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444 lines
17 KiB
HTML
444 lines
17 KiB
HTML
<!DOCTYPE html>
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<!-- Mirrored from www.mathsisfun.com/numbers/nature-golden-ratio-fibonacci.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:58:59 GMT -->
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<title>Nature, The Golden Ratio and Fibonacci Numbers</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Nature, The Golden Ratio,<br>
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and Fibonacci too ...</h1>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/sunflower.jpg" alt="sunflower" height="283" width="400"></p>
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<p>Plants can grow new cells in spirals, such as the pattern of seeds in this beautiful sunflower.</p>
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<p>The spiral happens naturally because each new cell is formed after a turn.</p>
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<p class="center"><i>"New cell, then turn,<br>
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then another cell, then turn, ..."</i></p>
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<div style="clear:both"></div>
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<h2>How Far to Turn?</h2>
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<p>So, if you were a plant, how much of a turn would you have in between new cells?</p>
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<table align="center" width="60%" border="0">
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<tbody>
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<tr>
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<td style="text-align:center;">If you don't turn at all, you get a straight line.</td>
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</tr>
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<tr>
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<td style="text-align:center;"><img src="images/seeds-straight-line.jpg" alt="seeds straight line" height="22" width="241"></td>
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</tr>
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<tr>
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<td style="text-align:center;">But that is a very poor design ... you want something <b>round</b> that will hold together with <b>no gaps</b>.</td>
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</tr>
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</tbody></table>
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<p><b>Why not try to find the best value for yourself?</b></p>
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<p class="center larger">Try different values, like <b>0.75</b>, <b>0.9</b>, <b>3.1416</b>, <b>0.62</b>, etc.</p>
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<p>Remember, you are trying to make a pattern with no gaps from start to end:</p>
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<div class="script" style="height: 430px;">
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images/golden-ratio-packing.js
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</div>
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<p> </p>
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<p>(By the way, it doesn't matter about the whole number part, like <b>1.</b> or <b>5.</b> because they are full revolutions that point us back in the same direction.)</p>
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<h2>What Did You Get?</h2>
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<p>If you got something that ends like <b>0.618</b> (or 0.382, which is 1 − 0.618) then <i>"Congratulations, you are a successful member of the plant kingdom!"</i></p>
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<table align="center" width="80%" border="0">
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<tbody>
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<tr>
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<td><img src="images/phi-flower.jpg" alt="phi flower" height="163" width="161"></td>
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<td>
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<p>That is because the <a href="golden-ratio.html">Golden Ratio</a> (<b>1.61803</b>...) is the best solution, and the Sunflower has found this out in its own natural way.</p>
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<p>Try it ... it should look like this.</p>
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</td>
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</tr>
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</tbody></table>
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<h2>Why?</h2>
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<p>Any number that is a simple fraction (example: 0.75 is 3/4, and 0.95 is 19/20, etc) will, after a while, make a pattern of lines stacking up, which makes gaps.</p>
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<p style="float:left; margin: 0 20px 5px 0;"><img src="images/phi.svg" alt="phi" height="95" width="84"></p>
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<p>But the Golden Ratio (its symbol is the Greek letter Phi, shown at left) is an expert at <b>not being any fraction</b>.</p>
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<p>It is an <a href="../irrational-numbers.html">Irrational Number</a> (meaning we cannot write it as a simple fraction), but more than that ... it is as far as we can get from being near any fraction.</p>
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<div style="clear:both"></div>
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<p> </p>
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<div class="simple">
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<table align="center" width="85%" border="0">
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<tbody>
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<tr>
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<th colspan="2">Just being irrational is not enough</th>
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</tr>
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<tr>
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<td><img src="images/pi1.svg" alt="pi symbol" height="100" width="100"></td>
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<td>
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<p>Pi (<b>3.141592654</b>...), which is also irrational.</p>
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<p>Unfortunately it has a decimal very close to 1/7 (= 0.142857...), so it ends up with 7 arms.</p>
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</td>
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</tr>
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<tr>
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<td><img src="images/e1.svg" alt="e symbol" height="100" width="100"></td>
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<td><b><i>e</i></b> (<b>2.71828...</b>) also irrational, does not work either because its decimal is close to 5/7 (0.714285...), so it also ends up with 7 arms.</td>
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</tr>
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</tbody></table></div>
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<h2>So, How Does the Golden Ratio Work?</h2><br>
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<table align="center" width="85%" border="0">
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<tbody>
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<tr>
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<td colspan="2">One of the special properties of the Golden Ratio is that it can be defined in terms of itself, like this:</td>
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</tr>
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<tr>
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<td style="text-align:right;"><img src="../images/style/right-arrow.gif" alt="right arrow" height="46" width="46"></td>
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<td><img src="images/phi-1p1onphi.png" alt="phi = 1+1/phi" height="23" width="112"></td>
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</tr>
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<tr>
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<td> </td>
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<td><i>(In numbers: 1.61803... = 1 + 1/1.61803...)</i></td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td colspan="2">That can be expanded into this fraction that goes on for ever (called a <i>"continued fraction"</i>):</td>
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</tr>
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<tr>
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<td style="text-align:right;"><img src="../images/style/right-arrow.gif" alt="right arrow" height="46" width="46"></td>
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<td><img src="images/phi-continued-fraction.png" alt="phi = 1+1/(1+1/(1+1/(1+1/..." height="56" width="161"></td>
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</tr>
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</tbody></table><br>
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<p class="center large">So, it neatly slips in between simple fractions.</p>
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<h2>Fibonacci Numbers</h2>
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<p>There is a special relationship between the Golden Ratio and <a href="fibonacci-sequence.html">Fibonacci Numbers</a> <i>(0, 1, 1, 2, 3, 5, 8, 13, 21, ... etc, each number is the sum of the two numbers before it)</i>.</p>
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<p>When we take any two successive <i>(one after the other)</i> Fibonacci Numbers, their ratio is very close to the Golden Ratio:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<th width="50">
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<div align="right">A </div>
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</th>
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<th width="50">
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<div align="right">B </div>
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</th>
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<th width="20"> </th>
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<th width="100">
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<div align="left">B / A</div>
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</th>
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</tr>
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<tr>
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<td style="width:50px;">
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<div align="right">2</div>
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</td>
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<td style="width:50px;">
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<div align="right">3</div>
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</td>
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<td style="width:20px;"> </td>
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<td style="width:100px;">1.5</td>
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</tr>
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<tr>
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<td style="width:50px;">
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<div align="right">3</div>
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</td>
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<td style="width:50px;">
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<div align="right">5</div>
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</td>
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<td style="width:20px;"> </td>
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<td style="width:100px;">1.666666666...</td>
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</tr>
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<tr>
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<td style="width:50px;">
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<div align="right">5</div>
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</td>
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<td style="width:50px;">
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<div align="right">8</div>
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</td>
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<td style="width:20px;"> </td>
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<td style="width:100px;">1.6</td>
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</tr>
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<tr>
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<td style="width:50px;">
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<div align="right">8</div>
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</td>
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<td style="width:50px;">
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<div align="right">13</div>
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</td>
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<td style="width:20px;"> </td>
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<td style="width:100px;">1.625</td>
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</tr>
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<tr>
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<td style="width:50px;">
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<div align="right">13</div>
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</td>
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<td style="width:50px;">
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<div align="right">21</div>
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</td>
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<td style="width:20px;"> </td>
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<td style="width:100px;">1.615384615...</td>
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</tr>
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<tr>
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<td height="14" width="50">
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<div align="right">...</div>
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</td>
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<td height="14" width="50">
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<div align="right">...</div>
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</td>
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<td height="14" width="20"> </td>
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<td height="14" width="100">...</td>
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</tr>
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<tr>
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<td style="width:50px;">
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<div align="right">144</div>
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</td>
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<td style="width:50px;">
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<div align="right">233</div>
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</td>
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<td style="width:20px;"> </td>
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<td style="width:100px;">1.618055556...</td>
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</tr>
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<tr>
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<td style="width:50px;">
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<div align="right">233</div>
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</td>
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<td style="width:50px;">
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<div align="right">377</div>
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</td>
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<td style="width:20px;"> </td>
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<td style="width:100px;">1.618025751...</td>
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</tr>
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<tr>
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<td height="14" width="50">
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<div align="right">...</div>
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</td>
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<td height="14" width="50">
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<div align="right">...</div>
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</td>
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<td height="14" width="20"> </td>
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<td height="14" width="100">...</td>
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</tr>
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</tbody></table><br>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/phi-flower2.jpg" alt="phi flower" height="163" width="161"></p>
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<p>So, just like we naturally get seven arms when we use 0.142857 (1/7), we tend to get Fibonacci Numbers when we use the Golden Ratio.</p>
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<p>Try counting the spiral arms - the "left turning" spirals, and then the "right turning" spirals ... what numbers did you get?</p>
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<div style="clear:both"></div>
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<h2>Spiral Leaf Growth</h2>
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<p style="float:left; margin: 0 20px 5px 0;"><img src="images/succulent-top.jpg" alt="succulent top view" height="187" width="250"></p>
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<p>This interesting behavior is not just found in sunflower seeds.</p>
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<p>Leaves, branches and petals can grow in spirals, too.</p>
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<p>Why? So that new leaves don't block the sun from older leaves, or so that the maximum amount of rain or dew gets directed down to the roots.</p>
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<p> </p>
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<p>In fact, when a plant has spirals the rotation tends to be a fraction made with two successive (one after the other) Fibonacci Numbers, for example:</p>
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<ul>
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<li>A half rotation is 1/2 (1 and 2 are Fibonacci Numbers)</li>
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<li>3/5 is also common (both Fibonacci Numbers), and</li>
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<li>5/8 also (you guessed it!)</li>
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</ul>
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<p>all getting closer and closer to the Golden Ratio.</p>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td style="text-align:right;">
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<p>And that is why Fibonacci Numbers are very common in plants.<br>
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1, 2, 3, 5, 8, 13, 21, ... etc occur in an amazing number of places.</p>
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<p>Here is a daisy with 21 petals<br>
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(but expect a few more or less, because<br>
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some may have dropped off or be just growing)</p>
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</td>
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<td><img src="images/daisy-21-petals.jpg" alt="daisy 21 petals" height="120" width="120"></td>
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</tr>
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</tbody></table>
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<p><b>But we don't see this in all plants</b>, as nature has many different methods of survival.</p>
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<p><br></p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/golden-angle.svg" alt="golden angle" height="188" width="180"></p>
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<h2>Golden Angle</h2>
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<p>So far we have been talking about "turns" (full rotations).</p>
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<p>The equivalent of 0.61803... rotations is 222.4922... degrees, or about 222.5°.</p>
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<p>In the other direction it is about <b>137.5°</b>, called the "Golden Angle".</p>
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<p> </p>
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<p class="center"><i class="larger">So, next time you are walking in the garden, look for the Golden Angle, and count petals and leaves to find Fibonacci Numbers,<br>
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and discover how clever the plants are ... !</i></p>
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<h2>Exercise</h2>
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<p>Why don't you go into the garden or park right now, and start counting leaves and petals, and measuring rotations to see what you find.</p>
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<p>You can write your results on this form:</p>
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<div class="simple">
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<table align="center" border="0">
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<tbody>
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<tr>
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<td colspan="3"><b>Plant Name or Description:</b></td>
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</tr>
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<tr>
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<td> </td>
|
||
<td rowspan="1" colspan="2"> </td>
|
||
|
||
</tr>
|
||
<tr>
|
||
<td colspan="3" rowspan="1"><b>Do the Leaves Grow in Spirals?</b> Y / N </td>
|
||
|
||
</tr>
|
||
<tr>
|
||
<td colspan="2">Count a group of Leaves:</td>
|
||
<td width="70"> </td>
|
||
</tr>
|
||
<tr>
|
||
<td> </td>
|
||
<td style="text-align:right;">How many leaves (a) ?</td>
|
||
<td> </td>
|
||
</tr>
|
||
<tr>
|
||
<td> </td>
|
||
<td style="text-align:right;">How many full rotations (b) ?</td>
|
||
<td> </td>
|
||
</tr>
|
||
<tr>
|
||
<td> </td>
|
||
<td style="text-align:right;">Rotation per leaf (b/a) :</td>
|
||
<td> </td>
|
||
</tr>
|
||
<tr>
|
||
<td> </td>
|
||
<td style="text-align:right;">Rotation Angle (360 × b/a) :</td>
|
||
<td> </td>
|
||
</tr>
|
||
<tr>
|
||
<td> </td>
|
||
<td> </td>
|
||
<td> </td>
|
||
</tr>
|
||
<tr>
|
||
<td colspan="3" rowspan="1"><b>Are There Flowers?</b> Y / N </td>
|
||
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;"> </td>
|
||
<td style="text-align:right;">How many petals on Flower 1:</td>
|
||
<td> </td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;"> </td>
|
||
<td style="text-align:right;">Flower 2:</td>
|
||
<td> </td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;"> </td>
|
||
<td style="text-align:right;">Flower 3:</td>
|
||
<td> </td>
|
||
</tr>
|
||
</tbody></table>
|
||
|
||
</div>
|
||
|
||
<p class="center">(But remember: nature has its own rules, and it does not have to follow mathematical patterns. But when it does it is awesome to see.)</p>
|
||
<p class="center"><br></p>
|
||
<div class="formal center80">
|
||
<h3>* Notes About the Animation</h3>
|
||
<p>Sunflower seeds grow from the center outwards, but on
|
||
the animation I found it easier to draw the younger seeds first and add
|
||
on the older ones.</p>
|
||
<p>The animation should continue longer to be the same
|
||
as the sunflower - this would result in 55 clockwise spirals and 34
|
||
counterclockwise spirals (successive Fibonacci Numbers). I just didn't
|
||
want it to take too long.</p>
|
||
<p>The spirals are not programmed into it - they occur
|
||
naturally as a result of trying to place the seeds as close to each
|
||
other as possible while keeping them at the correct rotation.</p></div>
|
||
<p> </p>
|
||
|
||
<div class="related">
|
||
<a href="golden-ratio.html">Golden Ratio</a>
|
||
<a href="fibonacci-sequence.html">Fibonacci Sequence</a>
|
||
<a href="../irrational-numbers.html">Irrational Numbers</a>
|
||
</div>
|
||
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