depth</code> – the current nesting level of the fractal.</p>
<div class="sourceCode" id="cb8"><pre class="sourceCode julia"><code class="sourceCode julia"><span id="cb8-1"><a href="examples-in-julia.html#cb8-1" aria-hidden="true" tabindex="-1"></a><span class="fu">triangle</span>(x, y, scale) <span class="op">=</span> <span class="fu">Shape</span>([x, x<span class="op">+</span>scale, x<span class="op">+</span>scale<span class="op">/</span><span class="fl">2</span>, x],</span>
<span id="cb8-2"><a href="examples-in-julia.html#cb8-2" aria-hidden="true" tabindex="-1"></a>      [y, y, <span class="fu">y+scale*sqrt</span>(<span class="fl">3</span>)<span class="op">/</span><span class="fl">2</span>, y])</span>
<span id="cb8-3"><a href="examples-in-julia.html#cb8-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb8-4"><a href="examples-in-julia.html#cb8-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Recursive function to draw a Sierpiński triangle. </span></span>
<span id="cb8-5"><a href="examples-in-julia.html#cb8-5" aria-hidden="true" tabindex="-1"></a><span class="co"># If depth=1, we draw a triangle using the triangle() function, </span></span>
<span id="cb8-6"><a href="examples-in-julia.html#cb8-6" aria-hidden="true" tabindex="-1"></a><span class="co"># if depth&gt;1, we draw three Sierpiński triangles.</span></span>
<span id="cb8-7"><a href="examples-in-julia.html#cb8-7" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">sierpinski</span>(x, y, scale, depth<span class="op">=</span><span class="fl">1</span>)</span>
<span id="cb8-8"><a href="examples-in-julia.html#cb8-8" aria-hidden="true" tabindex="-1"></a>  <span class="cf">if</span> depth<span class="op">==</span><span class="fl">0</span></span>
<span id="cb8-9"><a href="examples-in-julia.html#cb8-9" aria-hidden="true" tabindex="-1"></a>    <span class="fu">plot!</span>(<span class="fu">triangle</span>(x,y,scale),color<span class="op">=:</span>black,legend<span class="op">=:</span><span class="cn">false</span>)</span>
<span id="cb8-10"><a href="examples-in-julia.html#cb8-10" aria-hidden="true" tabindex="-1"></a>  <span class="cf">else</span></span>
<span id="cb8-11"><a href="examples-in-julia.html#cb8-11" aria-hidden="true" tabindex="-1"></a>    <span class="fu">sierpinski</span>(x, y, scale<span class="op">/</span><span class="fl">2</span>, depth<span class="op">-</span><span class="fl">1</span>)</span>
<span id="cb8-12"><a href="examples-in-julia.html#cb8-12" aria-hidden="true" tabindex="-1"></a>    <span class="fu">sierpinski</span>(x<span class="op">+</span>scale<span class="op">/</span><span class="fl">2</span>, y, scale<span class="op">/</span><span class="fl">2</span>, depth<span class="op">-</span><span class="fl">1</span>)</span>
<span id="cb8-13"><a href="examples-in-julia.html#cb8-13" aria-hidden="true" tabindex="-1"></a>    <span class="fu">sierpinski</span>(x<span class="op">+</span>scale<span class="op">/</span><span class="fl">4</span>,<span class="fu">y+sqrt</span>(<span class="fl">3</span>)<span class="op">*</span>scale<span class="op">/</span><span class="fl">4</span>,scale<span class="op">/</span><span class="fl">2</span>,depth<span class="op">-</span><span class="fl">1</span>)</span>
<span id="cb8-14"><a href="examples-in-julia.html#cb8-14" aria-hidden="true" tabindex="-1"></a>  <span class="cf">end</span></span>
<span id="cb8-15"><a href="examples-in-julia.html#cb8-15" aria-hidden="true" tabindex="-1"></a><span class="kw">end</span></span>
<span id="cb8-16"><a href="examples-in-julia.html#cb8-16" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb8-17"><a href="examples-in-julia.html#cb8-17" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(<span class="fl">0</span>, xlim<span class="op">=</span>(<span class="fl">0</span>,<span class="fl">1</span>), ylim<span class="op">=</span>(<span class="fl">0</span>,<span class="fl">1</span>), axis<span class="op">=</span><span class="cn">nothing</span>)</span>
<span id="cb8-18"><a href="examples-in-julia.html#cb8-18" aria-hidden="true" tabindex="-1"></a><span class="fu">sierpinski</span>(<span class="fl">0</span>, <span class="fl">0</span>, <span class="fl">1</span>, <span class="fl">6</span>)</span></code></pre></div>
<div class="figure">
<img src="images/julia_20_02.png" alt="" />
<p class="caption">The result of executing the following instructions</p>
</div>
</div>
<div id="sierpinski-carpet-3" class="section level3 hasAnchor" number="1.6.3">
<h3><span class="header-section-number">1.6.3</span> Sierpinski carpet<a href="examples-in-julia.html#sierpinski-carpet-3" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>The fractal structure proposed by Wacław Sierpiński can be repeated for other shapes.</p>
<div class="sourceCode" id="cb9"><pre class="sourceCode julia"><code class="sourceCode julia"><span id="cb9-1"><a href="examples-in-julia.html#cb9-1" aria-hidden="true" tabindex="-1"></a><span class="fu">square</span>(x, y, w) <span class="op">=</span> <span class="fu">Shape</span>([x, x<span class="op">+</span>w, x<span class="op">+</span>w, x], [y, y, y<span class="op">+</span>w, y<span class="op">+</span>w])   </span>
<span id="cb9-2"><a href="examples-in-julia.html#cb9-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-3"><a href="examples-in-julia.html#cb9-3" aria-hidden="true" tabindex="-1"></a><span class="co"># We use the square function defined above to draw a square.</span></span>
<span id="cb9-4"><a href="examples-in-julia.html#cb9-4" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">carpet</span>(x, y, scale, depth<span class="op">=</span><span class="fl">1</span>)</span>
<span id="cb9-5"><a href="examples-in-julia.html#cb9-5" aria-hidden="true" tabindex="-1"></a>  <span class="cf">if</span> depth<span class="op">==</span><span class="fl">0</span></span>
<span id="cb9-6"><a href="examples-in-julia.html#cb9-6" aria-hidden="true" tabindex="-1"></a>    <span class="fu">plot!</span>(<span class="fu">square</span>(x,y,scale), color<span class="op">=:</span>black, legend<span class="op">=:</span><span class="cn">false</span>) </span>
<span id="cb9-7"><a href="examples-in-julia.html#cb9-7" aria-hidden="true" tabindex="-1"></a>  <span class="cf">else</span></span>
<span id="cb9-8"><a href="examples-in-julia.html#cb9-8" aria-hidden="true" tabindex="-1"></a>    <span class="fu">carpet</span>(x, y, scale<span class="op">/</span><span class="fl">3</span>, depth<span class="op">-</span><span class="fl">1</span>)</span>
<span id="cb9-9"><a href="examples-in-julia.html#cb9-9" aria-hidden="true" tabindex="-1"></a>    <span class="fu">carpet</span>(x, y<span class="op">+</span>scale, scale<span class="op">/</span><span class="fl">3</span>, depth<span class="op">-</span><span class="fl">1</span>)</span>
<span id="cb9-10"><a href="examples-in-julia.html#cb9-10" aria-hidden="true" tabindex="-1"></a>    <span class="fu">carpet</span>(x, y<span class="op">+</span><span class="fl">2</span>scale, scale<span class="op">/</span><span class="fl">3</span>, depth<span class="op">-</span><span class="fl">1</span>)  </span>
<span id="cb9-11"><a href="examples-in-julia.html#cb9-11" aria-hidden="true" tabindex="-1"></a>    <span class="fu">carpet</span>(x<span class="op">+</span>scale, y, scale<span class="op">/</span><span class="fl">3</span>, depth<span class="op">-</span><span class="fl">1</span>)</span>
<span id="cb9-12"><a href="examples-in-julia.html#cb9-12" aria-hidden="true" tabindex="-1"></a>    <span class="fu">carpet</span>(x<span class="op">+</span>scale, y<span class="op">+</span><span class="fl">2</span>scale, scale<span class="op">/</span><span class="fl">3</span>, depth<span class="op">-</span><span class="fl">1</span>)</span>
<span id="cb9-13"><a href="examples-in-julia.html#cb9-13" aria-hidden="true" tabindex="-1"></a>    <span class="fu">carpet</span>(x<span class="op">+</span><span class="fl">2</span>scale, y, scale<span class="op">/</span><span class="fl">3</span>, depth<span class="op">-</span><span class="fl">1</span>)</span>
<span id="cb9-14"><a href="examples-in-julia.html#cb9-14" aria-hidden="true" tabindex="-1"></a>    <span class="fu">carpet</span>(x<span class="op">+</span><span class="fl">2</span>scale, y<span class="op">+</span>scale, scale<span class="op">/</span><span class="fl">3</span>, depth<span class="op">-</span><span class="fl">1</span>)</span>
<span id="cb9-15"><a href="examples-in-julia.html#cb9-15" aria-hidden="true" tabindex="-1"></a>    <span class="fu">carpet</span>(x<span class="op">+</span><span class="fl">2</span>scale, y<span class="op">+</span><span class="fl">2</span>scale, scale<span class="op">/</span><span class="fl">3</span>, depth<span class="op">-</span><span class="fl">1</span>)</span>
<span id="cb9-16"><a href="examples-in-julia.html#cb9-16" aria-hidden="true" tabindex="-1"></a>  <span class="cf">end</span></span>
<span id="cb9-17"><a href="examples-in-julia.html#cb9-17" aria-hidden="true" tabindex="-1"></a><span class="kw">end</span></span>
<span id="cb9-18"><a href="examples-in-julia.html#cb9-18" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-19"><a href="examples-in-julia.html#cb9-19" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(<span class="fl">0</span>, xlim<span class="op">=</span>(<span class="fl">0</span>,<span class="fl">3</span>), ylim<span class="op">=</span>(<span class="fl">0</span>,<span class="fl">3</span>), axis<span class="op">=</span><span class="cn">nothing</span>)</span>
<span id="cb9-20"><a href="examples-in-julia.html#cb9-20" aria-hidden="true" tabindex="-1"></a><span class="fu">carpet</span>(<span class="fl">0.0</span>, <span class="fl">0.0</span>, <span class="fl">1.0</span>, <span class="fl">5</span>)</span></code></pre></div>
<div class="figure">
<img src="images/julia_20_03.png" alt="" />
<p class="caption">The result of executing the following instructions</p>
</div>

</div>
</div>
<!-- </div> -->
            </section>

          </div>
        </div>
      </div>
<a href="examples-in-r.html" class="navigation navigation-prev " aria-label="Previous page"><i class="fa fa-angle-left"></i></a>
<a href="fractal-that-is-fixed-point.html" class="navigation navigation-next " aria-label="Next page"><i class="fa fa-angle-right"></i></a>
    </div>
  </div>
<script src="libs/gitbook/js/app.min.js"></script>
<script src="libs/gitbook/js/clipboard.min.js"></script>
<script src="libs/gitbook/js/plugin-search.js"></script>
<script src="libs/gitbook/js/plugin-sharing.js"></script>
<script src="libs/gitbook/js/plugin-fontsettings.js"></script>
<script src="libs/gitbook/js/plugin-bookdown.js"></script>
<script src="libs/gitbook/js/jquery.highlight.js"></script>
<script src="libs/gitbook/js/plugin-clipboard.js"></script>
<script>
gitbook.require(["gitbook"], function(gitbook) {
gitbook.start({
"sharing": {
"github": false,
"facebook": true,
"twitter": true,
"linkedin": false,
"weibo": false,
"instapaper": false,
"vk": false,
"whatsapp": false,
"all": ["facebook", "twitter", "linkedin", "weibo", "instapaper"]
},
"fontsettings": {
"theme": "white",
"family": "sans",
"size": 2
},
"edit": {
"link": null,
"text": null
},
"history": {
"link": null,
"text": null
},
"view": {
"link": null,
"text": null
},
"download": null,
"search": {
"engine": "fuse",
"options": null
},
"toc": {
"collapse": "subsection",
"toc_depth": 3
}
});
});
</script>

<!-- dynamically load mathjax for compatibility with self-contained -->
<script>
  (function () {
    var script = document.createElement("script");
    script.type = "text/javascript";
    var src = "true";
    if (src === "" || src === "true") src = "https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.9/latest.js?config=TeX-MML-AM_CHTML";
    if (location.protocol !== "file:")
      if (/^https?:/.test(src))
        src = src.replace(/^https?:/, '');
    script.src = src;
    document.getElementsByTagName("head")[0].appendChild(script);
  })();
</script>
</body>

</html>