{"id":310,"date":"2016-06-05T11:21:31","date_gmt":"2016-06-05T16:21:31","guid":{"rendered":"https:\/\/trigonography.com\/?p=310"},"modified":"2022-06-25T03:10:34","modified_gmt":"2022-06-25T08:10:34","slug":"special-angles-are-golden","status":"publish","type":"post","link":"https:\/\/trigonography.com\/?p=310","title":{"rendered":"Special Angles are Golden"},"content":{"rendered":"\n<p><style>div.article { width:480pt;margin:auto; margin-top:10pt; }<\/style><\/p>\n\n\n\n<div class=\"article\" style=\"width:720px; padding-top: 20pt; font-size:12pt;\"><br>This is <a href=\"https:\/\/en.wikipedia.org\/wiki\/George_Phillips_Odom,_Jr.\">George Odom<\/a>&#8216;s elegant construction of the ever-fascinating <a href=\"https:\/\/en.wikipedia.org\/wiki\/Golden_ratio\">golden ratio<\/a>, \\(\\phi\\):<br>\n<a href=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/07\/trigonograph-goldenanglesA.png\"><img loading=\"lazy\" decoding=\"async\" style=\"float: right; margin-top:0pt;\" src=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/07\/trigonograph-goldenanglesA.png\" alt=\"trigonograph-goldenanglesA\" width=\"360\" height=\"360\"><\/a>\n<p>&nbsp;<\/p>\n<p style=\"border: 1px solid #ccc; padding: 20pt; width: 180pt; margin-left: 20pt; padding-bottom: 10pt;\">If \\(a\\) measures a &#8220;midpoint segment&#8221; of an equilateral triangle, and if \\(b\\) measures the extension of that segment meeting the triangle&#8217;s circumcircle, then $$\\frac{a}{b} = \\phi = 1.618\\dots$$<\/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;\">As it turns out, Odom&#8217;s construction generalizes very nicely when we replace <em>&#8220;midpoint segment&#8221;<\/em> with <em>&#8220;trisecting segment&#8221;<\/em> or <em>&#8220;quadrisecting segment&#8221;<\/em>, and <em>&#8220;equilateral triangle&#8221;<\/em> with <em>&#8220;square&#8221;<\/em> or <em>&#8220;regular hexagon&#8221;<\/em>.<\/div>\n\n\n\n<a href=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/07\/trigonograph-goldenanglesB.png\"><img decoding=\"async\" src=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/07\/trigonograph-goldenanglesB.png\" alt=\"trigonograph-goldenanglesC\"\/><\/a><a href=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/07\/trigonograph-goldenanglesC.png\"><img decoding=\"async\" src=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/07\/trigonograph-goldenanglesC.png\" alt=\"trigonograph-goldenanglesC\"\/><\/a>\n\n\n\n<div class=\"article\" style=\"width:720px; font-size:12pt;\">I became aware of the square result while contemplating <a href=\"http:\/\/math.stackexchange.com\/questions\/1810718\/golden-ratio-flower-numerous-golden-ratios-in-five-circles-and-grid-of-squares\">this post to Math Stack Exchange<\/a> by Dr. Elliot McGucken (aka, &#8220;Astrophysics Math&#8221;). This prompted my own (re?-)discovery of the hexagon result, which simply falls out of the straightforward analysis I provide below.<br>It&#8217;s possible \u2014even likely\u2014 that all of these observations already appear somewhere in<br>the vast lore on the golden ratio, but I&#8217;m too lazy to do a literature search.<\/div>\n\n\n\n<div class=\"article\" style=\"width:720px; border: 1px solid #ccc;margin-left: auto; margin-right: auto; padding: 10pt; font-size:12pt;\"><b>Update:<\/b> The essence of the square construction appears in an isosceles-right-triangle construction <br><a href=\"http:\/\/forumgeom.fau.edu\/FG2015volume15\/FG201506.pdf\">published in 2015 by Q. H. Tran (PDF)<\/a>. Curiously, despite noting the Odom-esque flavor of his result,<br>Tran doesn\u2019t seem to realize that his figure features an obvious, and even-more-Odom-esque,<br>segment trisector. (The missed connection with the square is less surprising, given the semi-circular<br>context.) Nevertheless, it appears appropriate to assign priority to what I\u2019ll call \u201cTran\u2019s Square\u201d.<\/div>\n\n\n\n<div class=\"article\" style=\"font-size:12pt;\">Be that as it may &#8230;<\/div>\n\n\n\n<div class=\"article\" style=\"width:720px; font-size:12pt\"><a href=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/07\/trigonograph-goldenanglesN.png\"><img loading=\"lazy\" decoding=\"async\" style=\"float: left; margin-bottom:40px;\" src=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/07\/trigonograph-goldenanglesN.png\" alt=\"trigonograph-goldenanglesN\" width=\"360\" height=\"360\"><\/a><br>To see the underlying connections here, consider an inscribed angle of half-measure \\(\\theta\\), and suppose the &#8220;\\((n+1)\\)-secting segment&#8221; through the sides of this angle is such that the ratio of its length to that of its circle-meeting extension is \\(r : 1\\), for some not-necessarily-golden ratio \\(r\\).<\/div>\n\n\n\n<div class=\"article\" style=\"width:720px; font-size:12pt\">By the chord-chord aspect of the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Power_of_a_point\">Power of a Point theorem<\/a>, we have<br>$$n\\cdot 1 = x \\cdot ( x + r x )\\quad\\to\\quad x = \\sqrt{\\frac{n}{1+r}}$$<br>But, clearly, we also have \\(r x = 2\\sin\\theta\\), so that<br>$$\\sin\\theta = \\frac{rx}{2} = \\frac{\\sqrt{n}}{2}\\cdot\\frac{r}{\\sqrt{1+r}}\\qquad(\\star)$$<\/div>\n\n\n\n<div style=\"clear: both;\">&nbsp;<\/div>\n\n\n\n<div class=\"article\" style=\"width:720px; font-size:12pt\">Now, recall that what makes the <em>particular<\/em> ratio \\(\\phi\\) &#8220;golden&#8221; is that the square of its value is precisely \\(1\\) unit greater than the value itself; that is, \\(\\phi^2 = 1 + \\phi \\). Consequently, when \\(r = \\phi\\), the second factor in \\((\\star)\\) vanishes, leaving &#8230;\n<p>&nbsp;<\/p>\n<div style=\"display: inline-block; vertical-align: middle;\">$$\\sin\\theta = \\frac{\\sqrt{n}}{2}$$<\/div>\n<div style=\"display: inline-block; width: 180pt; margin: 0 8pt; vertical-align: middle;\">&#8230; which <em>should<\/em> be familiar to you <br>from the common mnemonic <br>involving trig&#8217;s &#8220;special angles&#8221; &#8230;<\/div>\n<div style=\"display: inline-block; vertical-align: middle;\">$$\\begin{align}<br>\\sin \\phantom{0}0^\\circ &amp;= \\sqrt{0}\\,\/2 \\\\[4pt]<br>\\sin \\color{blue}{30^\\circ} &amp;= \\sqrt{\\color{blue}{1}}\\,\/2 \\quad\\to\\quad\\text{Odom&#8217;s Triangle}\\\\[4pt]<br>\\sin \\color{red}{45^\\circ} &amp;= \\sqrt{\\color{red}{2}}\\,\/2 \\quad\\to\\quad\\text{Tran&#8217;s Square}\\\\[4pt]<br>\\sin \\color{violet}{60^\\circ} &amp;= \\sqrt{\\color{violet}{3}}\\,\/2 \\quad\\to\\quad\\text{Someone&#8217;s Hexagon} \\\\[4pt]<br>\\sin 90^\\circ &amp;= \\sqrt{4}\\,\/2<br>\\end{align}$$<\/div>\n<\/div>\n\n\n\n<div class=\"article\" style=\"width:720px; font-size:12pt;\">In this very peculiar sense, then, <b>special angles are indeed golden<\/b>.<\/div>\n\n\n\n<div class=\"article\" style=\"padding: 0pt; margin-top: 20pt; width:720px; font-size:12pt;\"><a href=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/07\/trigonograph-goldenanglesD.png\"><img loading=\"lazy\" decoding=\"async\" style=\"float: right; margin-top: 20pt;\" src=\"https:\/\/trigonography.com\/blog\/wp-content\/uploads\/2016\/07\/trigonograph-goldenanglesD.png\" alt=\"trigonograph-goldenanglesD\" width=\"360\" height=\"360\"><\/a>\n<p>&nbsp;<\/p>\n<p>Our sine relation establishes that \\(n\\) cannot exceed four. Therefore, the triangle, square, and hexagon above represent all available (non-degenerate) Odom-like constructions for integer \\(n\\). The composite view at right shows a bonus feature: <em>All the dividing points lie on an ellipse.<\/em> (No, the ellipse isn&#8217;t &#8220;golden&#8221;: neither its eccentricity, nor its aspect ratio, is \\(\\phi\\).)<\/p>\n<p>Of course, no one requires \\(n\\) to be an integer,<br>so there remains more to say. For instance, <em>the Odom-like construction for the regular pentagon, which already admits so many connections with the golden ratio, has \\(n = \\phi^2 = \\phi+1\\)<\/em>. Also, <em>\\(n = \\phi\\) implies \\(\\theta \\approx 39.49^\\circ \\).<\/em> (Is there anything (else) remarkable about that angle?) And so on.<\/p>\n<p>I&#8217;ll leave further considerations to the reader.<\/p>\n<\/div>\n\n\n\n<div style=\"clear: both;\">&nbsp;<\/div>\n","protected":false},"excerpt":{"rendered":"<p>This is George Odom&#8216;s elegant construction of the ever-fascinating golden ratio, \\(\\phi\\): &nbsp; If \\(a\\) measures a &#8220;midpoint segment&#8221; of an equilateral triangle, and if \\(b\\) measures the extension of that segment meeting the triangle&#8217;s circumcircle, then $$\\frac{a}{b} = \\phi = 1.618\\dots$$ &nbsp; As it turns out, Odom&#8217;s construction generalizes very nicely when we replace [&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-310","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts\/310","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=310"}],"version-history":[{"count":10,"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts\/310\/revisions"}],"predecessor-version":[{"id":1231,"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts\/310\/revisions\/1231"}],"wp:attachment":[{"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=310"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=310"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=310"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}