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*An exploration with the golden ratio offers opportunities to connect an understanding the conceptions of ratio and proportion to geometry. The mathematical connections between geometry and algebra can be highlighted by connecting Phi to the Fibonacci numbers and some golden figures. Also, the golden ratio is a good topic to introduce historic and aesthetic elements to a mathematical concept, because we can find that not a few artists and architects were connected with the golden ratio of their works through much of the art history.*

- Most Memorable Golden Ratio Examples in Modern Art
- A designer's guide to the Golden Ratio
- How to Use the Golden Ratio to Create Gorgeous Graphic Designs
- How to Use the Golden Ratio to Create Gorgeous Graphic Designs

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## Most Memorable Golden Ratio Examples in Modern Art

This section introduces you to some of the occurrences of the Fibonacci series and the Golden Ratio in architecture, art and music. The ancient Greeks knew of a rectangle whose sides are in the golden proportion 1 : 1. It occurs naturally in some of the proportions of the Five Plato nic Solids as we have already seen.

A construction for the golden section point is found in Euclid 's Elements. The golden rectangle is supposed to appear in many of the proportions of that famous ancient Greek temple, the Parthenon, in the Acropolis in Athens , Greece but there is no original documentary evidence that this was deliberately designed in. There is a replica of the original building accurate to one-eighth of an inch!

Its most famous monument is the Parthenon, a temple to the goddess Athena built around or BC. It is largely in ruins but is now undergoing some restoration see the photos at Roy George's site in the link above. Again there are no original plans of the Parthenon itself. It appears to be built on a design of golden rectangles and root-5 rectangles: the front view see diagram above : a golden rectangle, Phi times as wide as it is high the plan view: 5 as long as the front is wide so the floor area is a square-root-of-5 rectangle However, due to the top part being missing and the base being curved to counteract an optical illusion of level lines appearing bowed, these are only an approximate measures but reasonably good ones.

The Panthenon image here shows clear golden sections in the placing of the three horizontal lines but the overall shape and the other prominent features are not golden section ratios. David Silverman's page on the Parthenon has lots of information.

Look at the plan of the Parthenon. The dividing partition in the inner temple seems to be on the golden section both of the main temple and the inner temple.

Apart from that, I cannot see any other clear golden sections - can you? Austell , between Plymouth and Penzance in SW England and 50 miles from Land's End, has some wonderfully impressive greenhouses based on geodesic domes called biomes built in an old quarry.

It marks the Millenium in the year and is now one of the most popular tourist attractions in the SW of England. The logo shows the pattern of panels on the roof. What is million years old, weights 70 tonnes and is the largest of its type in the world? Peter Randall-Paige's design is based on the spirals found in seeds and sunflowers and pinecones. California Polytechnic Engineering Plaza The College of Engineering at the California Polytechnic State University have plans for a new Engineering Plaza based on the Fibonacci numbers and several geometric diagrams you will also have seen on other pages here.

There is also a page of images of the new building. The designer of the Plaza and former student of Cal Poly, Jeffrey Gordon Smith, says As a guiding element, we selected the Fibonacci series spiral, or golden mean, as the representation of engineering knowledge. The start of construction is currently planned for late or early in The United Nations Building in New York The architect Le Corbusier deliberately incorporated some golden rectangles as the shapes of windows or other aspects of buildings he designed.

Although you will read in some books that "the upright part of the L has sides in the golden ratio, and there are distinctive marks on this taller part which divide the height by the golden ratio", when I looked at photos of the building, I could not find these measurements. Can you? It has many images of the Parthenon, pictures of its friezes and other details. Note: the images cannot be copied or even made into links, only viewed on their page!

June Komisar's page of architectural links from the University of Michigan. She points to the Great Building Collection which has some excellent photo images on their Parthenon page. You can take your own virtual walk through the Parthenon! The Kings Tomb in Egypt and the golden section. Many books on oil painting and water colour in your local library will point out that it is better to position objects not in the centre of the picture but to one side or "about one-third" of the way across, and to use lines which divide the picture into thirds.

This seems to make the picture design more pleasing to the eye and relies again on the idea of the golden section being "ideal". It seems to have been designed with some clear golden sections as I've shown on Rob's picture: Show golden sections on the picture : The figure of Christ is framed by an oval with a flattened top.

At the golden section point vertically is the navel indicated at the narrowest part of the waist and also the lower edge of the girdle belt or waist-band , shown by blue arrows.

The bottom the the girdle waist-band is also at a golden section point for the whole figure from the top of the head to the soles of the feet, shown by purple arrows. Since this is also the position of the navel in the human body, this indicates the figure is standing. The top of the girdle and the line of the chest are at golden sections between the base of the girdle and the top of the garment the shoulders shown by red arrows.

The face also has several golden sections in it, the line of the eyes and the nostrils being at the major golden sections, shown by yellow lines. The two ovals forming the apron and the face are positioned vertically at golden section points apart and at golden sections in size as shown by the green arrows. The other two ovals, the sleeves, have a width that is 0.

Can you find any more golden sections? Links: More information on the tapestry. Take a virtual tour of the Cathedral. There is a very useful set of mathematical links to Art and Music web resources from Mathematics Archives that is worth looking at. Links to major sources of Art on the Web : Top9. Highly recommended! It includes art from Ancient to Modern, from paintings to ceramics and textiles, from all over the world as well as America. This site has links to several sources and images of his works and some links to sites on the golden section.

Why not visit the Leonardo Museum in the town of Vinci Italy itself from which town Leonardo is named, of course.

Let your mouse rest on their names to see their email addresses. Their two designs are based on the pattern in the middle where the strips in the lower half are of widths 1, 2, 3, 5, 8 and 13 in brown which are alternated with lighter strips of the same widths but in decreasing order.

Woolly Thoughts is Steve Plummer and Pat Ashforth's web site with many maths inspired knitting and crochet projects, including designs based on Fibonacci numbers, the golden spiral, pythagorean triangles, flexagons and much much more. They have worked for many years in schools giving a new twist to mathematics with their hands-on approach to design using school maths.

An excellent resource for teachers who want to get students involved in maths in a new way and also for mathematicians interested in knitting and crochet. Billie Ruth Sudduth is a North American artist specialising in basket work that is now internationally known.

Mathematics Teaching in the Middle School has a good online introduction to her work January Kees van Prooijen of California has used a similar series to the Fibonacci series - one made from adding the previous three terms, as a basis for his art.

Fibonacci and Phi for fashioning Furniture Pietro Malusardi and Karen Wallace have a web page showing some elegant applications of the golden section in furniture design. Custom Furniture Solutions have a Media cabinet designed using golden section proportions. Fletcher Cox is a craftsman in wood who has used the golden section in his birds-eye maple wooden plate.

In Sanskrit poetry syllables are are either long or short. In English we notice this in some words but not generally - all the syllables in the song above take about the same length of time to say whether they are stressed or not, so all the lines take the same amount of time to say.

However cloudy sky has two words and three syllables CLOW-dee SKY , but the first and third syllables are stressed and take a longer to say then the other syllable. Let's assume that long syllables take just twice as long to say as short ones. So we can ask the question: in Sanskrit poetry, if all lines take the same amount of time to say, what combinations of short S and long L syllables can we have?

This is just another puzzle of the same kind as on the Simple Fibonacci Puzzles page at this site. Four would seem reasonable - but wrong! It's five! Acarya Hemacandra and the so-called Fibonacci Numbers Int. Duckworth argues that Virgil consciously used Fibonacci numbers to structure his poetry and so did other Roman poets of the time.

Trudi H Garland's [see below] points out that on the 5-tone scale the black notes on the piano , the 8-tone scale the white notes on the piano and the notes scale a complete octave in semitones, with the two notes an octave apart included.

However, this is bending the truth a little, since to get both 8 and 13, we have to count the same note twice C C in both cases. Yes, it is called an oct ave, because we usually sing or play the 8th note which completes the cycle by repeating the starting note "an octave higher" and perhaps sounds more pleasing to the ear.

But there are really only 12 different notes in our octave, not 13! Baginsky's method of constructing violins is also based on golden sections.

Did Mozart use the Golden mean? He reports on John Putz's analysis of many of Mozart's sonatas. John Putz found that there was considerable deviation from golden section division and that any proximity to golden sections can be explained by constraints of the sonata form itself, rather than purposeful adherence to golden section division.

The Mathematics Magazine Vol 68 No. Phi in Beethoven's Fifth Symphony? This he does by ignoring the final 20 bars that occur after the final appearance of the motto and also ignoring bar Have a look at the full score for yourself at The Hector Berlioz website on the Berlioz: Predecessors and Contemporaries page, if you follow the Scores Available link.

A browser plug-in enables you to hear it also. Note that the repeated bars at the beginning are not included in the bar counts on the musical score. Tim Benjamin for points out that But there are bars and not ! Therefore the golden section points actually occur at bars shown as bar as the counts do not include the repeat and similarly marked as bar As UK composer Tim Benjamin points out: The bars are comprised of a repeated section of bars - so that's the first bars in the repeated section, the "exposition" - followed by of "development" section, then a 24 bar "recapitulation" standard "first movement form".

Therefore there can't really be anything significant at , because that moment happens twice. However at , there is something pretty odd - this inversion of the main motto. You have some big orchestral activity, then silence, then this quiet inversion of the motto, then silence, then big activity again. Also you have to bear in mind that bar numbers start at 1, and not 0, so you would need to look for something happening at This is in fact what happens - the strange inversion runs from But bar is precisely one that Haylock singles out to ignore!

So is it Beethoven's " phi -fth" or just plain old "Fifth"? Concert pianist Roy Howat's Web site has more information on his Debussy in Proportion book and others works and links.

## A designer's guide to the Golden Ratio

They all have one simple concept in common. 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 golden ratio —also known as the divine proportion, the golden mean, and the golden section. 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 probably best understood as the proportions Of course, the mathematical equation at work here is much more complicated than that.

In mathematics , two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The golden ratio is also called the golden mean or golden section Latin : sectio aurea. Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets , in some cases based on dubious fits to data. These often appear in the form of the golden rectangle , in which the ratio of the longer side to the shorter is the golden ratio.

Many works of art are claimed to have been designed using the golden ratio. However, many of these claims are disputed, or refuted by measurement. The golden ratio , an irrational number , is approximately 1. Various authors have claimed that early monuments have golden ratio proportions, often on conjectural interpretations, using approximate measurements, and only roughly corresponding to 1. These predate by some 1, years the Greek mathematicians first known to have studied the golden ratio.

A ratio is “golden” if the relationship of the larger to the smaller number is the same as the ratio between the two numbers added together and the larger number. Numerically, the golden ratio is roughly to 1. But for your art, you don't need to worry about the exact number.

## How to Use the Golden Ratio to Create Gorgeous Graphic Designs

Something deep in the core of all of us regards the golden ratio as beautiful , a fact that many artists and architects have employed for thousands of years. The different golden ratio examples and the use of this formula, viewed to help create the most pleasing images to the eye, aids numerous artists, architects, designers, and even musicians, towards a perfectly balanced harmony. The value of the golden ratio in contemporary art is possibly not as rich as is the case with examples of High Renaissance paintings by Leonardo Da Vinci, Michelangelo, Raphael, or much later, artists of the early 20th century, but its importance as one of the compositional tools can not be dismissed.

Bonus Download: New to painting? Start with my free Beginner's Guide to Painting. The golden ratio is the ratio of approximately 1 to 1. These are extremely important numbers to mathematicians.

The Golden Ratio is a mathematical ratio that's commonly found in nature. It can be used to create visually-pleasing, organic-looking compositions in your design projects or artwork. Whether you're a graphic designer, illustrator or digital artist, the Golden Ratio, also known as the Golden Mean, The Golden Section, or the Greek letter phi, can be used to bring harmony and structure to your projects. This guide will explain what it is, and how you can use it.

### How to Use the Golden Ratio to Create Gorgeous Graphic Designs

This section introduces you to some of the occurrences of the Fibonacci series and the Golden Ratio in architecture, art and music. The ancient Greeks knew of a rectangle whose sides are in the golden proportion 1 : 1. It occurs naturally in some of the proportions of the Five Plato nic Solids as we have already seen. A construction for the golden section point is found in Euclid 's Elements. The golden rectangle is supposed to appear in many of the proportions of that famous ancient Greek temple, the Parthenon, in the Acropolis in Athens , Greece but there is no original documentary evidence that this was deliberately designed in. There is a replica of the original building accurate to one-eighth of an inch! Its most famous monument is the Parthenon, a temple to the goddess Athena built around or BC.

If you take away the big square on the left, what remains is yet another golden rectangle. The appearance of this ratio in music, in patterns of human behavior, even in the proportion of the human body, all point to its universality as a principle of good structure and design. Used in art, the golden ratio is the most mysterious of all compositional strategies. Some scholars argue that the Egyptians applied the golden ratio when building the great pyramids, as far back as B.

#### Using the golden ratio in graphic design

Что. - Я вызываю агентов безопасности. - Нет, коммандер! - вскрикнула Сьюзан. - Нет. Хейл сжал ее горло. - Если вы вызовете службу безопасности, она умрет. Стратмор вытащил из-под ремня мобильник и набрал номер.

Мы погибли. - Теряем фильтры Протокола! - раздался чей-то голос. - Открылся третий уровень защиты! - Люди в комнате засуетились. На экране агент с короткой стрижкой безнадежно развел руками. - Сэр, ключа здесь .

Беккер поднялся на ноги, пытаясь выровнять дыхание. Попробовал добрести до двери. Меган скрылась во вращающейся секции, таща за собой сумку. Беккер почти вслепую приближался к двери. - Подожди! - крикнул .

В комнате зашушукались. С одного из столов на пол упали подставка для бумаг и стакан с карандашами, но никто даже не пошевельнулся, чтобы их поднять. Лишь едва слышно шуршали лопасти вентиляторов охлаждения мониторов да доносилось ровное дыхание Дэвида в микрофон, почти прижатый к его рту.

- Давайте ближе к сути дела. Агент Колиандер нажал несколько кнопок, и кадры стали сменяться быстрее. Люди на подиуме с нетерпением ждали, когда на экране появится их бывший сослуживец Энсей Танкадо. Ускоренное проигрывание видеозаписи придавало изображению некоторую комичность.

* Тридцать секунд. - Давайте же, - прошептал Фонтейн.*

Глаза Сьюзан сузились. Она терпеть не могла, когда он называл ее Сью. Вообще-то она ничего не имела против этого имени, но Хейл был единственным, кто его использовал, и это было ей неприятно. - Почему бы мне не помочь тебе? - предложил Хейл.

Ее прозрачный куполообразный потолок в центральной части поднимался на 120 футов. Купол из плексигласа имел ячеистую структуру - защитную паутину, способную выдержать взрыв силой в две мегатонны. Солнечные лучи, проходя сквозь этот экран, покрывали стены нежным кружевным узором. Крошечные частички пыли, пленницы мощной системы деионизации купола, простодушно устремлялись вверх широкой спиралью.

Других слов для него у нее не. Стратмор оторвался от перил и переложил пистолет в правую руку. Не произнеся ни слова, он шагнул в темноту, Сьюзан изо всех сил держалась за его плечо. Если она потеряет с ним контакт, ей придется его позвать, и тогда Хейл может их услышать. Удаляясь от таких надежных ступенек, Сьюзан вспомнила, как в детстве играла в салки поздно ночью, и почувствовала себя одинокой и беззащитной, ТРАНСТЕКСТ был единственным островом в открытом черном море.

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PDF | Citation Information: "Beauty of Golden Ratio in paintings, architecture, music, nature and technology", published in September

This ratio also can be seen in natural division, arts, music and architecture. It is measured as a standard for the aesthetic and beauty of the.