{"id":563,"date":"2016-06-22T01:13:50","date_gmt":"2016-06-22T06:13:50","guid":{"rendered":"https:\/\/trigonography.com\/?p=563"},"modified":"2022-06-25T03:39:57","modified_gmt":"2022-06-25T08:39:57","slug":"special-angles-are-golden-ii","status":"publish","type":"post","link":"https:\/\/trigonography.com\/?p=563","title":{"rendered":"Special Angles are Golden, II"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\"><style>div.article { width:480pt;margin:auto; margin-top:10pt; min-width:480pt}<\/style><\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\" style=\"font-size:16px\"><em>I&#8217;ve been encouraged to apply the type of analysis from <a href=\"https:\/\/trigonography.com\/2016\/06\/05\/special-angles-are-golden\/\">my previous post<\/a> to another situation.<br>I&#8217;ll follow the same format. (Did I mention that I&#8217;m lazy?)<\/em><\/p>\n\n\n\n<div class=\"article\" style=\"padding-top: 22pt; width:720pt; font-size:12pt;\"><p>The figure shows one of <a href=\"http:\/\/forumgeom.fau.edu\/FG2016volume16\/FG201632index.html\">\u0110\u00e0o Thanh Oai&#8217;s constructions<\/a> of the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Golden_ratio\">golden ratio<\/a>, \\(\\phi = 1.618\\dots\\).<\/p>\n<a href=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/08\/trigonograph-goldenanglesIIA-1.png\"><img loading=\"lazy\" decoding=\"async\" style=\"float:right; margin-top: 0pt;margin-right:120px;\" src=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/08\/trigonograph-goldenanglesIIA-1.png\" alt=\"trigonograph-goldenanglesIIA\" width=\"360\" height=\"360\"><\/a>\n<p style=\"margin-top:32px; width:240px; margin-left:120pt; text-align:center;padding-left:20px; padding-right:20px;bgcolor:yellow;\">Paraphrasing &#8230;<\/p>\n<p style=\"border: 1px solid #ccc; padding: 20px; width: 240px; margin-left:120pt; padding-bottom: 10pt; text-align:center;\">The circle through a vertex of an equilateral triangle, and through points one-fifth of the way up the adjacent sides, separates the third side into segments whose lengths are in the proportion $$1 \\;:\\; \\phi \\;:\\; 1$$<\/p>\n<\/div>\n\n\n\n<div style=\"clear: both;\">&nbsp;<\/div>\n\n\n\n<div class=\"article\" style=\"width:720px; font-size:12pt;\">\u0110\u00e0o provides a companion result involving a &#8220;two-fifths circle&#8221; and an isosceles right triangle, although depicting it on a square is perhaps more evocative. As in <a href=\"https:\/\/trigonography.com\/2016\/06\/05\/special-angles-are-golden\/\">my previous post<\/a>, a (new?) hexagonal counterpart \u2014with a pattern-furthering &#8220;three-fifths circle&#8221;\u2014 arises from the analysis below.<\/div>\n\n\n\n<a href=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/08\/trigonograph-goldenanglesIIB.png\"><img decoding=\"async\" src=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/08\/trigonograph-goldenanglesIIB.png\" alt=\"trigonograph-goldenanglesIIB\"\/><\/a> <a href=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/08\/trigonograph-goldenanglesIIC.png\"><img decoding=\"async\" src=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/08\/trigonograph-goldenanglesIIC.png\" alt=\"trigonograph-goldenanglesIIB\"\/><\/a>\n\n\n\n<div class=\"article\" style=\"width:720px; font-size:12pt\"><a href=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/08\/trigonograph-goldenanglesIIA.png\"><img loading=\"lazy\" decoding=\"async\" style=\"float: left; margin-bottom:40px;\" src=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/08\/trigonograph-goldenanglesIIA.png\" alt=\"trigonograph-goldenanglesIIA\" width=\"360\" height=\"360\"><\/a><br>For the general analysis, we consider an isosceles triangle with legs of unit length and vertex angle of half-measure \\(\\theta\\). And we suppose that the circle through the vertex, and through points at distance \\(s\\) up the legs, separates the base in proportion \\(1:r:1\\) (for not-necessarily-golden ratio \\(r\\)), as shown. Calculating the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Power_of_a_point\">power<\/a> of a base vertex with respect to the circle in two ways (along the leg, along the base) gives &#8230;<br>$$s \\cdot 1 = 2 x \\cdot (2 x + 2 x r ) = 4 x^2 (1 + r)$$<br>Then, since \\(\\sin\\theta = 2 x + x r = x (2 +r)\\),<br>we can write &#8230;<br>$$s = 4 \\sin^2\\theta\\;\\frac{1+r}{(2+r)^2} \\qquad(\\star)$$<br>In the particular case \\(r = \\phi\\), the fraction in \\((\\star)\\) doesn&#8217;t <em>completely vanish<\/em> as it did before;<br>it reduces to \\(1\/5\\):\n<p>&nbsp;<\/p>\n<div style=\"display: inline-block; vertical-align: middle;\">$$\\begin{array}{c}<br>\\displaystyle \\qquad\\quad s = \\frac{4}{5}\\sin^2\\theta\\qquad\\text{or}\\qquad \\sin\\theta = \\frac{\\sqrt{s}}{2}\\cdot\\frac{\\sqrt{5}}{5}<br>\\end{array}$$<\/div>\n<div style=\"width:720px; text-align:center\">&#8230; yielding familiar &#8220;special&#8221; cases &#8230;<\/div>\n<div style=\"width:720px;\">$$\\begin{align}<br>\\theta &amp;= \\;\\;0^\\circ :\\; s = 0\/5 \\\\[4pt]<br>\\theta &amp;= \\color{blue}{30^\\circ} :\\; s = \\color{blue}{1}\/5 \\quad\\to\\quad\\text{\u0110\u00e0o&#8217;s Triangle}\\\\[4pt]<br>\\theta &amp;= \\color{red}{45^\\circ} :\\; s = \\color{red}{2}\/5 \\quad\\to\\quad\\text{\u0110\u00e0o&#8217;s Square}\\\\[4pt]<br>\\theta &amp;= \\color{violet}{60^\\circ} :\\; s = \\color{violet}{3}\/5 \\quad\\to\\quad\\text{Someone&#8217;s Hexagon}\\\\[4pt]<br>\\theta &amp;= 90^\\circ :\\; s = 4\/5<br>\\end{align}$$<\/div>\n<\/div>\n\n\n\n<div style=\"clear: both;\">&nbsp;<\/div>\n\n\n\n<div class=\"article\" style=\"padding-top: 40pt; width:720px; font-size:12pt\"><a href=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/08\/trigonograph-goldenanglesIIX.png\"><img loading=\"lazy\" decoding=\"async\" style=\"float: right;\" src=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/08\/trigonograph-goldenanglesIIX.png\" alt=\"trigonograph-goldenanglesIIX\" width=\"360\" height=\"360\"><\/a><br>Along with the degenerate cases for \\(0^\\circ\\) and \\(90^\\circ\\), the triangle, square, and hexagon configurations exhaust all possibilities of \u0110\u00e0o-like constructions involving an &#8220;\\(n\\)-fifths circle&#8221; for integer \\(n\\).\n<p>&nbsp;<\/p>\n<p>The diagram at right combines the corresponding figures within a common circumcircle, and with a companion ellipse through separation points on appropriate chords. In this context, the chord for the \\(90^\\circ\\) figure collapses to a point, which actually obscures an important phenomenon.<\/p>\n<\/div>\n\n\n\n<div style=\"clear: both;\">&nbsp;<\/div>\n\n\n\n<div class=\"article\" style=\"width:720px; font-size:12pt\"><a href=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/08\/trigonograph-goldenanglesIIL.png\"><img decoding=\"async\" style=\"float: left; margin-top:0pt; margin-right:20px;\" src=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/08\/trigonograph-goldenanglesIIL.png\" alt=\"trigonograph-goldenanglesIIL\" width=\"320\"><\/a>\n<p>Holding the edges to a constant length (and focusing on the isosceles triangles formed with the chords), we find that the \\(90^\\circ\\) case, in and of itself, is problematic: the corresponding &#8220;four-fifths circle&#8221; is actually a <em>straight line<\/em> that coincides with the flat triangle&#8217;s legs and its base, making the locations of the separation points <em>undefined<\/em>. The problem resolves, however, when this case is viewed as the limiting form in the <em>continuous family<\/em> of triangles. (An analogous observation can\/should be added to <a href=\"https:\/\/trigonography.com\/2016\/06\/05\/special-angles-are-golden\/\">my previous article<\/a>. <em>(Lazy!)<\/em>&nbsp;)<\/p>\n<\/div>\n\n\n\n<div style=\"clear: both;\">&nbsp;<\/div>\n","protected":false},"excerpt":{"rendered":"<p>I&#8217;ve been encouraged to apply the type of analysis from my previous post to another situation.I&#8217;ll follow the same format. (Did I mention that I&#8217;m lazy?) The figure shows one of \u0110\u00e0o Thanh Oai&#8217;s constructions of the golden ratio, \\(\\phi = 1.618\\dots\\). Paraphrasing &#8230; The circle through a vertex of an equilateral triangle, and through [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-563","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts\/563","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=563"}],"version-history":[{"count":10,"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts\/563\/revisions"}],"predecessor-version":[{"id":1235,"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts\/563\/revisions\/1235"}],"wp:attachment":[{"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=563"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=563"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=563"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}