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geometry.html
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<!DOCTYPE HTML>
<html>
<head>
<!-- Required meta tags -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no">
<!-- Bootstrap CSS -->
<link rel="stylesheet" href="https://stackpath.bootstrapcdn.com/bootstrap/4.5.2/css/bootstrap.min.css"
integrity="sha384-JcKb8q3iqJ61gNV9KGb8thSsNjpSL0n8PARn9HuZOnIxN0hoP+VmmDGMN5t9UJ0Z" crossorigin="anonymous">
<title>Geometry</title>
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1, user-scalable=no" />
<link rel="stylesheet" href="assets/css/main.css" />
</head>
<body class="single is-preload">
<!-- Wrapper -->
<div id="wrapper">
<!-- Header -->
<header id="header">
<h1><a href="index1.html">Golden Ratio</a></h1>
<nav class="links">
<ul>
<li><a href="index1.html"><b>Home</b></a></li>
<li><a href="art.html">Art</a></li>
<li><a href="nature.html">Nature</a></li>
<li><a href="angle.html">Golden Angle</a></li>
<li><a href="rule.html">Rule of Thirds</a></li>
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<li>
<a href="art.html">
<h3>Golden Ratio And Art</h3>
</a>
</li>
<li>
<a href="archi.html">
<h3>Golden Ratio And Architecture</h3>
</a>
</li>
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<a href="nature.html">
<h3>Golden Ratio And Nature</h3>
</a>
</li>
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<a href="angle.html">
<h3>Relation with Golden Angle</h3>
</a>
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<h3>Golden Ratio and Rule of Thirds</h3>
</a>
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<h3>Fibonacci and Golden Ratio</h3>
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<article class="post">
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<div class="title">
<h2><a href="#">Geometrical usage</a></h2>
</div>
<div class="meta">
<a href="#" class="author"><span class="name">Jane Doe</span><img src="images/avatar.jpg"
alt="" /></a>
</div>
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<span class="image featured"><img src="images/clem-onojeghuo-T7gi_cyrkdg-unsplash.jpg" alt="" /></span>
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<li><a href="#">General</a></li>
<li><a href="#" class="icon solid fa-heart">28</a></li>
<li><a href="#" class="icon solid fa-comment">128</a></li>
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</article>
<div class="container">
<div class="row featurette d-flex justify-content-center align-items-center">
<div class="col-md-7">
<h2 class="featurette-heading">Introduction to Geometry <span class="text-muted"></span></h2>
<p class="lead">The number φ turns up frequently in geometry, particularly in figures with
pentagonal symmetry. The length of a regular pentagon's diagonal is φ times its side. The
vertices of a regular icosahedron are those of three mutually orthogonal golden rectangles.
</p>
<p class="lead">There is no known general algorithm to arrange a given number of nodes evenly on
a sphere, for any of several definitions of even distribution. However, a useful
approximation results from dividing the sphere into parallel
bands of equal surface area and placing one node in each band at longitudes spaced by a
golden section of the circle, i.e. 360°/φ ≅ 222.5°. This method was used to arrange the 1500
mirrors of the student-participatory satellite Starshine-3.
</p>
</div>
<div class="col-md-5">
<img class="bd-placeholder-img bd-placeholder-img-lg featurette-image img-fluid mx-auto"
width="500" height="500" src="images/FakeRealLogSpiral.svg.png" alt="">
</div>
</div>
<div class="row featurette d-flex justify-content-center align-items-center">
<div class="col-md-7 order-md-2">
<h2 class="featurette-heading">Golden triangle<span class="text-muted"></span></h2>
<p class="lead">The golden triangle can be characterized as an isosceles triangle ABC with the
property that bisecting the angle C produces a new triangle CXB which is a similar triangle
to the original.
If angle BCX = α, then XCA = α because of the bisection, and CAB = α because of the similar
triangles; ABC = 2α from the original isosceles symmetry, and BXC = 2α by similarity. The
angles in a triangle add up to 180°, so 5α = 180, giving α = 36°. So the angles of the
golden triangle are thus 36°-72°-72°. The angles of the remaining obtuse isosceles triangle
AXC (sometimes called the golden gnomon) are 36°-36°-108°.</p>
<p class="lead">Suppose XB has length 1, and we call BC length φ. Because of the isosceles
triangles XC=XA and BC=XC, so these are also length φ. Length AC = AB, therefore equals φ +
1. But triangle ABC is similar to triangle CXB, so AC/BC = BC/BX, AC/φ = φ/1, and so AC also
equals φ2. Thus φ2 = φ + 1, confirming that φ is indeed the golden ratio.
Similarly, the ratio of the area of the larger triangle AXC to the smaller CXB is equal to
φ, while the inverse ratio is φ − 1.</p>
</div>
<div class="col-md-5 order-md-1">
<img class="bd-placeholder-img bd-placeholder-img-lg featurette-image img-fluid mx-auto"
width="500" height="500" src="images/330px-Golden_triangle_(math).svg.png" alt="">
</div>
</div>
<div class="row featurette d-flex justify-content-center align-items-center">
<div class="col-md-7">
<h2 class="featurette-heading">Pentagram<span class="text-muted"></span></h2>
<p class="lead">The golden ratio plays an important role in the geometry of pentagrams. Each
intersection of edges sections other edges in the golden ratio. Also, the ratio of the
length of the shorter segment to the segment bounded by the two intersecting edges (a side
of the pentagon in the pentagram's center) is φ, as the four-color illustration shows.</p>
<p class="lead">The pentagram includes ten isosceles triangles: five acute and five obtuse
isosceles triangles. In all of them, the ratio of the longer side to the shorter side is φ.
The acute triangles are golden triangles. The obtuse isosceles triangles are golden gnomons.
</p>
</div>
<div class="col-md-5">
<img class="bd-placeholder-img bd-placeholder-img-lg featurette-image img-fluid mx-auto"
width="500" height="500" src="images/330px-Pentagram-phi.svg.png" alt="">
</div>
</div>
<div class="row featurette d-flex justify-content-center align-items-center">
<div class="col-md-7 order-md-2">
<h2 class="featurette-heading">Pentagon<span class="text-muted"></span></h2>
<p class="lead">In a regular pentagon the ratio of a diagonal to a side is the golden ratio,
while intersecting diagonals section each other in the golden ratio.</p>
</div>
<div class="col-md-5 order-md-1">
<img class="bd-placeholder-img bd-placeholder-img-lg featurette-image img-fluid mx-auto"
width="500" height="500" src="images/330px-Ptolemy_Pentagon.svg.png" alt="">
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