415 lines
41 KiB
HTML
415 lines
41 KiB
HTML
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<!DOCTYPE html><html lang="en"><head><meta charset="utf-8"/><meta content="width=device-width, initial-scale=1" name="viewport"/><title> <loganGatlin/> </title><link type="image/x-icon" rel="icon" href="resources/favicon.svg"/><style>
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/*!
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Pure v3.0.0
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Copyright 2013 Yahoo!
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Licensed under the BSD License.
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https://github.com/pure-css/pure/blob/master/LICENSE
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*/
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/*!
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normalize.css v | MIT License | https://necolas.github.io/normalize.css/
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Copyright (c) Nicolas Gallagher and Jonathan Neal
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*/
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/*! normalize.css v8.0.1 | MIT License | github.com/necolas/normalize.css */html{line-height:1.15;-webkit-text-size-adjust:100%}body{margin:0}main{display:block}h1{font-size:2em;margin:.67em 0}hr{box-sizing:content-box;height:0;overflow:visible}pre{font-family:monospace,monospace;font-size:1em}a{background-color:transparent}abbr[title]{border-bottom:none;text-decoration:underline;-webkit-text-decoration:underline dotted;text-decoration:underline dotted}b,strong{font-weight:bolder}code,kbd,samp{font-family:monospace,monospace;font-size:1em}small{font-size:80%}sub,sup{font-size:75%;line-height:0;position:relative;vertical-align:baseline}sub{bottom:-.25em}sup{top:-.5em}img{border-style:none}button,input,optgroup,select,textarea{font-family:inherit;font-size:100%;line-height:1.15;margin:0}button,input{overflow:visible}button,select{text-transform:none}[type=button],[type=reset],[type=submit],button{-webkit-appearance:button}[type=button]::-moz-focus-inner,[type=reset]::-moz-focus-inner,[type=submit]::-moz-focus-inner,button::-moz-focus-inner{border-style:none;padding:0}[type=button]:-moz-focusring,[type=reset]:-moz-focusring,[type=submit]:-moz-focusring,button:-moz-focusring{outline:1px dotted ButtonText}fieldset{padding:.35em .75em .625em}legend{box-sizing:border-box;color:inherit;display:table;max-width:100%;padding:0;white-space:normal}progress{vertical-align:baseline}textarea{overflow:auto}[type=checkbox],[type=radio]{box-sizing:border-box;padding:0}[type=number]::-webkit-inner-spin-button,[type=number]::-webkit-outer-spin-button{height:auto}[type=search]{-webkit-appearance:textfield;outline-offset:-2px}[type=search]::-webkit-search-decoration{-webkit-appearance:none}::-webkit-file-upload-button{-webkit-appearance:button;font:inherit}details{display:block}summary{display:list-item}template{display:none}[hidden]{display:none}html{font-family:sans-serif}.hidden,[hidden]{display:none!important}.pure-img{max-width:100%;height:auto;display:block}.pure-g{display:flex;flex-flow:row wrap;align-content:flex-start}.pure-u{display:inline-block;vertical-align:top}.pure-u-1,.pure-u-1-1,.pure-u-1-12,.pure-u-1-2,.pure-u-1-24,.pure-u-1-3,.pure-u-1-4,.pure-u-1-5,.pure-u-1-6,.pure-u-1-8,.pure-u-10-24,.pure-u-11-12,.pure-u-11-24,.pure-u-12-24,.pure-u-13-24,.pure-u-14-24,.pure-u-15-24,.pure-u-16-24,.pure-u-17-24,.pure-u-18-24,.pure-u-19-24,.pure-u-2-24,.pure-u-2-3,.pure-u-2-5,.pure-u-20-24,.pure-u-21-24,.pure-u-22-24,.pure-u-23-24,.pure-u-24-24,.pure-u-3-24,.pure-u-3-4,.pure-u-3-5,.pure-u-3-8,.pure-u-4-24,.pure-u-4-5,.pure-u-5-12,.pure-u-5-24,.pure-u-5-5,.pure-u-5-6,.pure-u-5-8,.pure-u-6-24,.pure-u-7-12,.pure-u-7-24,.pure-u-7-8,.pure-u-8-24,.pure-u-9-24{display:inline-block;letter-spacing:normal;word-spacing:normal;vertical-align:top;text-rendering:auto}.pure-u-1-24{width:4.1667%}.pure-u-1-12,.pure-u-2-24{width:8.3333%}.pure-u-1-8,.pure-u-3-24{width:12.5%}.pure-u-1-6,.pure-u-4-24{width:16.6667%}.pure-u-1-5{width:20%}.pure-u-5-24{width:20.8333%}.pure-u-1-4,.pure-u-6-24{width:25%}.pure-u-7-24{width:29.1667%}.pure-u-1-3,.pure-u-8-24{width:33.3333%}.pure-u-3-8,.pure-u-9-24{width:37.5%}.pure-u-2-5{width:40%}.pure-u-10-24,.pure-u-5-12{width:41.6667%}.pure-u-11-24{width:45.8333%}.pure-u-1-2,.pure-u-12-24{width:50%}.pure-u-13-24{width:54.1667%}.pure-u-14-24,.pure-u-7-12{width:58.3333%}.pure-u-3-5{width:60%}.pure-u-15-24,.pure-u-5-8{width:62.5%}.pure-u-16-24,.pure-u-2-3{width:66.6667%}.pure-u-17-24{width:70.8333%}.pure-u-18-24,.pure-u-3-4{width:75%}.pure-u-19-24{width:79.1667%}.pure-u-4-5{width:80%}.pure-u-20-24,.pure-u-5-6{width:83.3333%}.pure-u-21-24,.pure-u-7-8{width:87.5%}.pure-u-11-12,.pure-u-22-24{width:91.6667%}.pure-u-23-24{width:95.8333%}.pure-u-1,.pure-u-1-1,.pure-u-24-24,.pure-u-5-5{width:100%}.pure-button{display:inline-block;line-height:normal;white-space:nowrap;vertical-align:middle;text-align:center;cursor:pointer;-webkit-user-drag:none;-webkit-user-select:none;user-select:none;box-sizing:border-box}.pure-button::-moz-focus-inner{padding:0;border:0}.pure-button-group{letter-spacing:-.31em;text-rendering:optimizespeed}.opera-only :-o-
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</style><style>
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@font-face {
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font-family: Cascadia;
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src: url("/resources/CascadiaMonoPL.woff2");
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}
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body {
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color: #444;
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font-size: 1.2em;
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}
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.link {
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text-decoration: none;
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color: #1e90ff;
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}
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.link :visited {
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color: #1e90ff;
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}
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table {
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display: block;
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margin-top: 1em;
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margin-bottom: 1em;
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margin-left: auto;
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margin-right: auto;
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width: 60%;
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text-align: left;
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border-collapse: collapse;
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}
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th {
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padding: 0.25em 0.25em 0.25em 0.25em;
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border: 2px solid;
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border-collapse: collapse;
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}
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td {
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padding: 0.25em 0.25em 0.25em 0.25em;
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border: 2px solid;
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border-collapse: collapse;
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}
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/* https://css-loaders.com/spinner/ */
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/* HTML: <div class="loader"></div> */
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.loading {
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width: 512px;
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padding: 8px;
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aspect-ratio: 1;
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border-radius: 50%;
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background: #25b09b;
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--_m:
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conic-gradient(#0000 10%,#000),
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linear-gradient(#000 0 0) content-box;
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-webkit-mask: var(--_m);
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mask: var(--_m);
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-webkit-mask-composite: source-out;
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mask-composite: subtract;
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animation: l3 1s infinite linear;
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}
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@keyframes l3 {to{transform: rotate(1turn)}}
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.centered {
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display: block;
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margin-top: 1em;
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margin-bottom: 1em;
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margin-left: auto;
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margin-right: auto;
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width: 50%;
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}
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.pure-img-responsive { max-width: 100%; height: auto;
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}
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.cascadia {
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font-family: Cascadia;
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}
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.home-menu {
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padding: 0.5em;
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text-align: center;
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box-shadow: 0 1px 1px rgba(0,0,0, 0.10);
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}
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.home-menu {
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background: #333;
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}
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.home-menu a {
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color: #ccc;
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}
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.home-menu li a:hover,
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.home-menu li a:focus {
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animation: menu-hover-anim 0.25s ease-out;
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animation-fill-mode: forwards;
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background: none;
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border: none;
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}
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.link {
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color: #00f;
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text-decoration: none;
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}
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.video {
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text-align: center;
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}
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.video iframe {
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width: 32em;
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height: 18em;
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frameborder: "0";
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}
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/*
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Add transition to containers so they can push in and out.
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*/
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#layout,
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#menu,
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.menu-link {
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-webkit-transition: all 0.2s ease-out;
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-moz-transition: all 0.2s ease-out;
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-ms-transition: all 0.2s ease-out;
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-o-transition: all 0.2s ease-out;
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transition: all 0.2s ease-out;
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}
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/*
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This is the parent `<div>` that contains the menu and the content area.
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*/
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#layout {
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position: relative;
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left: 0;
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padding-left: 0;
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}
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#layout.active #menu {
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left: 150px;
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width: 150px;
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}
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#layout.active .menu-link {
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left: 150px;
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}
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/*
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The content `<div>` is where all your content goes.
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*/
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.content {
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margin: 0 auto;
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padding: 0 2em;
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max-width: 800px;
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margin-bottom: 50px;
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line-height: 1.6em;
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text-align: justify;
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}
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.header {
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margin: 0;
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color: #444;
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text-align: center;
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padding: 2.5em 4em 0;
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border-bottom: 1px solid #eee;
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}
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.header h1 {
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margin: 0.2em 0;
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font-size: 2.5em;
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font-weight: bold;
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}
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.header h2 {
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font-weight: 300;
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color: #888;
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padding: 0;
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margin-top: 0;
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}
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.distinct {
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background-color: #444;
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color: #fff;
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padding: 0.5em 0.5em 0.5em 0.5em;
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}
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@keyframes card-hover-anim {
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to {background-color: #ccc;}
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}
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@keyframes card-text-hover-anim {
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to {color: #222;}
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}
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.spaced {
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padding: 0.5em 0.5em 0.5em 0.5em;
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margin: 1em 0em 1em 0em;
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}
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h1 a svg {
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vertical-align: middle;
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}
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.card {
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background-color: #eee;
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text-decoration: none;
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text-align: left;
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}
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.card h1 {
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color: #444;
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}
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.card h2 {
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color: #777;
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}
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.card h3 {
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color: #999;
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}
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.card a {
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text-decoration: none;
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display: block;
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}
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.card:hover {
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animation: card-hover-anim 0.25s ease-out;
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animation-fill-mode: forwards;
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}
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/* .card:hover h3 { */
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/* animation: card-text-hover-anim 0.25s ease-out; */
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/* animation-fill-mode: forwards; */
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/* } */
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.card img {
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vertical-align: bottom;
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float: right;
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padding: 0em 0.1em 0em 0.1em;
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width: 60px;
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}
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.card svg {
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vertical-align: bottom;
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float: right;
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padding: 0em 0.1em 0em 0.1em;
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width: 3em;
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height: auto;
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}
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.content-subhead {
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margin: 50px 0 20px 0;
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font-weight: 300;
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color: #444;
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}
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code {
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font-family: Cascadia;
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}
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.code-block {
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/* background-color: #fdf6e3; */
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background-color: #eee;
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color: #002b36;
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font-family: Cascadia;
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font-size: 0.85em;
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padding: 0em 0.5em 0em 0.5em;
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margin: 1em 0em 1em 0em;
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page-break-inside: avoid;
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display: block;
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overflow: auto;
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word-wrap: break-word;
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max-width: 100%;
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}
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.code-block pre {
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display: block;
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margin: 0 0 0 0;
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padding: 0 0 0 0;
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}
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.code-block pre code {
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font-family: Cascadia;
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display: block;
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white-space: pre-wrap;
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}
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.code-block span {
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/* background-color: #fdf6e3; */
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background-color: #eee;
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}
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.keyword {
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color: #6c71c4;
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}
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.constant {
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color: #cb4b16;
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}
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.function {
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color: #268bd2;
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}
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.operator {
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color: #6c71c4;
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}
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.punctuation {
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color: #586e75;
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}
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.string {
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color: #859900;
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}
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.type {
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color: #2aa198;
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}
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.property {
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color: #586e75;
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}
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</style></head><body><div class="pure-g"><div class="pure-u-1-1"><div class="header"><h1> Fractal Explorer <a href="https://github.com/Xterminate1818/rowdyhacks-2023" target="_blank"><svg style="width: 1em; padding-bottom: 0.2em;" viewBox="0 0 128 128"><title> View source on GitHub </title><g fill="#181616"><path fill-rule="evenodd" clip-rule="evenodd" d="M64 5.103c-33.347 0-60.388 27.035-60.388 60.388 0 26.682 17.303 49.317 41.297 57.303 3.017.56 4.125-1.31 4.125-2.905 0-1.44-.056-6.197-.082-11.243-16.8 3.653-20.345-7.125-20.345-7.125-2.747-6.98-6.705-8.836-6.705-8.836-5.48-3.748.413-3.67.413-3.67 6.063.425 9.257 6.223 9.257 6.223 5.386 9.23 14.127 6.562 17.573 5.02.542-3.903 2.107-6.568 3.834-8.076-13.413-1.525-27.514-6.704-27.514-29.843 0-6.593 2.36-11.98 6.223-16.21-.628-1.52-2.695-7.662.584-15.98 0 0 5.07-1.623 16.61 6.19C53.7 35 58.867 34.327 64 34.304c5.13.023 10.3.694 15.127 2.033 11.526-7.813 16.59-6.19 16.59-6.19 3.287 8.317 1.22 14.46.593 15.98 3.872 4.23 6.215 9.617 6.215 16.21 0 23.194-14.127 28.3-27.574 29.796 2.167 1.874 4.097 5.55 4.097 11.183 0 8.08-.07 14.583-.07 16.572 0 1.607 1.088 3.49 4.148 2.897 23.98-7.994 41.263-30.622 41.263-57.294C124.388 32.14 97.35 5.104 64 5.104z"></path><path d="M26.484 91.806c-.133.3-.605.39-1.035.185-.44-.196-.685-.605-.543-.906.13-.31.603-.395 1.04-.188.44.197.69.61.537.91zm2.446 2.729c-.287.267-.85.143-1.232-.28-.396-.42-.47-.983-.177-1.254.298-.266.844-.14 1.24.28.394.426.472.984.17 1.255zM31.312 98.012c-.37.258-.976.017-1.35-.52-.37-.538-.37-1.183.01-1.44.373-.258.97-.025 1.35.507.368.545.368 1.19-.01 1.452zm3.261 3.361c-.33.365-1.036.267-1.552-.23-.527-.487-.674-1.18-.343-1.544.336-.366 1.045-.264 1.564.23.527.486.686 1.18.333 1.543zm4.5 1.951c-.147.473-.825.688-1.51.486-.683-.207-1.13-.76-.99-1.238.14-.477.823-.7 1.512-.485.683.206 1.13.756.988 1.237zm4.943.361c.017.498-.563.91-1.28.92-.723.017-1.308-.387-1.315-.877 0-.503.568-.91 1.29-.924.717-.013 1.306.387 1.306.88zm4.598-.782c.086.485-.413.984-1.126 1.117-.7.13-1.35-.172-1.44-.653-.086-.498.422-.997 1.122-1.126.714-.123 1.354.17 1.444.663zm0 0"></path></g></svg></a></h1><h2> Parallel Algorithms - Optimization - Hackathon Finalist </h2></div><div class="content"><h2 class="distinct"> Motivation </h2>
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In March of 2023 I attended RowdyHacks 8, my first hackathon. I was
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in the middle of my Calculus II class, which I was enjoying a lot. With
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Math at the front of my mind, I decided to write a program to visualize
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the Mandelbrot Fractal. I had written a similar program in Snap, a visual
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block-based programming language, so I was confident I could achieve results
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within the 24 hour time limit. I wanted to specifically optimize the program
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for speed without using GPU acceleration, because the laptop I was using did
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not have one.
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<h2 class="distinct"> Approach </h2>
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The premise behind the Mandelbrot Fractal is to take a complex number, and
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insert it into a recursive function. If the function does not diverge to
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infinity, then that number is part of the Mandelbrot Set, and it gets plotted
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on the screen.
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Calculating the Mandelbrot Set is deceptively simple. The hard part is doing so
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with a very large degree of precision, and repeating the calculations millions of
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times (prefferably in parallel.)
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<div class="code-block">
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<pre><code>
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<span class="comment">// Snippet from my initial naive algorithm</span>
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<span class="comment">//</span>
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<span class="comment">// This will produce the value of one pixel on the screen</span>
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<span class="keyword">pub</span> <span class="keyword function">fn</span> <span class="function">fractal</span><span class="punctuation bracket">(</span><span class="variable parameter">val</span>: <span class="type">Complex</span><span class="punctuation delimiter">,</span> <span class="variable parameter">max_iterations</span>: <span class="type builtin">usize</span><span class="punctuation bracket">)</span> <span class="operator">-></span> <span class="type builtin">usize</span> <span class="punctuation bracket">{</span>
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<span class="keyword storage">let</span> <span class="keyword storage modifier">mut</span> <span class="variable">iterations</span> <span class="operator">=</span> <span class="constant numeric integer">0</span><span class="punctuation delimiter">;</span>
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<span class="keyword storage">let</span> <span class="keyword storage modifier">mut</span> <span class="variable">last</span> <span class="operator">=</span> <span class="type">Complex</span><span class="punctuation delimiter">::</span><span class="function">new</span><span class="punctuation bracket">(</span><span class="constant numeric float">0.0</span><span class="punctuation delimiter">,</span> <span class="constant numeric float">0.0</span><span class="punctuation bracket">)</span><span class="punctuation delimiter">;</span>
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<span class="keyword storage">let</span> <span class="keyword storage modifier">mut</span> <span class="variable">squared</span> <span class="operator">=</span> <span class="type">Complex</span><span class="punctuation delimiter">::</span><span class="function">new</span><span class="punctuation bracket">(</span><span class="constant numeric float">0.0</span><span class="punctuation delimiter">,</span> <span class="constant numeric float">0.0</span><span class="punctuation bracket">)</span><span class="punctuation delimiter">;</span>
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<span class="comment">// Values exceeding 4 usually diverge. If the number</span>
|
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<span class="comment">// does not diverge after `max_iterations`, it probably</span>
|
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<span class="comment">// never will.</span>
|
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<span class="keyword control repeat">while</span> <span class="variable">squared</span><span class="punctuation delimiter">.</span><span class="variable other member">re</span> <span class="operator">+</span> <span class="variable">squared</span><span class="punctuation delimiter">.</span><span class="variable other member">im</span> <span class="operator"><=</span> <span class="constant numeric float">4.0</span> <span class="operator">&&</span>
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<span class="variable">iterations</span> <span class="operator"><</span> <span class="variable parameter">max_iterations</span> <span class="punctuation bracket">{</span>
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<span class="keyword storage">let</span> <span class="variable">im</span> <span class="operator">=</span> <span class="constant numeric float">2.0</span> <span class="operator">*</span> <span class="variable">last</span><span class="punctuation delimiter">.</span><span class="variable other member">re</span> <span class="operator">*</span> <span class="variable">last</span><span class="punctuation delimiter">.</span><span class="variable other member">im</span> <span class="operator">+</span> <span class="variable parameter">val</span><span class="punctuation delimiter">.</span><span class="variable other member">im</span><span class="punctuation delimiter">;</span>
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<span class="keyword storage">let</span> <span class="variable">re</span> <span class="operator">=</span> <span class="variable">squared</span><span class="punctuation delimiter">.</span><span class="variable other member">re</span> <span class="operator">-</span> <span class="variable">squared</span><span class="punctuation delimiter">.</span><span class="variable other member">im</span> <span class="operator">+</span> <span class="variable parameter">val</span><span class="punctuation delimiter">.</span><span class="variable other member">re</span><span class="punctuation delimiter">;</span>
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<span class="variable">last</span> <span class="operator">=</span> <span class="constructor">Complex</span> <span class="punctuation bracket">{</span> <span class="variable other member">re</span><span class="punctuation delimiter">,</span> <span class="variable other member">im</span> <span class="punctuation bracket">}</span><span class="punctuation delimiter">;</span>
|
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<span class="variable">squared</span><span class="punctuation delimiter">.</span><span class="variable other member">re</span> <span class="operator">=</span> <span class="variable">re</span><span class="punctuation delimiter">.</span><span class="function">powi</span><span class="punctuation bracket">(</span><span class="constant numeric integer">2</span><span class="punctuation bracket">)</span><span class="punctuation delimiter">;</span>
|
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<span class="variable">squared</span><span class="punctuation delimiter">.</span><span class="variable other member">im</span> <span class="operator">=</span> <span class="variable">im</span><span class="punctuation delimiter">.</span><span class="function">powi</span><span class="punctuation bracket">(</span><span class="constant numeric integer">2</span><span class="punctuation bracket">)</span><span class="punctuation delimiter">;</span>
|
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|
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<span class="variable">iterations</span> <span class="operator">+=</span> <span class="constant numeric integer">1</span><span class="punctuation delimiter">;</span>
|
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|
|
<span class="punctuation bracket">}</span>
|
||
|
|
<span class="variable">iterations</span>
|
||
|
|
<span class="punctuation bracket">}</span>
|
||
|
|
</code></pre>
|
||
|
|
</div>
|
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While researching the Mandelbrot set, I found a faster solution called the derivative
|
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bail algorithm. This algorithm considers the first derivative of the function rather
|
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than the functions value itself, which is more accurate and performant most of the time.
|
||
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|
<div class="code-block">
|
||
|
|
<pre><code>
|
||
|
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<span class="comment">// Snippet from my derivative bail implementation</span>
|
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<span class="keyword">pub</span> <span class="keyword function">fn</span> <span class="function">get_dbail</span><span class="punctuation bracket">(</span><span class="variable parameter">val</span>: <span class="type">Complex</span><span class="punctuation delimiter">,</span> <span class="variable parameter">max_dvt</span>: <span class="type builtin">f64</span><span class="punctuation delimiter">,</span> <span class="variable parameter">max_iter</span>: <span class="type builtin">usize</span><span class="punctuation bracket">)</span> <span class="operator">-></span> <span class="type builtin">usize</span> <span class="punctuation bracket">{</span>
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<span class="keyword storage">let</span> <span class="variable parameter">max_dvt</span> <span class="operator">=</span> <span class="variable parameter">max_dvt</span><span class="punctuation delimiter">.</span><span class="function">powi</span><span class="punctuation bracket">(</span><span class="constant numeric integer">2</span><span class="punctuation bracket">)</span><span class="punctuation delimiter">;</span>
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<span class="keyword storage">let</span> <span class="keyword storage modifier">mut</span> <span class="variable">it</span> <span class="operator">=</span> <span class="constant numeric integer">0</span><span class="punctuation delimiter">;</span>
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||
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|
<span class="keyword storage">let</span> <span class="keyword storage modifier">mut</span> <span class="variable">last</span> <span class="operator">=</span> <span class="type">Complex</span><span class="punctuation delimiter">::</span><span class="function">new</span><span class="punctuation bracket">(</span><span class="constant numeric float">0.0</span><span class="punctuation delimiter">,</span> <span class="constant numeric float">0.0</span><span class="punctuation bracket">)</span><span class="punctuation delimiter">;</span>
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||
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|
<span class="keyword storage">let</span> <span class="keyword storage modifier">mut</span> <span class="variable">squared</span> <span class="operator">=</span> <span class="type">Complex</span><span class="punctuation delimiter">::</span><span class="function">new</span><span class="punctuation bracket">(</span><span class="constant numeric float">0.0</span><span class="punctuation delimiter">,</span> <span class="constant numeric float">0.0</span><span class="punctuation bracket">)</span><span class="punctuation delimiter">;</span>
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||
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|
<span class="keyword storage">let</span> <span class="keyword storage modifier">mut</span> <span class="variable">deriv</span> <span class="operator">=</span> <span class="type">Complex</span><span class="punctuation delimiter">::</span><span class="function">new</span><span class="punctuation bracket">(</span><span class="constant numeric float">0.0</span><span class="punctuation delimiter">,</span> <span class="constant numeric float">0.0</span><span class="punctuation bracket">)</span><span class="punctuation delimiter">;</span>
|
||
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<span class="keyword control repeat">while</span> <span class="variable">squared</span><span class="punctuation delimiter">.</span><span class="variable other member">re</span> <span class="operator">+</span> <span class="variable">squared</span><span class="punctuation delimiter">.</span><span class="variable other member">im</span> <span class="operator"><=</span> <span class="constant numeric float">4.0</span>
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||
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<span class="operator">&&</span> <span class="variable">deriv</span><span class="punctuation delimiter">.</span><span class="variable other member">re</span><span class="punctuation delimiter">.</span><span class="function">powi</span><span class="punctuation bracket">(</span><span class="constant numeric integer">2</span><span class="punctuation bracket">)</span> <span class="operator">+</span> <span class="variable">deriv</span><span class="punctuation delimiter">.</span><span class="variable other member">im</span><span class="punctuation delimiter">.</span><span class="function">powi</span><span class="punctuation bracket">(</span><span class="constant numeric integer">2</span><span class="punctuation bracket">)</span> <span class="operator"><=</span> <span class="variable parameter">max_dvt</span>
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||
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<span class="operator">&&</span> <span class="variable">it</span> <span class="operator"><=</span> <span class="variable parameter">max_iter</span>
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||
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<span class="punctuation bracket">{</span>
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||
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<span class="variable">deriv</span><span class="punctuation delimiter">.</span><span class="variable other member">re</span> <span class="operator">=</span> <span class="constant numeric float">2.0</span> <span class="operator">*</span> <span class="punctuation bracket">(</span><span class="variable">deriv</span><span class="punctuation delimiter">.</span><span class="variable other member">re</span> <span class="operator">*</span> <span class="variable parameter">val</span><span class="punctuation delimiter">.</span><span class="variable other member">re</span> <span class="operator">-</span> <span class="variable">deriv</span><span class="punctuation delimiter">.</span><span class="variable other member">im</span> <span class="operator">*</span> <span class="variable parameter">val</span><span class="punctuation delimiter">.</span><span class="variable other member">im</span><span class="punctuation bracket">)</span><span class="punctuation delimiter">;</span>
|
||
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|
<span class="variable">deriv</span><span class="punctuation delimiter">.</span><span class="variable other member">im</span> <span class="operator">=</span> <span class="constant numeric float">2.0</span> <span class="operator">*</span> <span class="punctuation bracket">(</span><span class="variable">deriv</span><span class="punctuation delimiter">.</span><span class="variable other member">re</span> <span class="operator">*</span> <span class="variable parameter">val</span><span class="punctuation delimiter">.</span><span class="variable other member">im</span> <span class="operator">+</span> <span class="variable">deriv</span><span class="punctuation delimiter">.</span><span class="variable other member">im</span> <span class="operator">*</span> <span class="variable parameter">val</span><span class="punctuation delimiter">.</span><span class="variable other member">re</span><span class="punctuation bracket">)</span><span class="punctuation delimiter">;</span>
|
||
|
|
<span class="keyword storage">let</span> <span class="variable">im</span> <span class="operator">=</span> <span class="constant numeric float">2.0</span> <span class="operator">*</span> <span class="variable">last</span><span class="punctuation delimiter">.</span><span class="variable other member">re</span> <span class="operator">*</span> <span class="variable">last</span><span class="punctuation delimiter">.</span><span class="variable other member">im</span> <span class="operator">+</span> <span class="variable parameter">val</span><span class="punctuation delimiter">.</span><span class="variable other member">im</span><span class="punctuation delimiter">;</span>
|
||
|
|
<span class="keyword storage">let</span> <span class="variable">re</span> <span class="operator">=</span> <span class="variable">squared</span><span class="punctuation delimiter">.</span><span class="variable other member">re</span> <span class="operator">-</span> <span class="variable">squared</span><span class="punctuation delimiter">.</span><span class="variable other member">im</span> <span class="operator">+</span> <span class="variable parameter">val</span><span class="punctuation delimiter">.</span><span class="variable other member">re</span><span class="punctuation delimiter">;</span>
|
||
|
|
<span class="variable">last</span> <span class="operator">=</span> <span class="constructor">Complex</span> <span class="punctuation bracket">{</span> <span class="variable other member">re</span><span class="punctuation delimiter">,</span> <span class="variable other member">im</span> <span class="punctuation bracket">}</span><span class="punctuation delimiter">;</span>
|
||
|
|
<span class="variable">squared</span><span class="punctuation delimiter">.</span><span class="variable other member">re</span> <span class="operator">=</span> <span class="variable">re</span><span class="punctuation delimiter">.</span><span class="function">powi</span><span class="punctuation bracket">(</span><span class="constant numeric integer">2</span><span class="punctuation bracket">)</span><span class="punctuation delimiter">;</span>
|
||
|
|
<span class="variable">squared</span><span class="punctuation delimiter">.</span><span class="variable other member">im</span> <span class="operator">=</span> <span class="variable">im</span><span class="punctuation delimiter">.</span><span class="function">powi</span><span class="punctuation bracket">(</span><span class="constant numeric integer">2</span><span class="punctuation bracket">)</span><span class="punctuation delimiter">;</span>
|
||
|
|
<span class="variable">it</span> <span class="operator">+=</span> <span class="constant numeric integer">1</span><span class="punctuation delimiter">;</span>
|
||
|
|
<span class="punctuation bracket">}</span>
|
||
|
|
<span class="variable">it</span>
|
||
|
|
<span class="punctuation bracket">}</span>
|
||
|
|
</code></pre>
|
||
|
|
</div>
|
||
|
|
Deciding whether a number does or does not fall into the set is more of a heuristic
|
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|
|
than an analytic process. A good algorithm balances accuracy with efficiency. The
|
||
|
|
most glaring limitation of any calculation is the precision of the floating point
|
||
|
|
numbers used, especially when zooming in to the fractal. I deliberated the pros and
|
||
|
|
cons of using a floating point library with very high, or even arbitrary precision.
|
||
|
|
After extensive benchmarking, I was unable to get any of these to be efficient enough.
|
||
|
|
<h2 class="distinct"> Multi-Threading </h2>
|
||
|
|
In order to display the fractal, I needed an array of pixel data I could push to the
|
||
|
|
screen. The X and Y coordinates represent real and imaginary components respectively.
|
||
|
|
This means I need to run the function for every pixel on the screen. The Mandelbrot
|
||
|
|
Fractal is an obvious candidate for multi-threading, because the result of every
|
||
|
|
calculation is independent, and there are many calculations that need to be done.
|
||
|
|
I achieve this by breaking the pixel buffer into N partitions, and spawning a thread
|
||
|
|
to handle each (where N can be changed at runtime). This allowed me to rapidly iterate,
|
||
|
|
and find the optimal number of threads for the best performance.
|
||
|
|
<h2 class="distinct"> Results </h2>
|
||
|
|
I finished the project right before the deadline. This was my first time competing at
|
||
|
|
a hackathon, and the most code I had ever written in a 24 hour period. I was absolutely
|
||
|
|
exhausted, with no real idea of what to expect. I presented my project to the judges and
|
||
|
|
the other hackers at the event. The reception was incredibly positive, and I met many
|
||
|
|
brilliant people I stay in contact with to this day. My project won 2nd prize overall
|
||
|
|
at the event, for which I am extremely honored and greatful.
|
||
|
|
</div></div></div></body></html>
|