e39465ad2f
new file: Files/flashplayer_32_sa.exe
new file: favicon.ico
new file: globe.gif
new file: imgs/download.png
new file: imgs/zuck.jpg
new file: index.html
new file: other.ico
new file: script.js
new file: site.webmanifest
new file: sitemap.html
new file: styles/backround.css
new file: styles/border.css
new file: styles/fonts/Titillium_Web/OFL.txt
new file: styles/fonts/Titillium_Web/TitilliumWeb-Black.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Bold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-BoldItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLight.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Italic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Light.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-LightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Regular.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBoldItalic.ttf
new file: styles/fonts/webfontkit-20221027-163353/generator_config.txt
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-demo.html
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff2
new file: styles/fonts/webfontkit-20221027-165950/generator_config.txt
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-demo.html
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff2
new file: styles/style.css
new file: tools/2048/.gitignore
new file: tools/2048/.jshintrc
new file: tools/2048/CONTRIBUTING.md
new file: tools/2048/LICENSE.txt
new file: tools/2048/README.md
new file: tools/2048/Rakefile
new file: tools/2048/favicon.ico
new file: tools/2048/index.html
new file: tools/2048/js/animframe_polyfill.js
new file: tools/2048/js/application.js
new file: tools/2048/js/bind_polyfill.js
new file: tools/2048/js/classlist_polyfill.js
new file: tools/2048/js/game_manager.js
new file: tools/2048/js/grid.js
new file: tools/2048/js/html_actuator.js
new file: tools/2048/js/keyboard_input_manager.js
new file: tools/2048/js/local_storage_manager.js
new file: tools/2048/js/tile.js
new file: tools/2048/meta/apple-touch-icon.png
new file: tools/webretro/cores/neocd_libretro.js
new file: tools/webretro/cores/neocd_libretro.wasm
new file: tools/webretro/cores/nestopia_libretro.js
new file: tools/webretro/cores/nestopia_libretro.wasm
new file: tools/webretro/cores/o2em_libretro.js
new file: tools/webretro/cores/o2em_libretro.wasm
new file: tools/webretro/cores/opera_libretro.js
new file: tools/webretro/cores/opera_libretro.wasm
506 lines
23 KiB
HTML
506 lines
23 KiB
HTML
<!DOCTYPE html>
|
||
<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
|
||
|
||
<!-- Mirrored from www.mathsisfun.com/numbers/golden-ratio.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:38:12 GMT -->
|
||
<head>
|
||
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
|
||
|
||
|
||
<!-- #BeginEditable "doctitle" -->
|
||
<title>Golden Ratio</title>
|
||
<meta name="description" content="The golden ratio is a special number approximately equal to 1.618 that appears many times in mathematics, geometry, art, architecture and other areas.">
|
||
|
||
<style>
|
||
.golden {color:goldenrod; font-size:140%}
|
||
</style>
|
||
|
||
|
||
<link rel="stylesheet" type="text/css" href="../stylejs.css">
|
||
<!-- #EndEditable -->
|
||
<meta name="keywords" content="math, maths, mathematics, school, homework, education">
|
||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
|
||
<meta name="HandheldFriendly" content="true">
|
||
<meta name="referrer" content="always">
|
||
<link rel="preload" href="../images/style/font-champ-bold.ttf" as="font" type="font/ttf" crossorigin="">
|
||
<link rel="preload" href="../style4.css" as="style">
|
||
<link rel="preload" href="../main4.js" as="script">
|
||
<link rel="stylesheet" href="../style4.css">
|
||
<script src="../main4.js" defer="defer"></script>
|
||
<!-- Global site tag (gtag.js) - Google Analytics -->
|
||
<script async="" src="https://www.googletagmanager.com/gtag/js?id=UA-29771508-1"></script>
|
||
<script>
|
||
window.dataLayer = window.dataLayer || [];
|
||
function gtag(){dataLayer.push(arguments);}
|
||
gtag('js', new Date());
|
||
gtag('config', 'UA-29771508-1');
|
||
</script>
|
||
</head>
|
||
|
||
<body id="bodybg">
|
||
|
||
<div id="stt"></div>
|
||
<div id="adTop"></div>
|
||
<header>
|
||
<div id="hdr"></div>
|
||
<div id="tran"></div>
|
||
<div id="adHide"></div>
|
||
<div id="cookOK"></div>
|
||
</header>
|
||
|
||
<div class="mid">
|
||
|
||
<nav>
|
||
<div id="menuWide" class="menu"></div>
|
||
<div id="logo"><a href="../index.html"><img src="../images/style/logo.svg" alt="Math is Fun"></a></div>
|
||
|
||
<div id="search" role="search"></div>
|
||
<div id="linkto"></div>
|
||
|
||
<div id="menuSlim" class="menu"></div>
|
||
<div id="menuTiny" class="menu"></div>
|
||
</nav>
|
||
|
||
<div id="extra"></div>
|
||
|
||
<article id="content" role="main">
|
||
|
||
<!-- #BeginEditable "Body" -->
|
||
|
||
|
||
<h1 class="center">Golden Ratio</h1>
|
||
|
||
<div style="float:left; margin: -40px 10px 5px 0;"><span class="golden" style="font:120px Arial;">φ</span>
|
||
</div>
|
||
<p>The golden ratio <i>(symbol is the Greek letter "phi" shown at left)</i><br>
|
||
is a special number approximately equal to 1.618</p>
|
||
<p>It appears many times in geometry, art, architecture and other areas.</p>
|
||
|
||
|
||
<h2>The Idea Behind It</h2>
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td>
|
||
<p>We find the golden ratio when we divide a line into two parts so that:</p>
|
||
|
||
<div class="bgb1" style="text-align:center; padding: 3px;">the whole length divided by the long part</div>
|
||
<div style="text-align:center;"><b><i>is also equal to</i></b> </div>
|
||
<div class="bgb1" style="text-align:center; padding: 3px;">the long part divided by the short part</div> </td>
|
||
|
||
</tr>
|
||
</tbody></table><br>
|
||
<p class="center"><img src="images/golden-ratio.svg" alt="golden ratio (a+b)/a = a/b = 1.618..." height="261" width="375"></p>
|
||
<p>Have a try yourself (use the slider):</p>
|
||
|
||
<div class="script">images/golden-ratio.js</div>
|
||
|
||
|
||
<h2>Beauty</h2>
|
||
|
||
<p style="float:left; margin: 0 20px 5px 0;"><img src="images/golden-rectangle.svg" alt="golden rectangle" height="155" width="188"></p>
|
||
<p>This rectangle has been made using the Golden Ratio, Looks like a typical frame for a painting, doesn't it?</p>
|
||
<p>Some artists and architects believe the Golden Ratio makes the most pleasing and beautiful shape.</p>
|
||
<div style="clear:both"></div>
|
||
<div class="center80">
|
||
<p class="center">Do <b>you</b> think it is the "most pleasing rectangle"?</p>
|
||
<p class="center">Maybe you do or don't, that is up to you!</p>
|
||
</div><p style="float:right; margin: 0 0 5px 10px;"><img src="images/parthenon-golden-ratio.jpg" alt="parthenon golden ratio" height="201" width="325"></p>
|
||
<p> </p>
|
||
<p>Many buildings and artworks have
|
||
the Golden Ratio in them, such as the Parthenon in Greece, but it is not really known if it was designed that way.</p>
|
||
|
||
|
||
<h2>The Actual Value</h2>
|
||
|
||
<p>The Golden Ratio is equal to:</p>
|
||
<p class="golden" align="center"><b>1.61803398874989484820...</b> (etc.)</p>
|
||
<p>The digits just keep on going, with no pattern. In fact the Golden Ratio is known to be an <a href="../irrational-numbers.html">Irrational Number</a>, and I will tell you more about it later.</p>
|
||
|
||
|
||
<h2>Formula</h2>
|
||
|
||
<p>We saw above that the Golden Ratio has this property:</p>
|
||
<p class="center larger"><span class="intbl"><em>a</em><strong>b</strong></span> = <span class="intbl"><em>a + b</em><strong>a</strong></span></p>
|
||
<p>We can split the right-hand fraction like this:</p>
|
||
<p class="center larger"><span class="intbl"><em>a</em><strong>b</strong></span> = <span class="intbl"><em>a</em><strong>a</strong></span> + <span class="intbl"><em>b</em><strong>a</strong></span></p>
|
||
<p><span class="intbl"><em>a</em><strong>b</strong></span> is the Golden Ratio <span class="golden">φ</span>, <span class="intbl"><em>a</em><strong>a</strong></span>=1 and <span class="intbl"><em>b</em><strong>a</strong></span>=<span class="intbl"><em>1</em><strong><span class="golden">φ</span></strong></span>, which gets us:</p>
|
||
<div class="def">
|
||
<p class="center large">φ = 1 + <span class="intbl"><em>1</em><strong>φ</strong></span></p>
|
||
</div>
|
||
<p>So the Golden Ratio can be defined in terms of itself!</p>
|
||
<p>Let us test it using just a few digits of accuracy:</p>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left"><span class="golden">φ</span> =</span><span class="right">1 + <span class="intbl"><em>1</em><strong>1.618</strong></span></span></div>
|
||
<div class="row"><span class="left">=</span><span class="right">1 + 0.61805...</span></div>
|
||
<div class="row"><span class="left">=</span><span class="right">1.61805...</span></div>
|
||
</div>
|
||
<p>With more digits we would be more accurate.</p>
|
||
|
||
|
||
<h2>Calculating It</h2>
|
||
|
||
<p>You can use that formula to try and calculate <span class="golden">φ</span> yourself.</p>
|
||
<p>First <b>guess</b> its value, then do this calculation again and again:</p>
|
||
<ul>
|
||
<li>A) divide 1 by your value (=1/value)</li>
|
||
<li>B) add 1</li>
|
||
<li>C) now use <i>that</i> value and start again at A</li>
|
||
</ul>
|
||
<p>With a calculator, just keep pressing "1/x", "+", "1", "=", around and around.</p>
|
||
<p>I started with 2 and got this:</p>
|
||
<div class="simple">
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr style="text-align:center;">
|
||
<th>value</th>
|
||
<th>1/value</th>
|
||
<th>1/value + 1</th>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td><b>2</b></td>
|
||
<td>1/2 = 0.5</td>
|
||
<td> 0.5 + 1 = <b>1.5</b></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td><b>1.5</b></td>
|
||
<td>1/1.5 = 0.666...</td>
|
||
<td>0.666... + 1 = <b>1.666...</b></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td><b>1.666...</b></td>
|
||
<td>1/1.666... = 0.6</td>
|
||
<td>0.6 + 1 = <b>1.6</b></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td><b>1.6</b></td>
|
||
<td>1/1.6 = 0.625</td>
|
||
<td>0.625 + 1 = <b>1.625</b></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td><b>1.625</b></td>
|
||
<td>1/1.625 = 0.6153...</td>
|
||
<td>0.6154... + 1 = <b>1.6153...</b></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td><b>1.6153...</b></td>
|
||
<td> </td>
|
||
<td> </td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p>It gets closer and closer to <span class="golden">φ</span> the more we go.</p>
|
||
<p>But there are better ways to calculate it to thousands of decimal places quite quickly.</p>
|
||
<p style="float:right; margin: 0 0 25px 10px;"><img src="images/golden-ratio-construct-1.svg" alt="golden ratio construction step 1" height="221" width="188"></p>
|
||
|
||
|
||
<h2>Drawing It</h2>
|
||
|
||
<p>Here is one way to draw a rectangle with the Golden Ratio:</p>
|
||
<ul>
|
||
<li>Draw a square of size "1"</li>
|
||
<li>Place a dot half way along one side</li>
|
||
<li>Draw a line from that point to an opposite corner</li>
|
||
</ul>
|
||
<div style="clear:both"></div>
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/golden-ratio-construct.svg" alt="golden ratio construction" height="215" width="261"></p>
|
||
<ul>
|
||
<li>Now turn that line so that it runs along the square's side</li>
|
||
<li>Then you can extend the square to be a rectangle with the Golden Ratio!</li>
|
||
</ul>
|
||
<div style="clear:both"></div>
|
||
<p>(Where did <b><span class="intbl"><em>√5</em><strong>2</strong></span></b> come from? See footnote*)</p>
|
||
|
||
|
||
<h2>A Quick Way to Calculate</h2>
|
||
|
||
<p>That rectangle above shows us a simple formula for the Golden Ratio.</p>
|
||
<p>When the short side is <b>1</b>, the long side is <b><span class="intbl"><em>1</em><strong>2</strong></span>+<span class="intbl"><em>√5</em><strong>2</strong></span></b>, so:</p>
|
||
<div class="def">
|
||
<p class="center large">φ = <span class="intbl"><em>1</em><strong>2</strong></span> + <span class="intbl"><em>√5</em><strong>2</strong></span></p>
|
||
</div>
|
||
<p>The square root of 5 is approximately 2.236068, so the Golden Ratio is approximately 0.5 + 2.236068/2 = 1.618034. This is an easy way to calculate it when you need it.</p>
|
||
<div class="fun">
|
||
<p><b>Interesting fact</b>: the Golden Ratio is also equal to <b>2 × sin(54°)</b>, get your calculator and check!</p>
|
||
</div>
|
||
|
||
|
||
<h2>Fibonacci Sequence</h2>
|
||
|
||
<p>There is a special relationship between the Golden Ratio and the <a href="fibonacci-sequence.html">Fibonacci Sequence</a>:</p>
|
||
<p class="center"><span class="large">0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...</span></p>
|
||
<p class="center">(The next number is found by adding up the two numbers before it.)</p>
|
||
<p>And here is a surprise: when we take any two successive <i>(one after the other)</i> Fibonacci Numbers, <b>their ratio is very close to the Golden Ratio</b>.</p>
|
||
<p>In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few:</p>
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<th width="50">
|
||
<div align="right">A</div> </th>
|
||
<th width="50">
|
||
<div align="right">B</div> </th>
|
||
<th width="20"> </th>
|
||
<th width="100">B/A</th>
|
||
</tr>
|
||
<tr>
|
||
<td style="width:50px;">
|
||
<div align="right">2</div> </td>
|
||
<td style="width:50px;">
|
||
<div align="right">3</div> </td>
|
||
<td style="width:20px;"> </td>
|
||
<td style="width:100px;">1.5</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="width:50px;">
|
||
<div align="right">3</div> </td>
|
||
<td style="width:50px;">
|
||
<div align="right">5</div> </td>
|
||
<td style="width:20px;"> </td>
|
||
<td style="width:100px;">1.666666666...</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="width:50px;">
|
||
<div align="right">5</div> </td>
|
||
<td style="width:50px;">
|
||
<div align="right">8</div> </td>
|
||
<td style="width:20px;"> </td>
|
||
<td style="width:100px;">1.6</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="width:50px;">
|
||
<div align="right">8</div> </td>
|
||
<td style="width:50px;">
|
||
<div align="right">13</div> </td>
|
||
<td style="width:20px;"> </td>
|
||
<td style="width:100px;">1.625</td>
|
||
</tr>
|
||
<tr>
|
||
<td height="14" width="50">
|
||
<div align="right">...</div> </td>
|
||
<td height="14" width="50">
|
||
<div align="right">...</div> </td>
|
||
<td height="14" width="20"> </td>
|
||
<td height="14" width="100">...</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="width:50px;">
|
||
<div align="right">144</div> </td>
|
||
<td style="width:50px;">
|
||
<div align="right">233</div> </td>
|
||
<td style="width:20px;"> </td>
|
||
<td style="width:100px;">1.618055556...</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="width:50px;">
|
||
<div align="right">233</div> </td>
|
||
<td style="width:50px;">
|
||
<div align="right">377</div> </td>
|
||
<td style="width:20px;"> </td>
|
||
<td style="width:100px;">1.618025751...</td>
|
||
</tr>
|
||
<tr>
|
||
<td height="14" width="50">
|
||
<div align="right">...</div> </td>
|
||
<td height="14" width="50">
|
||
<div align="right">...</div> </td>
|
||
<td height="14" width="20"> </td>
|
||
<td height="14" width="100">...</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>We don't have to start with <b>2 and 3</b>, here I randomly chose <b>192 and 16</b> (and got the sequence <i>192, 16,208,224,432,656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, ...</i>):</p>
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<th width="50"> <div align="right">A </div></th>
|
||
<th width="50"> <div align="right">B </div></th>
|
||
<th width="20"> </th>
|
||
<th width="100"> <div align="left">B / A</div></th>
|
||
</tr>
|
||
<tr>
|
||
<td style="width:50px;">
|
||
<div align="right"><b>192</b></div></td>
|
||
<td style="width:50px;">
|
||
<div align="right"><b>16</b></div></td>
|
||
<td style="width:20px;"> </td>
|
||
<td style="width:100px;">0.08333333...</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="width:50px;">
|
||
<div align="right">16</div></td>
|
||
<td style="width:50px;">
|
||
<div align="right">208</div></td>
|
||
<td style="width:20px;"> </td>
|
||
<td style="width:100px;">13</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="width:50px;">
|
||
<div align="right">208</div></td>
|
||
<td style="width:50px;">
|
||
<div align="right">224</div></td>
|
||
<td style="width:20px;"> </td>
|
||
<td style="width:100px;">1.07692308...</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="width:50px;">
|
||
<div align="right">224</div></td>
|
||
<td style="width:50px;">
|
||
<div align="right">432</div></td>
|
||
<td style="width:20px;"> </td>
|
||
<td style="width:100px;">1.92857143...</td>
|
||
</tr>
|
||
<tr>
|
||
<td height="14" width="50">
|
||
<div align="right">...</div></td>
|
||
<td height="14" width="50">
|
||
<div align="right">...</div></td>
|
||
<td height="14" width="20"> </td>
|
||
<td height="14" width="100">...</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="width:50px;">
|
||
<div align="right">7408</div></td>
|
||
<td style="width:50px;">
|
||
<div align="right">11984</div></td>
|
||
<td style="width:20px;"> </td>
|
||
<td style="width:100px;">1.61771058...</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="width:50px;">
|
||
<div align="right">11984</div></td>
|
||
<td style="width:50px;">
|
||
<div align="right">19392</div></td>
|
||
<td style="width:20px;"> </td>
|
||
<td style="width:100px;">1.61815754...</td>
|
||
</tr>
|
||
<tr>
|
||
<td height="14" width="50">
|
||
<div align="right">...</div></td>
|
||
<td height="14" width="50">
|
||
<div align="right">...</div></td>
|
||
<td height="14" width="20"> </td>
|
||
<td height="14" width="100">...</td>
|
||
</tr>
|
||
</tbody></table>
|
||
|
||
|
||
<h2>The Most Irrational ...</h2>
|
||
|
||
<p>I believe the Golden Ratio is the <b>most</b> <a href="../irrational-numbers.html">irrational number</a>. Here is why ...</p>
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td>We saw before that the Golden Ratio can be defined in terms of itself,<br>
|
||
like this:</td>
|
||
</tr>
|
||
<tr>
|
||
|
||
<td class="center larger">φ = 1 + <span class="intbl"><em>1</em><strong>φ</strong></span></td>
|
||
</tr>
|
||
<tr>
|
||
|
||
<td><i>(In numbers: 1.61803... = 1 + 1/1.61803...)</i></td>
|
||
</tr>
|
||
<tr>
|
||
|
||
<td> </td>
|
||
</tr>
|
||
<tr>
|
||
<td>That can be expanded into this fraction that goes on for ever<br> (called a <i>"continued fraction"</i>):</td>
|
||
</tr>
|
||
<tr>
|
||
|
||
<td class="center larger"><img src="images/phi-continued-fraction.png" alt="phi continued fraction: phi = 1+1/(1+1/(1+1/(1+1/..." height="56" width="161"></td>
|
||
</tr>
|
||
</tbody></table><br>
|
||
<p class="center large">So, it neatly slips in between simple fractions.</p>
|
||
<p>Note: many other irrational numbers are close to rational numbers (such as <a href="pi.html">Pi</a> = 3.141592654... is pretty close to 22/7 = 3.1428571...)</p>
|
||
<p> </p>
|
||
<p style="float:left; margin: 0 10px 5px 0;"><img src="../geometry/images/pentagram-lengths.svg" alt="pentagram lengths" height="" width=""></p>
|
||
|
||
|
||
<h2>Pentagram</h2>
|
||
|
||
<p>No, not witchcraft! The pentagram is more famous as a magical or holy symbol. And it has the Golden Ratio in it:</p>
|
||
<ul>
|
||
<li>a/b = 1.618...</li>
|
||
<li>b/c = 1.618...</li>
|
||
<li>c/d = 1.618...</li>
|
||
</ul>
|
||
<p>Read more at <a href="../geometry/pentagram.html">Pentagram</a>.</p>
|
||
|
||
|
||
<h2>Other Names</h2>
|
||
|
||
<p>The Golden Ratio is also sometimes called the <b>golden section</b>, <b>golden mean</b>, <b>golden number</b>, <b>divine proportion</b>, <b>divine section</b> and <b>golden proportion</b>.</p>
|
||
|
||
|
||
<h2>Footnotes for the Keen</h2>
|
||
|
||
<div class="def">
|
||
|
||
<h3>* Where did √5/2 come from?</h3>
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/golden-ratio-r5-2.svg" alt="golden ratio square root of 5 over 2" height="223" width="189"></p>
|
||
<p>With the help of <a href="../pythagoras.html">Pythagoras</a>:</p>
|
||
<p class="so">c<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup></p>
|
||
<p class="so">c<sup>2</sup> = (<span class="intbl"><em>1</em><strong>2</strong></span>)<sup>2</sup> + 1<sup>2</sup></p>
|
||
<p class="so">c<sup>2</sup> = <span class="intbl"><em>1</em><strong>4</strong></span> + 1</p>
|
||
<p class="so">c<sup>2</sup> = <span class="intbl"><em>5</em><strong>4</strong></span></p>
|
||
<p class="so">c = √(<span class="intbl"><em>5</em><strong>4</strong></span>)</p>
|
||
<p class="so">c = <span class="intbl"><em>√5</em><strong>2</strong></span></p>
|
||
</div><p> </p>
|
||
<div class="def">
|
||
|
||
<h3>Solving using the Quadratic Formula</h3>
|
||
<p>We can find the value of <span class="golden">φ</span> this way:</p>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">Start with:</span><span class="right">φ = 1 + <span class="intbl"><em>1</em><strong>φ</strong></span></span></div>
|
||
<div class="row"><span class="left">Multiply both sides by φ:</span><span class="right">φ<sup>2</sup> = φ + 1</span></div>
|
||
<div class="row"><span class="left">Rearrange to:</span><span class="right">φ<sup>2</sup> − φ − 1 = 0</span></div>
|
||
</div>
|
||
<p>Which is a <a href="../algebra/quadratic-equation.html">Quadratic Equation</a> and we can use the Quadratic Formula:</p>
|
||
<p class="center larger">φ = <span class="intbl"> <em>−b ± √(b<sup>2 </sup>− 4ac)</em> <strong>2a</strong></span></p>
|
||
<p>Using <b>a=1</b>, <b>b=−1</b> and <b>c=−1</b> we get:</p>
|
||
<p class="center larger">φ = <span class="intbl"> <em>1 ± √(1+ 4)</em> <strong>2</strong></span></p>
|
||
<p>And the positive solution simplifies to:</p>
|
||
<p class="center larger">φ = <span class="intbl"><em>1</em><strong>2</strong></span> + <span class="intbl"><em>√5</em><strong>2</strong></span></p>
|
||
<p>Ta da!</p>
|
||
</div>
|
||
<p> </p>
|
||
<div class="def">
|
||
|
||
<h3>Kepler Triangle</h3>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">We saw above that:</span><span class="right">φ<sup>2</sup> = φ + 1</span></div>
|
||
<div class="row"><span class="left">And Pythagoras says a right-angled triangle has:</span><span class="right">c<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup></span></div>
|
||
</div>
|
||
<p>That inspired a man called Johannes Kepler to create this triangle:</p>
|
||
<p class="center"><img src="images/kepler-triangle.svg" alt="Kepler triangle" height="354" width="327"></p>
|
||
<p>It is really cool because:</p>
|
||
<ul>
|
||
<li>it has Pythagoras and <span class="golden">φ</span> together</li>
|
||
<li>the ratio of the sides is <b>1 : √φ : φ</b>, making a <a href="../algebra/sequences-sums-geometric.html">Geometric Sequence</a>.</li></ul></div>
|
||
<p> </p>
|
||
<div class="questions">2014, 2015, 15017, 8339, 8340, 8341, 8342, 15014, 15015, 8342</div>
|
||
|
||
<div class="related">
|
||
<a href="nature-golden-ratio-fibonacci.html">Nature and The Golden Ratio</a>
|
||
<a href="fibonacci-sequence.html">Fibonacci Sequence</a>
|
||
<a href="../geometry/pentagram.html">Pentagram</a>
|
||
<a href="../geometry/index.html">Geometry Index</a>
|
||
</div>
|
||
<!-- #EndEditable -->
|
||
|
||
</article>
|
||
|
||
<div id="adend" class="centerfull noprint"></div>
|
||
<footer id="footer" class="centerfull noprint"></footer>
|
||
<div id="copyrt">Copyright © 2022 Rod Pierce</div>
|
||
|
||
</div>
|
||
</body><!-- #EndTemplate -->
|
||
<!-- Mirrored from www.mathsisfun.com/numbers/golden-ratio.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:38:14 GMT -->
|
||
</html> |