Проект (10, 11 класс) на тему: Исследовательская работа на тему «Золотое сечение»

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р абота на тему: « Золотое сечение-Золотая пропорция»

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Цель Работы Доказать, что «золотое сечение- золотая пропорция» — верх совершенства гармонии в природе.

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Задачи: Изучить понятие «золотое сечение- золотая пропорция»; Определить и рассмотреть применение «золотого сечения » в архитектуре, искусстве, живой и неживой природе, в современном мире Исследовать и изучить присутствие золотого сечения в окружающей меня жизни

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Методы исследования: Работа с учебной и научно-популярной литературой, ресурсами сети Интернет. Социологический опрос. Наблюдение, сравнение, анализ, аналогия.

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Предмет исследования: им стал поиск закономерностей «золотой пропорции» в различных областях неживой и живой природы, и в моем окружении

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Социологический опрос Знаете ли вы, что такое «золотое сечение» или «золотая пропорция»?

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Золотое сечение в математике Деление отрезка в среднем и крайнем отношении называют золотым сечением . В истории утвердилось ещё одно название – « золотая пропорция». Пусть, С є АВ, и производит, как говорят, «золотое сечение» отрезка АВ, так что АС: АВ =СВ: АС Золотым сечением — называется такое делении отрезка, при котором большая часть так относится к целому, как меньшая часть к большей

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Числа Фибоначчи С Золотой пропорцией тесно связан ряд чисел Фибоначчи. 1,1,2,3,5,8,13,21,34,55,89 и т.д. Если взять калькулятор и разделить каждое из них на предыдущее, то получиться: 1:1=1; 2:1=2; 3:2=1,5; 5:3=1,666666; 8:5=1,6; 13:8=1,625; 21:13=1,615384;… 0,618

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Золотая пропорция и тело человека Пропорции мужского тела 13 : 8 = 1,625

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Золотая пропорция и тело человека пропорции женского тела 8 : 5 = 1,6.

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Золотая пропорция и тело человека пропорция новорожденного 1 : 1, к 13 годам она равна 1,6 а к 21 году равняется мужской.

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Золотое сечение в природе Раковина закручена по спирали Золотое сечение в спирали Архимеда.

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Золотое сечение в природе И в растительном, и в животном мире настойчиво пробивается тенденция природы — симметрия относительно направления роста и движения Было установлено, что числовой ряд чисел Фибоначчи характеризует структурную организацию многих живых систем. Например, винтовое листорасположение на ветке составляет дробь (число оборотов на стебле/число листьев в цикле, напр. 2/5; 3/8; 5/13), соответствующую рядам Фибоначчи.

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Золотое сечение в природе Цветки и семена подсолнуха, ромашки, чешуйки в плодах ананаса, хвойных шишках «упакованы» по логарифмическим (» золотым») спиралям , завивающимся навстречу друг другу, Причем числа «правых «и «левых» спиралей всегда относятся друг к другу, как соседние числа Фибоначчи

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Золотое сечение в природе Рассмотрим побег цикория. Если первый выброс принять за 100 единиц, то второй равен 62 единицам, третий – 38, четвертый – 24 и т.д. Длина лепестков тоже подчинена золотой пропорции. В росте, завоевании пространства растение сохраняло определенные пропорции. Импульсы его роста постепенно уменьшались в пропорции золотого сечения.

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Золотое сечение в природе У многих бабочек соотношение размеров грудной и брюшной части тела отвечает золотой пропорции. Сложив крылья, ночная бабочка образует правильный равносторонний треугольник. Но стоит развести крылья, и вы увидите тот же принцип членения тела на 2,3,5,8. Стрекоза также создана по законам золотой пропорции : отношение длин хвоста и корпуса равно отношению общей длины к длине хвоста.

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В ящерице с первого взгляда улавливаются приятные для нашего глаза пропорции — длина ее хвоста так относится к длине остального тела, как 62 к 38. . Золотое сечение в природе

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Золотое сечение в архитектуре Среди грандиозных пирамид Египта особое место занимает великая пирамида фараона Хеопса. Она самая крупная и наиболее хорошо изученная. Чего только не находили в ее пропорциях! Число «пи» и золотое сечение, число дней в году, расстояние до Солнца, диаметр Земли.

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Золотое сечение в архитектуре Золотое сечение в здании Большого театра Москвы.

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Портрет «Джоконды» Портрет «Джоконды»привлекает тем, что композиция рисунка построена на «золотых треугольниках» (точнее на треугольниках, являющихся кусками правильного звездчатого пятиугольника).

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Исследование №1 «Золотое сечение в природе »

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Таблица1.Результаты измерений. № Наименование растения АВ АС ВС АС / ВС ВС / АВ 1 Толстянка- Денежное дерево 4.5 1 3.5 0.286 0.777 2 Рео 10 6 4 1.5 0.4 3 Кротон 8 4.6 3.4 1.352 0.425 4 Малочай 4.5 2 2.5 0.8 0.555 5 Герань 4 1.5 2.5 0.6 0.625 6 Традесканция 6 2.5 3.5 0.714 0.583 7 Вьюнок 19 8 11 0.727 0.578 8 Вьющаяся традесканция 8.5 3 5.5 0.545 0.647 9 Китайская роза 6 2.5 3.5 0.714 0.583 10 Дефинбахия 14 6 8 0.750 0.571 я убедилась в том, что не все, но большенство из отношений стремятся к 0.618. Более близким к этому значению оказался цветок герань.

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Результат исследования №1 АВ=1,5 ВС=2,5 АС=4 АВ/ВС=0,6 ВС/АС=0,625 Φ =0,618 Герань

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Исследование №2 «Золотое сечение в пропорциях тела человека» Пропорции мужского тела 13 : 8 = 1,625 Пропорции женского тела 8 : 5 = 1,6.

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Результат исследования № 2 У Всех ребят, из 12-и исследуемых, не выявлено отклонений от золотого сечения. У них пропорции тела близки к «золотому сечению»

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А в детских игрушках, куклах важно соблюдать естественные пропорции частей тела? Сейчас если спросить у маленькой девочки, «Какую куклу ты хочешь?». Ответ последует незамедлительно «Барби!» или подобную ей куклу (которые сейчас стали особо популярными). С первого взгляды ничего негативного и угрожающего в такой кукле нет. Она красивая, у неё красивые волосы и наряды, но многие считают её вредной для детишек. Почему?

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кукла Барби не соответствует реальности во внешности. Есть три изъяна в её внешности : Непропорциональные ноги. Непропорциональные руки (слишком длинные, как и ноги) Обсолютно непропорциональная фигура (широкие плечи, большой бюст, очень тонкая талия и плоские бёдра) Фигуры являются гораздо более худыми по сравнению с традиционными Барби способствуют развитию у девочек чувства неуверенности относительно собственного образа Играя с ними, девочки закладывают себе неправильные представления о фигуре, и вырастая, они начинают изводить себя диетами, чтобы стать стройнее, добиться мнимого идеала.

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Заключение «Золотое сечение- золотая пропорция» давно перестало быть сокровищем только геометрии. Оно выходит далеко за рамки геометрии. Хотя это понятие нам не заметно, но всё-таки оно существует. И производителям детских игрушек , следует этого придерживаться

90000 Phi: The Golden Ratio | Live Science 90001 90002 The number phi, often known as the golden ratio, is a mathematical concept that people have known about since the time of the ancient Greeks. It is an irrational number like 90003 pi 90004 and e, meaning that its terms go on forever after the decimal point without repeating. 90005 90002 Over the centuries, a great deal of lore has built up around phi, such as the idea that it represents perfect beauty or is uniquely found throughout nature.But much of that has no basis in reality. 90005 90008 Definition of phi 90009 90002 Phi can be defined by taking a stick and breaking it into two portions. If the ratio between these two portions is the same as the ratio between the overall stick and the larger segment, the portions are said to be in the golden ratio. This was first described by the Greek mathematician Euclid, though he called it «the division in extreme and mean ratio,» according to mathematician 90003 George Markowsky 90004 of the University of Maine.2 = phi + 1 90005 90002 This representation can be rearranged into a 90003 quadratic equation 90004 with two solutions, (1 + √5) / 2 and (1 — √5) / 2. The first solution yields the positive irrational number 1.6180339887 … (the dots mean the numbers continue forever) and this is generally what’s known as phi. The negative solution is -0.6180339887 … (notice how the numbers after the decimal point are the same) and is sometimes known as little phi. 90005 90002 One final and rather elegant way to represent phi is as follows: 90005 90002 5 ^ 0.5 * 0.5 + 0.5 90005 90002 This is five raised to the one-half power, times one-half, plus one-half. 90005 90002 90031 Related: 90032 90003 90031 The 11 Most Beautiful Mathematical Equations 90032 90004 90005 90002 Phi is closely associated with the 90003 Fibonacci sequence 90004, in which every subsequent number in the sequence is found by adding together the two preceding numbers. This sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 and so on. It is also associated with many misconceptions.90005 90002 By taking the ratio of successive Fibonacci numbers, you can get closer and closer to phi. Interestingly, if you extend the Fibonacci sequence backward — that is, before the zero and into negative numbers — the ratio of those numbers will get you closer and closer to the negative solution, little phi -0.6180339887 … 90005 90008 Does the golden ratio exist in nature? 90009 90002 Though people have known about phi for a long time, it gained much of its notoriety only in recent centuries.Italian Renaissance mathematician Luca Pacioli wrote a book called «De Divina Proportione» ( «The Divine Proportion») in 1509 that discussed and popularized phi, according to Knott. 90005 90002 Pacioli used drawings made by 90003 Leonardo da Vinci 90004 that incorporated phi, and it is possible that da Vinci was the first to call it the «sectio aurea» (Latin for the «golden section»). It was not until the 1800s that American mathematician Mark Barr used the Greek letter Φ (phi) to represent this number.90005 90002 As evidenced by the other names for the number, such as the divine proportion and golden section, many wondrous properties have been attributed to phi. Novelist Dan Brown included a long passage in his bestselling book «The Da Vinci Code» (Doubleday, 2000), in which the main character discusses how phi represents the ideal of beauty and can be found throughout history. More sober scholars routinely debunk such assertions. 90005 90002 For instance, phi enthusiasts often mention that certain measurements of the 90003 Great Pyramid of Giza 90004, such as the length of its base and / or its height, are in the golden ratio.Others claim that the Greeks used phi in designing the Parthenon or in their beautiful statuary. 90005 90002 90005 Phi enthusiasts like to point out that the Pyramids of Giza, built between 2589 and 2504 BC, were constructed in the golden ratio. But measurements are inherently imprecise and arbitrary, so the pyramids are not accurate examples of the golden ratio. (Image credit: Dan Breckwoldt Shutterstock) 90002 But as Markowsky pointed out in his тисячі дев’ятсот дев’яносто дві paper in the 90003 College Mathematics Journal 90004, titled «Misconceptions About the Golden Ratio»: «measurements of real objects can only be approximations.Surfaces of real objects are never perfectly flat. «He went on to write that inaccuracies in the precision of measurements lead to greater inaccuracies when those measurements are put into ratios, so claims about ancient buildings or art conforming to phi should be taken with a heavy grain of salt. 90005 90002 The dimensions of architectural masterpieces are often said to be close to phi, but as Markowsky discussed, sometimes this means that people simply look for a ratio that yields 1.6 and call that phi.Finding two segments whose ratio is 1.6 is not particularly difficult. Where one chooses to measure from can be arbitrary and adjusted if necessary to get the values ​​closer to phi. 90005 90002 Attempts to find phi in the human body also succumb to similar fallacies. A recent study claimed to find the golden ratio in different proportions of the human skull. But as Dale Ritter, the lead human anatomy instructor for Alpert Medical School (AMS) at Brown University in Rhode Island, 90003 told Live Science 90004: 90005 90002 «I believe the overarching problem with this paper is that there is very little (perhaps no ) science in it … with so many bones and so many points of interest on those bones, I’d imagine there would be at least a few «golden ratios elsewhere in the 90003 human skeletal system 90004.90005 90002 90031 Related: 90032 90031 Photos: Large Numbers That Define the Universe 90032 90005 90002 And while phi is said to be common in nature, its significance is overblown. Flower petals often come in Fibonacci numbers, such as five or eight, and pine cones grow their seeds outward in spirals of Fibonacci numbers. But there are just as many plants that do not follow this rule as those that do, Keith Devlin, a mathematician at Stanford University, 90003 told Live Science 90004. 90005 90002 People have claimed that seashells, such as those of the nautilus, exhibit properties in which phi lurks.But as Devlin 90003 points out on his website 90004, «the nautilus does grow its shell in a fashion that follows a logarithmic spiral, ie, spiral that turns by a constant angle along its entire length, making it everywhere self-similar. But that constant angle is not the golden ratio. Pity, I know, but there it is. » 90005 90002 While phi is certainly an interesting mathematical idea, it is we humans who assign importance to things we find in the universe. An advocate looking through phi-colored glasses might see the golden ratio everywhere.But it’s always useful to step outside a particular perspective and ask whether the world truly conforms to our limited understanding of it. 90005 90002 90031 Additional resources: 90032 90005 .90000 Learn How to Rank in 48 Hours 90001 90002 Do you want to find keywords you can rank for in Google in the top 50 (or better) in a few days? 90003 90002 You can use a data-driven method called the Keyword Golden Ratio that will help you to find the low-hanging fruit among the keywords and leapfrog the competition. 90003 90002 90007 In this guide, you’ll find: 90008 90003 90010 90011 What is Keyword Golden Ratio and how it works 90012 90011 A case study of KGR success 90012 90011 Why 90016 ultra long tails 90017 help you skip the Google Sandbox 90012 90011 How to use a keyword tool to make the process easier 90012 90011 Some actual examples of KGR terms 90012 90011 Pitfalls and Frequently Asked Questions 90012 90025 90026 What is Keyword Golden Ratio? 90027 90002 The Keyword Golden Ratio (KGR) is a data-driven way to find keywords that are literally underserved on the internet.90003 90002 If you find a Keyword Golden Ratio phrase and you publish a piece of content targeting the keyword, your article should rank in the top 50 within a few days — sometimes hours. If your site has a decent authority and is not brand new, then you can rank in the top 15. 90003 90002 The KGR is helpful for three main reasons. 90003 90034 90011 90007 New website owners need to get traffic early to stay motivated. 90008 The KGR helps newbies get small wins faster. It’s amazing to get your first few visitors and sales.Nothing is more motivating than proving your concept by getting paid. 90012 90011 90007 You can prioritize what keywords and content you’ll publish on your site. 90008 If you’re staring at a list of thousands of keywords, then it’s pretty helpful to have a tool that can help you narrow down the list. 90012 90011 Since it’s a manual process, 90007 you have data that other people can not just pull up on a tool in a couple of minutes. 90008 That’s a valuable advantage that rewards hard work. 90012 90047 90002 By the time you finish this post, I believe you’ll be curious enough to try the KGR yourself.90003 90026 KGR success story 90027 90002 Before I dive into the specifics, let’s review a case study 90007 to prove that Keyword Golden Ratio can work to bring in more traffic and boost revenue. 90008 90003 90002 I primarily work on Amazon Affiliate sites so I can actually show earnings, not just traffic improvements. 90003 90002 In my initial full-scale test of the Keyword Golden Ratio, I published 200 posts focused on KGR keywords over five months and the results were amazing. 90003 90002 The website went 90007 from $ 100 to $ 14,853 in 12 months 90008 and 90007 the traffic increased by about 800%.90008 90003 90066 90002 Some details about the process: 90003 90010 90011 I tested the KGR with 20 posts and saw positive results so I wanted to scale up to see if I could 10x the results. 90012 90011 With the goal of 200 posts, I ramped up the publishing schedule slowly to keep out of pocket costs low. So in the first month, only 7 posts went live, but in the fifth month, I published over 60 posts. 90012 90011 The growth was gradual at first, then the traffic popped shortly after publishing all 200 posts.90012 90011 90007 No link building was done during the 12 months 90008, but the site did have some existing links beforehand via guest posting. 90012 90011 The earnings followed the traffic growth and grew to close to $ 15,000 in December. You can read the full case study here. 90012 90025 90083 90002 Curious enough? Let’s take a look at the details! 90003 90026 How is Keyword Golden Ratio calculated? 90027 90002 The Keyword Golden Ratio is this: 90003 90002 90007 The number of Google results that have the keyword phrase in the title divided by the monthly search volume, where the search volume is less than 250.90008 90003 90094 90002 90007 If the KGR is less than 0.25, then you should rank in the top 50 as soon as your page is indexed. 90008 90003 90002 You should still rank in the top 250 when the KGR is between 0.25 and 1 and it should be pretty fast. 90003 90002 90007 I strongly recommend to stick to the formula and look for KGR under 0.25 in the beginning. 90008 It is well-tested by numerous people. Later on, you can experiment with the numbers. 90003 90105 Details about the formula 90106 90002 I cap the search volume at 250 for a couple of reasons.90003 90034 90011 90007 Lower search volume terms can rank faster in Google based on my field studies and experience of others. 90008 So all else being equal, a keyword with 250 searches will rank faster than a keyword with 2500 searches per month. 90012 90011 90007 Most people target keywords with higher search volumes since there is greater profit potential. 90008 So there are more competitors in the game for the same short-tail keywords. 90012 90047 90002 The sweet spot for the KGR is under 0.25. When you multiply out the formula, it’s easier to see why you can rank in the top 50 quickly. 90003 90002 If you have a KGR of 0.25 and a maximum search volume of 250, you end up with about 63 90016 allintitle 90017 results. That means that there are about 63 pages on the web that are targeting that keyword phrase by using it in the title. 90003 90105 What is allintitle? 90106 90002 90016 Allintitle 90017 is an advanced Google search operator that outputs the results which have all the words in the meta title of the post.90003 90131 90002 90016 Allintitle 90017 search operator shows you the results that have all the searched words in the meta title 90003 90002 Do not worry if Google throttles your 90016 «allintitle» 90017 search. Usually, you can only do a certain number of them in a time period. Google does this to prevent automatic scraping of the data by robots. 90003 90002 If you exceed the limit, you’ll have to enter a Captcha to prove you’re not a bot. 90003 90142 90002 an example of a reCaptcha limitation after a certain number of 90016 allintitle 90017 searches 90003 90026 Ultra long-tail keywords and Google Sandbox 90027 90002 I assert that if you target long-tail keywords and ultra long-tail keywords, you’ll be able to get your site out of the sandbox faster.90150 90003 90002 90007 What is the sandbox? 90008 90003 90002 The Google Sandbox is the idea that it takes a while for Google to allow a new site to rank quickly. The idea has been around for years but has not been confirmed to actually exist. 90150 90003 90002 If you do not get out of the sandbox sooner, you’ll at least get traffic to your site faster. I can not prove the early exit from the sandbox since it’s not a confirmed ranking condition by Google. 90003 90002 90007 But I know dozens of people that used the KGR to get traffic to their brand new sites within weeks, not months.90008 90003 90105 Does the KGR really cut the sandbox? 90106 90002 Effectively, yes. Yet, it’s impossible to prove since there are so many variables in play. 90003 90002 But another thought experiment is to think about it the other way round: 90003 90002 90007 The high-volume keywords will be inherently more competitive. 90008 There will be established companies with websites ranking well. 90003 90175 90002 That means they are making money and they can reinvest that money to improve the content and get links and keep the ranking.90003 90002 I believe that higher search volume keywords take longer to rank even if the competition looks weak. 90003 90002 Maybe Google slows down the rate in which sites can rank based on the keyword search volume. 90003 90002 90007 Takeaway 90008 90003 90002 Go after low search volume keywords — the long tail and ultra long tail — to get traffic to your site faster. 90003 90002 Do you bust out of the Google Sandbox faster? It’s hard to say, but you will get traffic. 90003 90026 How to use a keyword tool to find KGR terms 90027 90002 Now I’ll show you a few real examples of keywords that fit the KGR.90003 90002 We’ll use an example of a niche that has been used in several public case studies — 90016 the shaver niche 90017. I try not to reveal anyone’s keywords or sites. It’s awesome to see that even in a niche that has public sites, lots of competition, and many copycats, 90007 there are many KGR options still available! 90008 90003 90105 1. Find relevant keyword ideas 90106 90002 One of my favorite ways to find KGR compliant terms is to use Google Autosuggest and this format: 90150 90003 90002 90007 90016 best 90017 90008 (product-type) 90007 90016 for 90017 90008 (application or user-type) 90003 90002 You can do this manually and it looks like this: 90003 90217 90002 But you can save a lot of time and effort by using a keyword tool such as KWFinder with advanced filtering options and the Keyword Difficulty (KD) metric.90003 90002 First, select Autocomplete and enter your search phrase, then click on the 90016 «Find Keywords» 90017 button: 90003 90224 90002 Then you’ll see the unfiltered list of results. We have quite a few, 324. 90007 We care about the Search Volume and Keyword Difficulty 90008 — the other metrics are not important now. 90003 90229 90002 For this example, I left the geography as «Anywhere,» but you can select your target location. 90003 90105 2. Filter out keywords with search volume higher than 250 90106 90002 Remember the KGR requires that 90007 the search volume is under 250.90008 That means we can set up the filter to get rid of the keywords that we do not care about. 90003 90002 Enter in 250 for the MAX search volume in the filter. 90003 90240 90002 Click the blue «Set filter» button and the list is refined by removing the higher search volumes. 90003 90243 90002 Now, we have plenty of options that have a search volume under 250. You could add more criteria to the filter such as a KD max of 15, 20, 25, etc … 90003 90002 I see the results for this Autosuggest search that 90016 «best electric shaver for ingrown hairs» 90017 has a favorable KD value at 17.The search volume is 150, which fulfills the requirement to be less than 250. 90003 90250 90002 You can hover over the row and click to copy the full keyword phrase to save time. 90003 90105 3. Find the number of websites with exact match titles 90106 90002 Next, you need to check the 90016 allintitle 90017, the advanced Google Search command. 90003 90002 90007 You type in allintitle: and the search phrase with no spaces and no quotes. 90008 Do not enter the quotes because that will lower the number of results which will throw off your calculations.90003 90002 Google returns the number of results which is the numerator in the KGR formula. 90003 90265 90002 We see there are 22 results so we can plug that into the formula. 90003 90002 KGR = (allintitle results) divided by (search volume) 90003 90002 90007 KGR = 22/90 = 0.14 90008 90003 90002 Great! That’s less than 0.14 so we have a KGR compliant term. 90003 90002 If you prefer video content, you can check out my demo of the KGR with KWFinder: 90003 90002 90279 90280 90003 90002 KWFinder also has great metrics for each keyword so you can do competition analysis without leaving the app.That’s a massive time saver! 90003 90284 90002 Looking at the same list, I found more keyword phrases. This is a very good thing to see. 90003 90287 90002 It means there are 0 90016 allintitle 90017 results. So the KGR is 0, clearly as far below 0.25 as you can get. The search volume is 20 so we’ve found a great keyword phrase. 90003 90002 Here’s one last example. 90003 90294 90002 There are 4 results. 90003 90002 KGR = (allintitle results) divided by (search volume) 90003 90002 90007 KGR = 4/30 = 0.13 90008 90003 90105 What about non-KGR terms? 90106 90002 I found a term, 90016 «best electric shaver for black men» 90017, with a low KD score of 20 which is very good. 90003 90309 90002 So I checked the allintitle. 90003 90312 90002 Only 50 results — that seems great, right ?? 90003 90002 KGR = (allintitle results) divided by (search volume) 90003 90002 90007 KGR = 50/1100 = 0.045 90008 90003 90002 90016 Is this a KGR Compliant Keyword? 90017 90003 90002 90007 The answer is no.90008 It is a fine keyword and the calculated KGR is really great. But 90007 it does not fullfill the criterium of search volume less than 250. 90008 However, part of the magic of the KGR is the ability to rank in Google quickly. 90003 90002 I believe one of the factors is the search volume so having a higher search volume will have an impact on ranking quickly. I’ll go deeper on this very topic in the FAQ section and there are exceptions to the rule. 90003 90105 Use my personal spreadsheet to check «allintitle» faster 90106 90002 You can see it’s cumbersome to click back and forth from KWFinder to Google, copy, paste, and all that.90003 90002 To help you, I created a spreadsheet to help you do this faster. 90003 90002 You can fill in the list of keywords and search volume from KWFinder. Then, 90007 the column is auto-populated with a hyperlink that contains the allintitle plus the keyword for you. 90008 90003 90002 After you fill in the number 90016 allintitle 90017 results, 90007 the KGR is automatically calculated. 90008 The cell will turn green, yellow, or red based on the KGR value. 90003 90349 90002 Do not forget to make your own copy so you can edit the spreadsheet.90003 90352 90002 Get the spreadsheet on Google Sheets 90003 90105 Keyword Difficulty vs. the Keyword Golden Ratio — which is better? 90106 90002 Both Keyword Difficulty and Keyword Golden Ratio are ways to help you find keywords you can rank for easily. So you may ask — which one is better? 90003 90002 The thing is that 90007 you do not have to choose between the two because the KD and KGR can both be correct. 90008 They look at different factors to gauge the competition for a given keyword.90003 90002 At the core, here is the difference: 90003 90002 90007 The KD metric from KWFinder is based on the Link Profile Strength (LPS) of the web pages that are currently ranking, while the KGR is based on sites that have the keyword in the title. 90008 90003 90002 So KD takes mainly the following into account: 90003 90010 90011 Domain Authority and Page Authority from MOZ 90012 90011 Citation Flow and Trust Flow from Majestic 90012 90025 90002 The KGR helps us see if a webmaster is specifically targeting the keyword phrase or not.If the webmaster does not put the phrase in the title, then it’s safe to assume they are not intentionally targeting the keyword. 90003 90002 90007 Both are useful metrics in gauging the competition even though they come at the solution from different directions. 90008 90003 90002 If you have a keyword phrase that is KGR compliant with a low KD metric, then you probably found a gem of a keyword. 90003 90002 If your keyword phrase has poor KGR and KD metrics, it’s probably a tough keyword phrase to rank.90003 90026 Pitfalls with the KGR 90027 90002 The KGR is a fantastic tool and can help you get traffic fast. But it’s not all rainbows and gummy worms. Here are a few things to consider and remember about the Keyword Golden Ratio. 90003 90105 Keyword stuffing 90106 90002 Do not use the keyword phrase too often in the content. Usually, KGR terms will be longer than normal keyword phrases, sometimes 4 or more words. 90003 90002 Longer phrases should not be used very often in the text. It would be really weird and unnatural to use a long keyword phrase 8 times in a post.90003 90002 For example, it would look weird if you used «best ballpoint pen for journaling» more than a couple times. 90003 90002 90007 My rule of thumb is to use a KGR phase once in the title, then once in the main content. 90008 You can use other partial match keywords, related phrases, and LSI keywords as normal. 90003 90002 That’s it. Do not overdo it. 90003 90002 90007 Pro tip: 90008 People often follow the keyword density suggestions from Yoast SEO plugin. However, it may suggest to use the keyword way too many times.Do not forget that it’s just a tool created for an average content creator. You are not average — the Keyword Golden Ratio is an advanced technique. 90150 90003 90105 Keywords that may not rank 90106 90002 Sometimes a KGR term is not going to rank well, even if all the metrics tell you it’s a good keyword. 90003 90002 90016 Why is that? 90017 90003 90002 Google has figured out what the end user, the searcher, is interested in. 90003 90002 For example, let’s say you have an affiliate site so you want to see a few similar sites on the first page of the search results.90003 90002 If you find a KGR term, then you should Google the term before you write and publish the content. Check out the results to see what Google is delivering to the searcher — what is the user’s search intent. 90003 90002 If you see mostly e-commerce sites in the results, then searchers tend to like those results better than affiliate sites. 90003 90426 90002 If you see YouTube videos in the majority of the results, that means searchers want to see a video. 90003 90002 If the term happens to be medical or health related, then you’re likely to see big authoritative sites like Mayo Clinic, WebMD, medical journals, and other sites with a huge amount of trust in the medical field.90003 90002 The takeaway is that Google tells you what sites are ranking in the search results. All you have to do is look carefully. 90003 90002 90007 If you do not see any sites that are similar to yours, then that’s a bad sign 90008 — a sign that even a KGR with great metrics is likely not going to rank. 90003 90105 Difficulty finding KGR complaint terms 90106 90002 Finding keywords is a skill, and it’s one that you can get better at. So, finding KGR terms is going to be hard at first.90003 90002 Using tools like KWFinder can make it much easier, of course, but it comes down to putting the time in. 90003 90002 Part of the magic of the KGR is that you can not just buy a tool that will do it. You have to look for the keywords manually. 90003 90105 Trying only one KGR compliant post 90106 90002 The KGR works really well at scale. I suggest you try 20 KGR terms to test out your site. 90003 90002 90007 I’ve found that about 5% of KGR terms perform better than expected, 15% worse than expected, and 80% rank somewhere in the top 30 — 50 results.90008 90003 90002 The 80% tend to move higher in rankings over time depending on the quality of the content, your site’s authority, and the relative competition. 90003 90002 90007 So if you only publish one KGR compliant post, it may not rank. 90008 You might think it’s a failure. But if you publish enough posts to give it a fair try, you’ll see the magic. 90003 90026 FAQs about the Keyword Golden Ratio 90027 90105 Can I automate finding KGR terms? Is there a tool that will do it for me? 90106 90002 No, not any tool I know of.The Captcha introduces an issue for developers looking to automate this. 90003 90002 Configuring the app would be cumbersome so support for such an app would kill the margins to make it profitable. Go for it if you’re interested! 90003 90105 What if keyword research tools have different search volumes? 90106 90002 Pick one tool and stick with it. All keyword search volumes are estimates — even if you get them directly from Google. 90003 90002 All keyword research tools use different algorithms to generate search volume estimates.So the exact search volume does not really matter too much. 90003 90002 90007 What is important is the relative volume compared to other keywords, i.e. 90016 keyword X 90017 gets 3 times more searches than 90016 keyword Y 90017. 90008 90003 90002 The Keyword Golden Ratio can perform in spite of the inexact nature of keyword search volumes. 90003 90105 I found a term with a KGR under 0.25, but the search volume is over 250. 90106 90002 Go for it. Technically, this is not a KGR term based on the definition, like the example above.90003 90002 However, you have found a keyword with a smaller amount of competition so that’s a good thing. 90003 90002 90007 You may not rank as quickly or at all, but that goes for just about any keyword where you are not looking at the competition levels. 90008 90003 90002 There are 2 main things that you need to think about: 90003 90034 90011 Let’s say the search volume for the term you found is 10,000. If the KGR is 0.25, that means that you have 2,500 other sites that have the keyword phrase in the title and probably intentionally targeting that keyword.90012 90011 With the strict KGR definition of targeting KGR values ​​under 0.25 and a search volume of under 250, you’re only competing with a maximum of 62.5. 90012 90047 90002 So there is way more competition and it’s simple math to see that. 90003 90002 I contend that higher search volume keywords take longer to rank for than low search volumes regardless of the competition numbers. 90003 90105 Are there exceptions for the KGR? 90106 90002 If your site is not brand new and has some authority via solid backlinks, then you can use the KGR as a guide, not a strict rule.90003 90002 You can be a bit looser with the formula and go for search volumes that are higher, like 1,000 searches per month. Or, go for KGR values ​​up to .8 or 1.5 — it’s up to you to test to see how your site will be treated. 90003 90105 Where can I learn more about the KGR? 90106 90002 I created a free masterclass on YouTube. It’s 100% free and organized in a playlist. 90003 90026 Conclusion 90027 90002 The Keyword Golden Ratio is one of the best SEO techniques for beginners. It can change the way you look at long-tail keywords so that you can be confident that your content will get the attention it deserves.90003 90002 90007 It takes skill and effort to find KGR terms, but it’s worthwhile and can help separate you from your competition. 90008 90003 90002 If you’re starting a brand new site, the KGR can help you get traffic to your site before it is out of the Google sandbox. 90003 90002 It’s free to try — so it’s up to you from here. 90003 .90000 How to Use the Golden Ratio in Design (with Examples) 90001 90002 90003 90004 90002 Want to be on the same creative level as Leonardo Da Vinci, Salvador Dali and the designers of the Parthenon? They all have one simple concept in common. 90004 90002 The Ancient Greeks were one of the first to discover a way to harness the beautiful asymmetry found in plants, animals, insects and other natural structures. They expressed this mathematical phenomenon with the Greek letter phi, but today, we call it the 90008 golden ratio 90009 -also known as the divine proportion, the golden mean, and the golden section.90004 90002 Much like the rule of thirds, this mathematical concept can be applied to your graphic designs to make them more visually appealing to the viewer. The golden ratio is a little more complicated, so we recommend you first read our guide to the rule of thirds if math is not your forte. 90004 90013 What is the golden ratio? 90014 90002 The golden ratio is probably best understood as the proportions 90008 1: 1.618 90009. Of course, the mathematical equation at work here is much more complicated than that.90004 90002 The ratio itself is derived from the 90008 Fibonacci sequence 90009, a naturally occurring sequence of numbers that can be found practically everywhere in nature, from the number of leaves on a tree to the spiral shape of a seashell. It can also be found in famous works of art and architecture and even in our own faces. 90004 90002 The Fibonacci sequence is easy to remember. Starting with 0 and 1, add the last number of the sequence to the number that came before it to create the next number in the sequence.So it goes 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on to infinity. From the Fibonacci sequence, the Greeks developed the golden ratio to better express the difference between any two numbers in succession within the sequence. 90004 90002 90026 90004 90002 The golden ratio is not exact when it comes to the Fibonnacci sequence-the difference between two numbers on the sequence is not always exactly equal to the golden ratio, but it’s pretty close. 90004 90030 Creating a golden rectangle 90031 90002 So now that we understand the basic numbers at play, here’s a more advanced technique for using those numbers in a more visual way.A «golden rectangle» is one that fits the parameters of the golden ratio-but the more times you divide a golden rectangle according to the golden ratio, the more useful it becomes. 90004 90002 Let’s start by creating a rectangle with golden proportions. To make this simple, we’ll start with a width of 1000 pixels and divide it by 1.618 to get a height of about 618 pixels. 90004 90002 90037 90004 90002 Now add a 618 x 618 square on the right side of the canvas, leaving behind a 382 x 618 rectangle on the left side-another golden rectangle! 90004 90002 90042 90004 90002 If you take that new rectangle and create another square within it, you’ll end up with another golden rectangle in the leftover space, which you can then divide up again, and so on and so forth.90004 90002 90047 90004 90002 Notice how each time you divide your golden rectangle, the largest dividing line kind of spirals in onto itself? That’s no accident-it forms the shape of a «golden spiral,» one of the more ubiquitous shapes that you’ll deal with when working with the golden ratio. 90004 90002 90052 90004 90030 Visualizing the golden ratio with other shapes 90030 90031 90031 90002 The golden rectangle is the simplest (and arguably the most useful) way to visualize the golden ratio, but you can also use circles and triangles in a very similar way.90004 90002 For instance, you can create an approximate golden spiral shape out of circles-and those circles fit perfectly inside a system of golden rectangles. 90004 90002 90063 90004 90002 There’s also such a thing as a «golden triangle,» an isosceles triangle with two equal sides and one distinctive side that are in golden proportion to one another. When these triangles are nested inside of one another, it creates the exact same «golden spiral» shape. 90004 90002 90068 90004 90030 Golden proportions are ubiquitous 90031 90002 The golden ratio is not just some lofty mathematical theory; it shows up all the time in the real world.While many of the most often-cited examples of the golden ratio have been debunked, there are still plenty of them throughout nature and in man-made works of art. 90004 90002 90075 90004 90002 Keep in mind that the golden ratio is based on irrational numbers, so many of these examples do not 90078 exactly 90079 conform to the golden ratio. And even when the golden ratio can be found in a work of art, that does not necessarily mean the artist intentionally included it as part of some grand design.90004 90002 Though there are those who would argue otherwise, the golden ratio probably does not have any mystical powers of beauty drawn from the primordial fabric of the universe. But it does seem likely that this ubiquitous pattern has some aesthetically appealing properties and tends to suggest a sense of natural balance and visual harmony. 90004 90013 Using the golden ratio in graphic design 90014 90085 90002 Photo Credit: The Fish of Life 90004 Building your graphic design around a golden rectangle or a Fibonacci sequence takes some real art mastery, but any designer can use it as a general guideline to add tweaks and improvements to their design.The more you get used to the golden ratio, the more you’ll find that it’s merely showing you where the best parts of your design are-it’s not necessarily dictating how you should design your piece. 90002 Sometimes you’ll find that the golden ratio suits a design perfectly, while other times, you’ll come up with clever ideas that do not necessarily adhere to the rules. That’s fine-for as many designs you’ll find out there in the world that fulfill the golden ratio, you’ll find nearly an equal amount of designs that do not.It’s not a make-or-break rule; it’s just another tool you can add to your designer’s toolkit. 90004 90030 Typography 90031 90002 The easiest way to start using the golden ratio is to implement it within your typographical graphic design elements. For example, let’s say that you’re using 10pt font for the body text. Using the golden ratio, you can determine the best size for the headings by multiplying by 1.618. 90004 90002 So, 10 × 1.618 = 16.18, which you can round down to 16pt font. 90004 90002 Or let’s say your headline text is 20pt and you want to find an appropriate size for your body text.Since the headline text is the bigger element, you would divide by 1.618 instead of multiplying. 90004 90002 Therefore, 20 / 1.618 = 12.36, which you could round down to 12pt body text (or up to 13pt, if you so choose). That’s the beauty of the golden ratio-you do not have to be perfect. 90004 90002 90101 90004 90030 Sizing / cropping images 90031 90002 Naturally, a simple way to incorporate the golden ratio into a design is to crop photos (or any other images you may use) into a golden rectangle shape.Again, that does not mean you should always do it for 90078 every 90079 photo, but you may want to consider it for an image that’s particularly central to your design. 90004 90002 90110 90004 90002 When you crop images with the golden ratio in mind (especially when working with photography), you might also consider using the golden spiral as a guide for the shot’s composition. For example, you could crop a photo to golden proportions in such a way that the main focal point of the image is at the center of the corresponding golden spiral.It adds interest in a way that’s very similar to the rule of thirds, but many consider it a more natural-looking, aesthetically pleasing choice. 90004 90002 90115 90004 90002 Using the golden ratio in your visual designs can also be as simple as applying it to the proportional size difference between two different elements-even elements that are not «golden» shapes themselves. 90004 90002 For example, say you had an image in your design that was 2 inches wide and you wanted to pair it with a smaller picture.A 2 inch image divided by 1.618 comes out to about 1.236 inches, which you could safely round down to 1.2 inches. 90004 90002 You could also add a larger image to the design, which would require you to multiply your 2 inch photo by the golden ratio to end up with roughly 3.2 inches. And now you’ve got yourself a Fibonacci sequence going, creating even more interest than before. 90004 90002 90124 90004 90030 Logo design 90031 90002 You can also use the golden ratio to add aesthetic appeal directly to a company’s branding.Even if the logo itself is not shaped like a golden rectangle or triangle, it can still employ elements that use golden proportions. 90004 90002 90131 90004 90030 General layout 90031 90002 Once again, using a golden spiral to inform your graphic design’s layout is a lot like using the rule of thirds grid-you want the focus of the design to be centered on the spiral, using the golden rectangles as division lines for the placement of visual elements. But unlike the rule of thirds grid, you can move the golden rectangle around to suit your needs.90004 90002 After all-if you could not move around a golden rectangle, then every design would look like a golden rectangle or spiral. Instead, think of your golden rectangle as a ruler-it does not change, but you can move it around the canvas to measure out the elements that are already there. 90004 90002 90140 90004 90002 Digital designers have to work within tighter constraints when it comes to creating a layout; they can not control the dimensions of the audience’s monitor, after all. A common trick in web design is to use the golden ratio to divide space between the body of the website and the sidebar.Taking the measurements of the space they’re working with, web designers can ensure that the body is 1.618 times larger than the sidebar by taking the total width of the canvas, dividing it by 1.618, and then subtracting that number from the overall width of the canvas. 90004 90002 90145 90004 90002 The same technique can still apply to print design-but you have to be careful. Web designers are working within a horizontal medium, and much of print design is vertically oriented. Of course, pretty much anything that is printed vertically can also be printed horizontally-but you will not always have the option.90004 90002 The advantage to working in print is that the size of the media itself can be measured out according to the golden ratio. However, there are also times when print designers are constrained to a standard size and unable to use custom print options. Thankfully, you can still apply the golden ratio to the layout of any print template; you just have to be smart about it. 90004 90002 The good thing about presentation folders and other print materials that open up is that they give you both a vertically and horizontally aligned canvas to work with.Whenever you open up a presentation folder, the combined interior creates one big horizontal rectangle. Therefore, it’s fairly easy to have interior elements based on the golden ratio, such as the size and placement of the pockets. 90004 90002 90154 90004 90002 You could also implement the same kind of sidebar idea that web designers use by having a custom-made info flap inside the folder. Even the position of your printed design elements inside the folder can have an aesthetic appeal to the viewer if you place them according to the golden proportion.90004 90013 Golden ratio tools 90014 90002 We did not want to let you go home from this golden ratio party without a few party favors to take home. We gathered up some of the best tools and apps to help you incorporate the golden ratio into your design. 90004 90162 90163 Atrise Golden Section — This handy add-on lets you place a golden proportion grid right over your design software of choice allowing you to measure out your design according to the golden ratio. The newest version of Photoshop already has golden ratio and spiral overlay options built-in, but this tool is very useful if you’re using an older version or a different program.90164 90163 Golden Calipers — Print is a physical medium, so if you like working with your hands, you might want to pick up a pair of golden calipers-a measurement tool designed specially to help you design according to the golden ratio. 90166 90167 90164 90163 90008 Golden Ratio App 90009 — For $ 2.99 on the Mac App Store, you can have this simple but effective app that lets you easily create golden ratio layouts. 90008 (Update: No longer available) 90009 90164 90163 Golden Ratio Typography App — Ensure that your typography is proportionate by creating columns of body text that follow the golden ratio.90164 90163 Phi Calculator — Designers who use the golden ratio often find themselves constantly reaching for their calculators. Reach for the right one-one that automatically makes golden ratio calculations. 90164 90163 Phi Matrix — This computer software for Windows and Mac gives you the power to apply golden ratio rectangles over any image you can conjure up on your computer with any software-even your web browser or word processor. 90164 90181 90002 The degree to which you rely on the golden ratio is up to you, but even the slightest application of its proportions can really add appeal to your designs.If you want to be extra sure your design is up to snuff, try measuring it up to both the golden ratio and the rule of thirds. If your design satisfies both, you could have something great on your hands. 90004 90078 Click the image below to view it full-size. 90079 90186 90187 90004 90002 The golden ratio is a really fascinating subject and we’re completely geeked out about how often it appears in nature, classic art, architecture, logo design and more. We want to see your best examples of the golden ratio in action, so 90008 leave a comment below 90009 with your samples-especially if they’re designs you created yourself.90004 .90000 The golden ratio and aesthetics 90001 90002 November 2002 90003 90002 90005 Mario Livio is a scientist and self-proclaimed «art fanatic» who owns many hundreds of art books. Recently, he combined his passions for science and art in two popular books, 90006 The Accelerating Universe 90005, which appeared in 2000, and 90006 The Golden Ratio 90005, reviewed in this issue of 90005 Plus 90006. The former book discusses «beauty» as an essential ingredient in fundamental theories of the universe.The latter discusses the amazing appearances of the peculiar number 1.618 … in nature, the arts, and psychology. Here he gives us a taster. 90006 90003 90014 The origins of the divine proportion 90015 90002 In the 90005 Elements 90006, the most influential mathematics textbook ever written, Euclid of Alexandria (ca. 300 BC) defines a proportion derived from a division of a line into what he calls its «extreme and mean ratio.» Euclid’s definition reads: 90003 90020 90005 A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.90006 90023 In other words, in the diagram below, point 90005 C 90006 divides the line in such a way that the ratio of 90005 AC 90006 to 90005 CB 90006 is equal to the ratio of 90005 AB 90006 to 90005 AC 90006. Some elementary algebra shows that in this case the ratio of 90005 AC 90006 to 90005 CB 90006 is equal to the irrational number 1.618 (precisely half the sum of 1 and the square root of 5). 90038 90002 C divides the line segment AB according to the Golden Ratio 90003 90002 Who could have guessed that this innocent-looking line division would have implications for numerous natural phenomena ranging from the leaf and seed arrangements of plants to the structure of the crystals of some aluminium alloys, and from the arts to the stock market? 90003 90043 90002 Sunflower seeds 90003 90002 In fact, with the increasing realization of the astonishing properties of this number over the centuries since Euclid’s definition, the number was given the honorifics «Divine Proportion» and «Golden Ratio.»90003 90002 Here I shall concentrate only on one of the surprising (claimed) attributes of the Golden Ratio — its presumed association with aesthetics, since it provides a wonderful example of an attempt to mingle mathematics with the arts. 90003 90014 The golden ratio in the arts 90015 90002 Many books claim that if you draw a rectangle around the face of Leonardo da Vinci’s 90005 Mona Lisa 90006, the ratio of the height to width of that rectangle is equal to the Golden Ratio. No documentation exists to indicate that Leonardo consciously used the Golden Ratio in the 90005 Mona Lisa 90006 ‘s composition, nor to where precisely the rectangle should be drawn.Nevertheless, one has to acknowledge the fact that Leonardo was a close personal friend of Luca Pacioli, who published a three-volume treatise on the Golden Ratio in 1509 (entitled 90005 Divina Proportione 90006). 90003 90060 90002 Sacrament of the Last Supper, by Salvador Dali 90003 90002 Another painter, about whom there is very little doubt that he actually did deliberately include the Golden Ratio in his art, is the surrealist Salvador Dali. The ratio of the dimensions of Dali’s painting 90005 Sacrament of the Last Supper 90006 is equal to the Golden Ratio.Dali also incorporated in the painting a huge dodecahedron (a twelve-faced Platonic solid in which each side is a pentagon) engulfing the supper table. The dodecahedron, which according to Plato is the solid «which the god used for embroidering the constellations on the whole heaven,» is intimately related to the Golden Ratio — both the surface area and the volume of a dodecahedron of unit edge length are simple functions of the Golden Ratio. 90003 90002 These two examples are only the tip of the iceberg in terms of the appearances of the Golden Ratio in the arts.The famous Swiss-French architect and painter Le Corbusier, for example, designed an entire proportional system called the «Modulor,» that was based on the Golden Ratio. The Modulor was supposed to provide a standardized system that would automatically confer harmonious proportions to everything, from door handles to high-rise buildings. But why would all of these artists (there are many more than mentioned above) even consider incorporating the Golden Ratio in their works? The attempts to answer this question have led to a long series of psychological experiments, designed to investigate a potential relationship between the human perception of «beauty» and mathematics.90003 90014 Is beauty in the eye of the beholder? 90015 90071 90002 The pioneering (rather crude) experiments in this field were conducted by the German physicist and psychologist Gustav Theodor Fechner in the 1860s. Fechner’s experiment was simple: ten rectangles varying in their length-to-width ratios were placed in front of a subject, who was asked to select the most pleasing one. The results showed that 76% of all choices centered on the three rectangles having ratios of 1.75, 1.62, and 1.50, with a peak at the «Golden Rectangle» (with ratio 1.62). Fechner went further and measured the dimensions of thousands of rectangular-shaped objects (windows, picture frames in the museums, books in the library), and claimed (in his book 90005 Vorschule der Aesthetik 90006) to have found the average ratio to be close to the Golden Ratio. 90003 90002 Many psychologists have repeated similar experiments since then, and obtained rather conflicting results. Michael Godkewitsch of the University of Toronto, for example, pointed out that 90005 average 90006 group preferences often do not reflect the most preferred choice.For example, the brand of tea that everybody ranks second-best may on the average be rated best, but nobody will ever buy it. Godkewitsch therefore noted that 90005 first choices 90006 provide a more meaningful measure of preference than averages. Godkewitsch concluded from a study conducted in 1974 that the preference for the Golden Rectangle reported in the earlier experiments was an artifact of the rectangle’s 90005 position 90006 in the range of rectangles presented to the subjects.He noted: «The basic question whether there is or is not, in the Western world, a reliable verbally expressed aesthetic preference for a particular ratio between length and width of rectangular shapes can probably be answered negatively. «90003 90002 Other experiments, however, gave different results. In particular, British psychologist Chris McManus concluded in 1980 that: «There is moderately good evidence for the phenomenon which Fechner championed.» Nevertheless, McManus acknowledged that «whether the Golden Section [another name for the Golden Ratio] 90005 per se 90006 is important, as opposed to similar ratios (e.g. 1.5, 1.6 or even 1.75), is very unclear. «90003 90088 90002 The entire topic received a new twist with a flurry of psychological attempts to determine the origin of facial attractiveness. For example, psychologist Judith Langlois of the University of Texas at Austin and her collaborators tested the idea that a facial configuration that is close to the population 90005 average 90006 is fundamental to attractiveness. Langlois digitized the faces of male and female students and mathematically averaged them, creating two-, four-, eight-, sixteen-, and thirty-two-face composites.College students were then asked to rate the individual and composite faces for attractiveness. Langlois found that the 16- and 32-averaged faces were rated significantly higher than individual faces. Langlois explained her findings as being broadly based on natural selection (Physical characteristics close to the mean having been selected during the course of evolution), and on «prototype theory» (prototypes being preferred over non-prototypes). 90003 90002 Science writer Eric Haseltine claimed (in an article in 90005 Discover 90006 magazine in September 2002) to have found that the distance from the chin to the eyebrows in Langlois’s 32-composite faces divides the face in a Golden Ratio.A similar claim was made in тисяча дев’ятсот дев’яносто чотири by orthodontist Mark Lowey, then at University College Hospital in London. Lowey made detailed measurements of fashion models ‘ faces. He asserted that the reason we classify certain people as beautiful is because they come closer to Golden Ratio proportions in the face than the rest of the population. 90003 90002 Many disagree with both Langlois’s and Lowey’s conclusions. Psychologist David Perret of the University of St. Andrews, for example, published in тисячі дев’ятсот дев’яносто чотири the results of a study that showed that 90005 individual 90006 attractive faces were preferred to the composites.Furthermore, when computers were used to exaggerate the shape differences away from the average, those too were preferred. Perret claimed to have found that his beautiful faces did have something in common: higher cheek bones, a thinner jaw, and larger eyes relative to the size of the face. 90003 90002 An even larger departure from the «averageness» hypothesis was found in a study by Alfred Linney from the Maxillo Facial Unit at University College Hospital. Using lasers to make precise measurements of the faces of top models, Linney and his colleagues found that the facial features of the models were just as varied as those in the rest of the population.90003 90002 I will certainly not attempt to make the ultimate sense of sex appeal in an article on the Golden Ratio. I would like to point out, however, that the human face provides us with hundreds of lengths to choose from. If you have the patience to juggle and manipulate the numbers in various ways, you are bound to come up with some ratios that are equal to the Golden Ratio. 90003 90002 Furthermore, I should note that the literature is bursting with false claims and misconceptions about the appearance of the Golden Ratio in the arts (e.g. in the works of Giotto, Seurat, Mondrian). The history of art has nevertheless shown that artists who have produced works of truly lasting value are precisely those who have departed from any formal canon for aesthetics. In spite of the Golden Ratio’s truly amazing mathematical properties, and its propensity to pop up where least expected in natural phenomena, I believe that we should abandon its application as some sort of universal standard for «beauty,» either in the human face or in the arts.90003 90107 90108 About the author 90109 90110 90002 Dr. Mario Livio is the Head of the Science Division at the Space Telescope Science Institute (STScI), the institute which conducts the scientific program of the Hubble Space Telescope. He received his Ph.D. in theoretical astrophysics from Tel Aviv University in Israel, was a professor in the Physics Dept. of the Technion-Israel Institute of technology from 1981 till 1991 року, and joined STScI in 1991. Dr Livio has published over 300 scientific papers and received numerous awards for research and for excellence in teaching.90003 90002 His interests span a broad range of topics in astrophysics, from cosmology to the emergence of intelligent life. Dr. Livio has done much fundamental work on the topic of accretion of mass onto black holes, neutron stars, and white dwarfs, as well as on the formation of black holes and the possibility to extract energy from them. During the past two years Dr. Livio’s research focused on supernova explosions and their use in cosmology to determine the rate of expansion of the universe.In particular, he has shown that in spite of some uncertainties that still exist in theoretical models for supernovae, it is very likely that the recent findings that the expansion of our universe is accelerating are correct. 90003 .