{"id":1414,"date":"2017-06-09T11:26:24","date_gmt":"2017-06-09T09:26:24","guid":{"rendered":"http:\/\/ccpics.de\/16\/?p=1414"},"modified":"2017-06-09T11:26:24","modified_gmt":"2017-06-09T09:26:24","slug":"business-shooting","status":"publish","type":"post","link":"https:\/\/www.ccpics.de\/2018\/business-shooting\/","title":{"rendered":"Die Fibonacci-Spirale"},"content":{"rendered":"<p>Die <b>Goldene <a title=\"Spirale\" href=\"https:\/\/de.wikipedia.org\/wiki\/Spirale\">Spirale<\/a><\/b> ist ein Sonderfall der <a title=\"Logarithmische Spirale\" href=\"https:\/\/de.wikipedia.org\/wiki\/Logarithmische_Spirale\">logarithmischen Spirale<\/a>. Diese Spirale l\u00e4sst sich mittels rekursiver Teilung eines Goldenen Rechtecks in je ein <a class=\"mw-redirect\" title=\"Quadrat (Geometrie)\" href=\"https:\/\/de.wikipedia.org\/wiki\/Quadrat_(Geometrie)\">Quadrat<\/a> und ein weiteres, kleineres Goldenes Rechteck konstruieren (siehe nebenstehendes Bild). Sie wird oft durch eine Folge von Viertelkreisen approximiert. Ihr Radius \u00e4ndert sich bei jeder 90\u00b0-Drehung um den Faktor <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\">{\\displaystyle \\Phi }<\/span><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/aed80a2011a3912b028ba32a52dfa57165455f24\" alt=\"\\Phi \" aria-hidden=\"true\" \/><\/span>.<sup id=\"cite_ref-10\" class=\"reference\"><a href=\"https:\/\/de.wikipedia.org\/wiki\/Goldener_Schnitt#cite_note-10\">[* 3]<\/a><\/sup><\/p>\n<p>Es gilt: <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\">{\\displaystyle \\textstyle r(\\varphi )=ae^{k\\varphi }}<\/span><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/99ded6e9c3202d926356f6ac7e11d7f2daaa7c66\" alt=\"\\textstyle r(\\varphi )=ae^{k\\varphi }\" aria-hidden=\"true\" \/><\/span> mit der Steigung <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\">{\\displaystyle \\textstyle k=\\pm {\\frac {\\ln {\\Phi }}{\\alpha _{\\llcorner }}}}<\/span><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b1898f50782f4f8e3343aa06ef836ffaa5b9465d\" alt=\"\\textstyle k=\\pm {\\frac {\\ln {\\Phi }}{\\alpha _{\\llcorner }}}\" aria-hidden=\"true\" \/><\/span>, wobei <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\">{\\displaystyle \\alpha _{\\llcorner }}<\/span><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f3b6a28352e0a8fe0e127cbf252ceff9d1b1b878\" alt=\"\\alpha _{\\llcorner }\" aria-hidden=\"true\" \/><\/span> hierbei der Zahlenwert f\u00fcr den <a title=\"Rechter Winkel\" href=\"https:\/\/de.wikipedia.org\/wiki\/Rechter_Winkel\">rechten Winkel<\/a>, also 90\u00b0 bzw. <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\">{\\displaystyle {\\tfrac {\\pi }{2}}}<\/span><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b4e31a202557dfbf326b44ebcc914ba3ab08fff1\" alt=\"{\\tfrac {\\pi }{2}}\" aria-hidden=\"true\" \/><\/span> ist, also <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\">{\\displaystyle \\textstyle k={\\frac {2\\ln(\\Phi )}{\\pi }}}<\/span><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a00ec829c60b580fbe02f5708db8638f44eabb43\" alt=\"\\textstyle k={\\frac {2\\ln(\\Phi )}{\\pi }}\" aria-hidden=\"true\" \/><\/span> mit der Goldenen Zahl <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\">{\\displaystyle \\textstyle \\Phi ={\\frac {{\\sqrt {5}}+1}{2}}}<\/span><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4ea067c79ef5c4012bf44fdee6a27bcdfc650b29\" alt=\"\\textstyle \\Phi ={\\frac {{\\sqrt {5}}+1}{2}}\" aria-hidden=\"true\" \/><\/span><\/p>\n<p>Mithin gilt f\u00fcr die Steigung: <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\">{\\displaystyle \\textstyle |k|\\approx 0{,}005346798\/{}^{\\circ }\\approx 0{,}30634896\/\\mathrm {rad} }<\/span><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1a10f6e44d9a71d6a5bfdf90dcf677cb4bc9751d\" alt=\"\\textstyle |k|\\approx 0{,}005346798\/{}^{\\circ }\\approx 0{,}30634896\/\\mathrm {rad} \" aria-hidden=\"true\" \/><\/span>.<\/p>\n<p>Die Goldene Spirale ist unter den logarithmischen Spiralen durch die folgende Eigenschaft ausgezeichnet. Seien <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\">{\\displaystyle P_{1},P_{2},P_{3},P_{4}}<\/span><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1b178861ae1b036b860aede198708a7682c64a2d\" alt=\"P_{1},P_{2},P_{3},P_{4}\" aria-hidden=\"true\" \/><\/span> vier auf der Spirale aufeinanderfolgende Schnittpunkte mit einer Geraden durch das Zentrum. Dann sind die beiden Punktepaare <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\">{\\displaystyle P_{1},P_{4}}<\/span><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4e9f7020a2c2d19f20c46805796506c88b9b3ba6\" alt=\"{\\displaystyle P_{1},P_{4}}\" aria-hidden=\"true\" \/><\/span> und <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\">{\\displaystyle P_{2},P_{3}}<\/span><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/898fae629cca84b7757189661a867ae1c240cec1\" alt=\"P_2,P_3\" aria-hidden=\"true\" \/><\/span> <a title=\"Harmonische Teilung\" href=\"https:\/\/de.wikipedia.org\/wiki\/Harmonische_Teilung\">harmonisch konjugiert<\/a>, d.h. das <a title=\"Doppelverh\u00e4ltnis\" href=\"https:\/\/de.wikipedia.org\/wiki\/Doppelverh%C3%A4ltnis\">Doppelverh\u00e4ltnis<\/a> <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\">{\\displaystyle (P_{1},P_{4};P_{2},P_{3})}<\/span><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c761c311d032b3643b0fdc69d49e8d1347485e0c\" alt=\"{\\displaystyle (P_{1},P_{4};P_{2},P_{3})}\" aria-hidden=\"true\" \/><\/span> ist <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\">{\\displaystyle -1}<\/span><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/704fb0427140d054dd267925495e78164fee9aac\" alt=\"-1\" aria-hidden=\"true\" \/><\/span>. <sup id=\"cite_ref-11\" class=\"reference\"><a href=\"https:\/\/de.wikipedia.org\/wiki\/Goldener_Schnitt#cite_note-11\">[8]<\/a><\/sup><\/p>\n<div id=\"attachment_1415\" style=\"width: 1034px\" class=\"wp-caption alignnone\"><img decoding=\"async\" aria-describedby=\"caption-attachment-1415\" class=\"size-large wp-image-1415\" src=\"http:\/\/ccpics.de\/16\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_1_resize-1024x690.jpg\" alt=\"\" width=\"1024\" height=\"690\" srcset=\"https:\/\/www.ccpics.de\/2018\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_1_resize-200x135.jpg 200w, https:\/\/www.ccpics.de\/2018\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_1_resize-300x202.jpg 300w, https:\/\/www.ccpics.de\/2018\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_1_resize-400x270.jpg 400w, https:\/\/www.ccpics.de\/2018\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_1_resize-600x404.jpg 600w, https:\/\/www.ccpics.de\/2018\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_1_resize-768x518.jpg 768w, https:\/\/www.ccpics.de\/2018\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_1_resize-800x539.jpg 800w, https:\/\/www.ccpics.de\/2018\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_1_resize-1024x690.jpg 1024w, https:\/\/www.ccpics.de\/2018\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_1_resize-1200x809.jpg 1200w, https:\/\/www.ccpics.de\/2018\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_1_resize.jpg 2061w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><p id=\"caption-attachment-1415\" class=\"wp-caption-text\">CCPICS \/ CARSTEN RIEDEL<\/p><\/div>\n<div id=\"attachment_1416\" style=\"width: 1034px\" class=\"wp-caption alignnone\"><img decoding=\"async\" aria-describedby=\"caption-attachment-1416\" class=\"size-large wp-image-1416\" src=\"http:\/\/ccpics.de\/16\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_2_resize-1024x690.jpg\" alt=\"\" width=\"1024\" height=\"690\" srcset=\"https:\/\/www.ccpics.de\/2018\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_2_resize-200x135.jpg 200w, https:\/\/www.ccpics.de\/2018\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_2_resize-300x202.jpg 300w, https:\/\/www.ccpics.de\/2018\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_2_resize-400x270.jpg 400w, https:\/\/www.ccpics.de\/2018\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_2_resize-600x404.jpg 600w, https:\/\/www.ccpics.de\/2018\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_2_resize-768x518.jpg 768w, https:\/\/www.ccpics.de\/2018\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_2_resize-800x539.jpg 800w, https:\/\/www.ccpics.de\/2018\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_2_resize-1024x690.jpg 1024w, https:\/\/www.ccpics.de\/2018\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_2_resize-1200x809.jpg 1200w, https:\/\/www.ccpics.de\/2018\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_2_resize.jpg 2061w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><p id=\"caption-attachment-1416\" class=\"wp-caption-text\">CCPICS \/ CARSTEN RIEDEL<\/p><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Die Goldene Spirale ist ein Sonderfall der logarithmischen Spirale. Diese Spirale l\u00e4sst sich mittels rekursiver Teilung eines Goldenen Rechtecks in je ein Quadrat und ein weiteres, kleineres Goldenes Rechteck konstruieren (siehe nebenstehendes Bild). Sie wird oft durch eine Folge von Viertelkreisen approximiert. Ihr Radius \u00e4ndert sich bei jeder 90\u00b0-Drehung um den Faktor {\\displaystyle \\Phi }.[* 3] Es gilt: {\\displaystyle \\textstyle r(\\varphi )=ae^{k\\varphi }} mit der [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":1415,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"gallery","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-1414","post","type-post","status-publish","format-gallery","has-post-thumbnail","hentry","category-allgemein","post_format-post-format-gallery"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Die Fibonacci-Spirale - ccpics<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.ccpics.de\/2018\/business-shooting\/\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Die Fibonacci-Spirale - ccpics\" \/>\n<meta name=\"twitter:description\" content=\"Die Goldene Spirale ist ein Sonderfall der logarithmischen Spirale. Diese Spirale l\u00e4sst sich mittels rekursiver Teilung eines Goldenen Rechtecks in je ein Quadrat und ein weiteres, kleineres Goldenes Rechteck konstruieren (siehe nebenstehendes Bild). Sie wird oft durch eine Folge von Viertelkreisen approximiert. Ihr Radius \u00e4ndert sich bei jeder 90\u00b0-Drehung um den Faktor {\\displaystyle \\Phi }.[* 3] Es gilt: {\\displaystyle \\textstyle r(\\varphi )=ae^{k\\varphi }} mit der [&hellip;]\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/www.ccpics.de\/2018\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_1_resize.jpg\" \/>\n<meta name=\"twitter:label1\" content=\"Verfasst von\" \/>\n\t<meta name=\"twitter:data1\" content=\"admin\" \/>\n\t<meta name=\"twitter:label2\" content=\"Gesch\u00e4tzte Lesezeit\" \/>\n\t<meta name=\"twitter:data2\" content=\"1\u00a0Minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/www.ccpics.de\\\/2018\\\/business-shooting\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.ccpics.de\\\/2018\\\/business-shooting\\\/\"},\"author\":{\"name\":\"admin\",\"@id\":\"https:\\\/\\\/www.ccpics.de\\\/2018\\\/#\\\/schema\\\/person\\\/74def6da2d3e20b88f9bba881e445c4e\"},\"headline\":\"Die Fibonacci-Spirale\",\"datePublished\":\"2017-06-09T09:26:24+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/www.ccpics.de\\\/2018\\\/business-shooting\\\/\"},\"wordCount\":205,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/www.ccpics.de\\\/2018\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/www.ccpics.de\\\/2018\\\/business-shooting\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/www.ccpics.de\\\/2018\\\/wp-content\\\/uploads\\\/2017\\\/06\\\/Holl\u00e4nder_1_resize.jpg\",\"inLanguage\":\"de\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/www.ccpics.de\\\/2018\\\/business-shooting\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/www.ccpics.de\\\/2018\\\/business-shooting\\\/\",\"url\":\"https:\\\/\\\/www.ccpics.de\\\/2018\\\/business-shooting\\\/\",\"name\":\"Die Fibonacci-Spirale - 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Diese Spirale l\u00e4sst sich mittels rekursiver Teilung eines Goldenen Rechtecks in je ein Quadrat und ein weiteres, kleineres Goldenes Rechteck konstruieren (siehe nebenstehendes Bild). Sie wird oft durch eine Folge von Viertelkreisen approximiert. Ihr Radius \u00e4ndert sich bei jeder 90\u00b0-Drehung um den Faktor {\\displaystyle \\Phi }.[* 3] Es gilt: {\\displaystyle \\textstyle r(\\varphi )=ae^{k\\varphi }} mit der [&hellip;]","twitter_image":"https:\/\/www.ccpics.de\/2018\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_1_resize.jpg","twitter_misc":{"Verfasst von":"admin","Gesch\u00e4tzte Lesezeit":"1\u00a0Minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/www.ccpics.de\/2018\/business-shooting\/#article","isPartOf":{"@id":"https:\/\/www.ccpics.de\/2018\/business-shooting\/"},"author":{"name":"admin","@id":"https:\/\/www.ccpics.de\/2018\/#\/schema\/person\/74def6da2d3e20b88f9bba881e445c4e"},"headline":"Die Fibonacci-Spirale","datePublished":"2017-06-09T09:26:24+00:00","mainEntityOfPage":{"@id":"https:\/\/www.ccpics.de\/2018\/business-shooting\/"},"wordCount":205,"commentCount":0,"publisher":{"@id":"https:\/\/www.ccpics.de\/2018\/#organization"},"image":{"@id":"https:\/\/www.ccpics.de\/2018\/business-shooting\/#primaryimage"},"thumbnailUrl":"https:\/\/www.ccpics.de\/2018\/wp-content\/uploads\/2017\/06\/Holl\u00e4nder_1_resize.jpg","inLanguage":"de","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/www.ccpics.de\/2018\/business-shooting\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/www.ccpics.de\/2018\/business-shooting\/","url":"https:\/\/www.ccpics.de\/2018\/business-shooting\/","name":"Die Fibonacci-Spirale - 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