Golden ratio calculator 1.618 for any length

What is the golden ratio and why is 1.618 called the golden ratio?

The golden ratio is the proportion you get when a line is divided so that the whole length relates to the longer part exactly as the longer part relates to the shorter one. Written with a for the long part and b for the short part, the condition is (a + b) / a = a / b, and the only positive number that satisfies it is φ = (1 + √5) / 2, approximately 1.6180339887. The Greek letter φ, phi, was proposed for it by the American mathematician Mark Barr at the start of the twentieth century in honour of the sculptor Phidias.

The word golden is a translation of the German goldener Schnitt, the golden cut, which appears in print for the first time in Martin Ohm's mathematics textbook of 1835. Before that the division went by the name Euclid gave it, the extreme and mean ratio, and the name Luca Pacioli gave it in 1509, the divine proportion. None of these names describes a property of the number. They record the esteem in which geometers held a division that keeps reproducing itself: cut the longer part of a golden section in the same ratio and the new pieces stand in the same relation as the old ones did.

In percentages, a golden section puts about 61.8% of the length into the longer part and about 38.2% into the shorter part. The number 1.618 is the ratio between the two parts, 0.618 is the share of the whole taken by the longer part, and 2.618 is the ratio between the whole and the shorter part. All three are powers of the same φ.

What is the golden ratio formula and where does it come from?

The golden ratio formula is φ = (1 + √5) / 2, and it comes from solving the defining proportion as a quadratic equation. Set the short part b to 1 and call the long part x. The condition (a + b) / a = a / b then reads (x + 1) / x = x, which rearranges to x² = x + 1. The quadratic formula gives x = (1 ± √5) / 2, and since a length has to be positive, the golden ratio is the root with the plus sign. The other root, (1 − √5) / 2 ≈ −0.618, is exactly −1/φ and turns up in the Fibonacci formula further down.

The equation x² = x + 1 is the source of every identity attached to the golden ratio. Divide both sides by φ and you get φ = 1 + 1/φ, so 1/φ = φ − 1 ≈ 0.618. Multiply instead and you get φ² = φ + 1 ≈ 2.618, then φ³ = 2φ + 1 ≈ 4.236, and in general φⁿ = F(n)·φ + F(n − 1), where F(n) is the nth Fibonacci number. The same relation makes φ the simplest of all continued fractions, 1 + 1 / (1 + 1 / (1 + …)), and the simplest nested square root, √(1 + √(1 + √(1 + …))).

Because √5 is irrational, φ is irrational too: its decimal expansion 1.6180339887498948… never repeats and never terminates. It is an algebraic number of degree two, so it can be written exactly with a square root, unlike π or e.

How to calculate the golden ratio for any length

To calculate the golden ratio for a given length, multiply the whole by 0.618 to get the longer part and by 0.382 to get the shorter part, or divide by φ and φ² respectively. A 100 cm shelf divided at the golden section has a long part of 61.8 cm and a short part of 38.2 cm, and a 1920 pixel wide screen splits into a main column of about 1187 pixels and a side column of about 733 pixels. In every unit the two parts add up to the whole and the long part divided by the short part gives 1.618.

The calculation runs in every direction. Starting from the longer part a, the whole is a × φ and the shorter part is a / φ. Starting from the shorter part b, the longer part is b × φ and the whole is b × φ², or b × 2.618. A photograph whose short side is 20 cm becomes a golden rectangle when its long side is 32.4 cm, and a headline set at 32 px pairs with body text at 32 / 1.618, which rounds to 20 px.

When the exact number is more precision than a layout needs, the 3:5:8 rule gives the same result in whole units. The Fibonacci numbers 3, 5 and 8 divide a length of 8 into parts of 5 and 3, with 8/5 = 1.6 and 5/3 ≈ 1.667, both within 3% of φ. The next triple, 5:8:13, is closer still, with 13/8 = 1.625.

What is a golden rectangle and how is the golden spiral drawn?

A golden rectangle is a rectangle whose long side is φ times its short side, so a rectangle of 1 by 1.618, 10 by 16.18 or 5 by 8 to a close approximation. Its defining property is that cutting off a square on the short side leaves a smaller rectangle with exactly the same proportions. Cut a square from a 1.618 by 1 rectangle and the remainder measures 0.618 by 1, and 1 / 0.618 is again 1.618. The process never ends, which is why the golden rectangle can be subdivided into an infinite sequence of ever smaller squares spiralling inward.

Drawing one with compass and straightedge takes three steps. Draw a square, mark the midpoint of one side, and set the compass from that midpoint to an opposite corner. The arc swung down to the extended base marks the far end of the golden rectangle, since the midpoint-to-corner distance is √5 / 2 for a unit square and 1/2 + √5 / 2 equals φ.

The golden spiral is the logarithmic spiral that grows by a factor of φ for every quarter turn, so by φ⁴, about 6.85, for every full turn. It passes through the corners of the nested squares inside a golden rectangle. The curve usually shown in diagrams is a close cousin, the Fibonacci spiral, made of quarter circles drawn inside squares with side lengths 1, 1, 2, 3, 5, 8 and 13. The two look alike at a glance but differ slightly, because the quarter circles change radius in jumps while the true spiral grows continuously. The same proportion hides inside the regular pentagon, whose diagonal is φ times its side, which is why Euclid needed the ratio in the first place.

How are the Fibonacci numbers connected to the golden ratio?

The Fibonacci numbers are connected to the golden ratio because the ratio of any two consecutive terms of the sequence approaches φ as the numbers grow. The sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 is built by adding the previous two terms, and the quotients 2/1 = 2, 3/2 = 1.5, 5/3 ≈ 1.667, 8/5 = 1.6, 13/8 = 1.625 and 21/13 ≈ 1.615 close in on 1.618 from above and below in turn. By 144/89 the error is below one part in ten thousand. Johannes Kepler noticed this convergence in the early seventeenth century and ranked the division into extreme and mean ratio with the theorem of Pythagoras as one of the two great treasures of geometry.

The reason is the recurrence itself. If consecutive terms settle towards a fixed ratio r, then F(n + 1) / F(n) = 1 + F(n − 1) / F(n) = 1 + 1 / r, which is again the equation r² = r + 1 whose positive root is φ. The exact link is Binet's formula, published by Jacques Binet in 1843 and known earlier to Abraham de Moivre and Leonhard Euler: F(n) = (φⁿ − (−1/φ)ⁿ) / √5. Because the second term shrinks towards zero, F(n) is simply φⁿ / √5 rounded to the nearest whole number.

The golden ratio is also, in a precise sense, the number that is hardest to approximate with fractions. Its continued fraction consists only of ones, so each convergent improves on the previous one by the smallest possible amount, and by Hurwitz's theorem φ is the worst case among all irrational numbers. That is why φ is sometimes called the most irrational number, and it is the reason the golden angle spaces seeds so evenly, as the next section explains.

Where does the golden ratio appear in nature?

The golden ratio appears in nature most reliably in phyllotaxis, the arrangement of leaves, seeds and scales around a plant stem or seed head, where successive elements are placed at the golden angle of about 137.5°, which is 360° / φ². Because φ is so poorly approximated by fractions, turning by that angle never brings a new seed directly over an earlier one, and the seeds fill the disc with no gaps and no radial spokes. The eye then picks out two families of spirals running in opposite directions, and their counts are consecutive Fibonacci numbers.

Sunflowers are the standard example, with spiral counts of 34 and 55 or 55 and 89 in a large head. Pine cones commonly show 8 and 13, pineapples 8, 13 and 21, and the florets of a romanesco or a daisy follow the same pattern. The claim has been tested by counting: in a citizen science project published by Jonathan Swinton and colleagues in Royal Society Open Science in 2016, 657 sunflower heads were examined and roughly three in four showed Fibonacci spiral counts, while the rest showed related sequences or irregular patterns. Fibonacci phyllotaxis is a strong tendency of plant growth, with exceptions.

Other popular examples hold up less well. The nautilus shell is a logarithmic spiral and is often drawn beside a golden spiral as if they were the same curve, but measured shells grow by a factor closer to 3 per full turn while a golden spiral grows by φ⁴ ≈ 6.85. Spiral galaxies and hurricanes are logarithmic spirals of various growth rates, none tied to φ. Claims about the human body, such as the navel dividing height in the golden ratio, depend on which individuals are measured and where the tape is placed, a point made in detail by Mario Livio in 2002 and by George Markowsky in 1992.

How is the golden ratio used in design, typography and photography?

In design the golden ratio is used as a rule of thumb for dividing space: a page or screen is split into a main area and a secondary area in the proportion 1.618 to 1, or in the Fibonacci approximations 5:8 and 3:5:8 that are easier to work with in whole units. A two-column layout of 8 and 5 units, a card with a 5 by 8 aspect ratio, and margins that relate as 3 to 5 all come from the same idea.

Typography applies the ratio to scale. A modular type scale built on 1.618 multiplies each size by φ to get the next: 16 px body text gives 25.9 px for a subheading and 41.9 px for a heading, typically rounded to 26 and 42. Many designers find that ratio too steep for long documents and use a Fibonacci-based scale such as 13, 21, 34 px or a milder multiplier instead.

Photography has two related grids. The rule of thirds divides the frame into three equal parts each way and places subjects on the lines or their intersections; it is a simplification that is easy to apply through the viewfinder. The phi grid divides the frame in the proportion 1 : 0.618 : 1 instead, so the lines sit closer to the centre, at 38.2% and 61.8% of the width and height. Both are conventions for placing a subject off centre, and neither has been shown to make a composition better than the other.

Twentieth-century modernism made deliberate use of the ratio. Le Corbusier published the Modulor in 1948, a system of measurements built from the golden ratio and the height of a human figure, and applied it to buildings such as the Unité d'Habitation in Marseille. Salvador Dalí painted The Sacrament of the Last Supper in 1955 on a canvas whose sides are in the golden ratio, with a dodecahedron hovering over the table. These are documented cases in which the artist chose the ratio, unlike the cases discussed next, where the ratio is read into a work by later measurement.

Who discovered the golden ratio and when was it named?

The golden ratio was first defined in writing by Euclid of Alexandria around 300 BC, in Book VI of the Elements, whose third definition states that a straight line is cut in extreme and mean ratio when the whole is to the greater segment as the greater is to the less. Euclid constructs the cut in Book II, uses it in Book IV to build the regular pentagon and in Book XIII to build the dodecahedron and icosahedron. The Pythagoreans, whose emblem was the pentagram, probably knew the proportion earlier, but no text of theirs survives.

The Franciscan friar and mathematician Luca Pacioli published De divina proportione in Venice in 1509, a treatise on the ratio and on the regular solids, with illustrations drawn by Leonardo da Vinci, his colleague at the court of Milan. Pacioli's title gave the ratio a theological framing that persisted for centuries. In the early seventeenth century Johannes Kepler studied the ratio in connection with the pentagon and the Fibonacci sequence, and his treatise On the Six-Cornered Snowflake of 1611 discusses the pentagonal symmetry of flowers in those terms.

The modern name arrived late. Martin Ohm, a German mathematician and brother of the physicist Georg Ohm, used the term goldener Schnitt in the second edition of his textbook Die reine Elementar-Mathematik in 1835, and the phrase spread through German mathematics before it was translated into English as the golden section and golden ratio. The symbol φ was proposed by Mark Barr around 1909 and recorded in Theodore Andrea Cook's The Curves of Life in 1914; the alternative symbol τ, from the Greek word for cut, is still used by some mathematicians.

Is the golden ratio really in the Parthenon, the pyramids and the Mona Lisa?

There is no documentary evidence that the Parthenon, the Great Pyramid of Giza or the Mona Lisa were designed on the golden ratio, and the measurements offered in support of those claims are chosen after the fact. George Markowsky examined the best known examples in his 1992 paper Misconceptions about the Golden Ratio and Mario Livio revisited them in his 2002 book The Golden Ratio, and both reach the same conclusion: with enough freedom to decide where a rectangle begins and ends, a golden rectangle can be fitted to almost any object, and the fit says nothing about the intention of its maker.

The Parthenon illustrates the problem. Drawings that show its façade inside a golden rectangle include the pediment and part of the steps, and different authors draw the rectangle from different edges to make the ratio come out; the architects left no record of using it. The Great Pyramid is a subtler case. The ratio of its slant height to half its base is close to φ, about 1.619, but the same proportion follows from the Egyptian practice of specifying a slope as a horizontal run per unit of rise, and there is no evidence that the builders knew √5.

For the Mona Lisa the argument rests on Leonardo's friendship with Pacioli and on rectangles drawn over the face by later writers. Leonardo drew the solids for De divina proportione, so he understood the ratio, but nothing in his notebooks records using it to compose a painting, and the rectangles overlaid on the portrait differ from one book to the next.

The idea that people prefer golden rectangles was tested by the psychologist Gustav Fechner in 1876, who asked subjects to pick the most pleasing rectangle from a set and reported a preference near 1.6. Later replications have given mixed results, and a review by Christopher Green in Perception in 1995 found the evidence for a special preference weak and sensitive to how the experiment is run. The golden ratio is a real and remarkable number, plants use it for reasons that are understood, and most of the claims about art and architecture are stories told about the number rather than facts about the works.

Why is the golden ratio called the divine proportion in sacred geometry?

The golden ratio is called the divine proportion because Luca Pacioli gave it that name in 1509, and in the tradition of sacred geometry it has been treated ever since as a symbol of harmony, growth and the unity of part and whole. Pacioli listed reasons for the name that were theological rather than mathematical: the ratio is a single value, like the one God; it is defined by three terms, like the Trinity; it cannot be expressed as a whole-number fraction, which he compared with the ineffability of the divine; and it is the key to the dodecahedron, which Plato had associated with the cosmos.

Sacred geometry is the practice of reading spiritual meaning into geometric forms, and within it the golden ratio, the pentagram and the golden spiral appear as emblems of life unfolding according to a hidden order. The pentagram carried that meaning for the Pythagoreans and for medieval and Renaissance writers on cosmology, and the spiral has been adopted by modern esoteric authors as a picture of growth from a centre. In these traditions the number is contemplated as a symbol, and the calculator makes no claims of that kind.

The mathematical facts and the symbolic tradition can be held apart without either one diminishing the other. The self-similarity of the golden section, in which every part repeats the proportion of the whole, is the property that made the number attractive to Pacioli and to later mystics, and it is the same property that makes φ the limit of the Fibonacci ratios and the spacing angle of sunflower seeds.