The golden ratio is a ratio used in mathematics, architecture, art and science. It describes the ratio between two sizes in which the smaller part is in the same proportion to the larger part as the larger part is to the whole. The division ratio is denoted by the Greek letter φ (Phi) and has the rounded value of 1.618 (or 1/φ = ~0.618).
A distance of length 1 is divided by the golden ratio rounded at 0.618. The larger part of the resulting two parts is called the major and the smaller part the minor. The difference between the golden ratio and other known division ratios becomes clearer in comparison. Here using the example of halving and thirds.
A section of length 1 is divided in half at 0.5. The resulting two parts are called halves.
A section of length 1 divides into thirds at ~0.33 and ~0.66. The resulting three parts are called thirds.
The name "golden ratio" refers to the fact that this division ratio is considered particularly harmonious and aesthetic in mathematics. Additionally, the golden ratio is commonly found in architecture, art, and nature, which contributes to its perception as something "golden" or "ideal." Alternative names for it include the "divine ratio," "Fibonacci ratio," or "divine proportion" (Italian: 'Divina proportione').
The golden ratio appears in many areas of nature and art.
There are many other examples of the occurrence of the golden ratio. It is often considered aesthetically pleasing and harmonious.
Geometry holds two great treasures: one is the Pythagorean Theorem, the other is the Golden Section. The first we can compare to a bushel of gold, the second we can call a precious jewel.
A proportion is the size ratio of two parts. The uniqueness of the Golden Ratio's proportion can be well demonstrated when compared to other size ratios. To illustrate this, the Golden Ratio is compared to halving and quartering, two other commonly used proportions.
When a segment of 1 is divided exactly in the middle, two equal parts are created. Each of the two parts is exactly half the length of the original segment, measuring 0.5 each (black/gray stripes).
What's special about the Golden Ratio is that the size ratio of the smaller part to the larger part is the same as that of the larger part to the original segment. This means the orange part relates to the blue part in the same way that the blue part relates to the sum of blue and orange.
Mathematically, it results in: Blue: ~0.618 Orange: 1 - 0.618 = ~0.382 Orange to Blue: 0.382 / 0.618 = 0.618 Blue to Black: 0.618 / 1 = 0.618
So, Orange to Blue and Blue to Black have the same proportion to each other.
The division in the Golden Ratio can be continued infinitely. Each part then maintains the same proportion to its parts. Therefore, it can also be said that the Golden Ratio is the proportion of infinity.
Instead of dividing the segment, it's also possible to enlarge it. In this case, it should be multiplied by 1 + 0.618 = 1.618. This results in a larger whole, with its parts appearing harmonious to each other because they share the same proportion. This only happens with division in the Golden Ratio.
To illustrate another proportion frequently used in architecture, the quartering is considered (red/green stripes).
When a segment is divided into quarters, the calculations are as follows: Red: ~0.25 (1/4) Green: 1 - 0.25 = 0.75 (3/4) Red to Green: 0.25 / 0.75 = 0.33333 (1/3) Green to Black: 0.75 / 1 = 0.75 (3/4)
As with halving, the parts here do not share the same size ratio with each other as the larger part does with the whole.
In a golden rectangle, the side lengths are in the Golden Ratio proportion. It is formed when the minor side of a Golden Ratio is rotated by 90 degrees.
A golden rectangle can always be divided into a square and a smaller golden rectangle (Mouseover/Tap).
A golden spiral is formed when the radius of a circle decreases by a factor of the Golden Ratio every 90 degrees of rotation around its center (Mouseover/Tap). Therefore, it is always bounded by a golden rectangle.
A common approximation construction involves repeatedly dividing a golden rectangle into smaller golden rectangles. Within the resulting squares, a quarter circle is drawn in a way that continues an existing quarter circle (yellow curve).
All golden rectangles share only two diagonals (dashed lines). Their intersection point is the center of the spiral. The spiral can also be drawn from the center outward, continually increasing in size.
The Golden Ratio can be applied not only to line segments but also to the circle. The golden angle is formed when a full circle of 360° is multiplied by the Golden Ratio. This results in two angles: 360° * 0.618 = ~222.5° and 360° - ~222.5° = ~137.5°. The smaller of the two angles is referred to as the golden angle (orange). Not only the angles but also the lengths of the resulting circular arcs are in the Golden Ratio to each other (Mouseover/Tap).
Pentagon (from Ancient Greek 'Five (corner) angles') is the Greek term for a five-sided figure. However, it usually refers to a regular pentagon, meaning one in which all sides are of equal length.
When the vertices are connected by diagonals, a pentagram (from Ancient Greek 'Five lines') is formed, also known as a five-pointed star.
Especially in the composition of paintings, the angles of the regular pentagon are frequently used:
Golden triangles are formed within a regular pentagon. These are isosceles triangles in which the lengths of two sides are in the Golden Ratio to each other. There can be exactly two variations (Mouseover/Tap).
A regular pentagon can only be constructed with an understanding of the Golden Ratio. In classical geometry, only a compass and an unmarked straightedge are allowed for this purpose.
In antiquity and the Middle Ages, the construction of a regular pentagon, and thus the Golden Ratio, was known to only a few. Pentagons were therefore considered mystical symbols for a long time, accessible only to those with access to books and knowledge.
The Platonic solids are named after the ancient philosopher Plato (4th century BC), who described them in his book Timaeus. Each of these solids is composed of the same regular polygons, which gives them a particularly beautiful symmetry. The Platonic solids are the tetrahedron (4 equilateral triangles), the cube (6 squares), the octahedron (8 equilateral triangles), the dodecahedron (12 pentagons), and the icosahedron (20 equilateral triangles). From a mathematical point of view, it is remarkable that there can be exactly these five solids if regular polygons are to fit together seamlessly.
The dodecahedron is clearly connected with the golden ratio, as it consists of 12 regular pentagons.
The golden ratio is also revealed, though less obviously, in the icosahedron. Five of its 20 triangles meet at a common point. In doing so they form regular pentagons (blue/orange lines).
Another connection with the golden ratio appears in the internal structure of an icosahedron, i.e. the lines between the most distant opposite vertices. An icosahedron is formed from three golden rectangles placed in space so that each is aligned along the x-, y-, and z-axis, with their center of gravity at a common point, the center of the icosahedron (blue rectangles, mouseover). When the corners of the golden rectangles are connected, all the vertices of the 20 equilateral triangles of an icosahedron are produced (gray lines).
An icosahedron is common in nature, as of all regular polyhedra — that is, the five Platonic solids — it fills a sphere of a given diameter most efficiently. For example, protein shells of viruses (capsids) in most cases have the shape of an icosahedron.
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The latitude of the pilgrimage city Mecca with the Kaaba, which is important in Islam, is located in the golden section between north and south pole (blue and orange arc). For geographical classification further places with mouseover
Illustration of the golden angle of 137.5°.Every 137.5 degrees, an increasing colored dot is placed on an increasing radius. What initially appears chaotic develops into a multi-armed spiral (here 13 arms). The pattern of the golden angle can be found, among others, in composite flowers (Mouseover)
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Mecca is the birthplace of the Prophet Muhammad and home to the central holiest site in Islam, the Kaaba. The geographical location of Mecca is closely related to the Golden Ratio.
Every location on Earth can be uniquely specified by latitude and longitude coordinates.
Latitude is determined by imagining a line from the center of the Earth to the equator, with this angle set to zero. The North and South Poles are now situated 90° above or below the equator (northern or southern latitude).
For longitude, in a top-down view of the Earth, a line is imagined from the North or South Pole to the equator. The angle zero was established as the Prime Meridian in 1884 and passes through Greenwich, England. Now, every location on Earth is situated up to 180° east or west of this point.
Since the Prime Meridian was arbitrarily chosen and could just as easily pass through Tokyo or San Francisco, longitude is not relevant in the context of Mecca's geographical position. However, latitude, which has natural reference points in the Earth's North and South Poles, is significant.
Mecca is located at 21.25° north latitude (green line). The distance of 1 degree of latitude on the Earth's surface is constant and approximately 111.7 kilometers.
The distance from Mecca to the South Pole (blue point) is:
(90° + 21.25°) * 111.7 km = 12,426.63 km (blue line)
The distance from Mecca to the North Pole (orange point) is:
(90° - 21.25°) * 111.7 km = 7,679.38 km (orange line)
Both distances are in the ratio of the Golden Ratio:
7,679.38 km / 12,426.63 km = ~ 0.618.
Around a central point, an angle of 137.5° is drawn. The point is marked. Starting from this point, the angle is drawn again from the center, and so on. With each new angle drawn, the radius is extended by a very short distance.
The numbers 8, 13, and 21 are part of the Fibonacci sequence. When every 8th, 13th, or 21st angle is marked, these markings create a spiral pattern. The spiral pattern occurs only with Fibonacci numbers
All members of the Asteraceae family exhibit the Golden Angle. The spirals are clearly visible in the flowers. Such an arrangement of petals ensures that even densely packed petals receive the maximum possible exposure area for light (Mouseover/Tap).
Some examples of Asteraceae include sunflowers, chrysanthemums, chamomile, dandelions, and lettuce.
It is important to mention that the Golden Ratio is just one of many proportions used in art, albeit considered one of the most aesthetically pleasing from a mathematical perspective. In addition to the Golden Ratio, various other proportions like thirds, quarters, and more are often used, sometimes within the same artwork. The goal is to maintain the viewer's interest by creating a diverse and engaging composition.
The Golden Ratio has been used in paintings since the Middle Ages, especially in the composition of images, such as the arrangement of figures. The renowned painter Leonardo da Vinci, considered one of the greatest artists of all time, created some of the most impressive compositions, and it's no wonder that he incorporated the Golden Ratio into his paintings. It's often said that the Golden Ratio was primarily used by Renaissance painters like Leonardo, Raphael, or Albrecht Dürer. However, the presence of the Golden Ratio can be observed throughout various artistic epochs as a part of the composition in the works of outstanding artists. This demonstrates that their paintings were not only inspired by divine intuition but also the product of mathematically inclined individuals.
Die Erschaffung Adams, Michelangelo, um 1508-1512, Sixtinische Kapelle (Deckengemälde)
With proportional divisions Michelangelo emphasizes two hands and four eyes of the depicted figures
Explanation of the five upper horizontal lines from top to bottom:
The use of classical geometric proportions can also be demonstrated with respect to the image height (mouseover). Explanation of the four vertical lines from left to right:
The Golden Ratio has been used in architecture since antiquity. Certain pyramids in Egypt are believed to exhibit these proportions, such as the ratio of the slant height to half the base side. However, as the pyramids have significantly deteriorated over time, and their originally thick white limestone casings have been largely removed, it's no longer possible to accurately determine their original dimensions to conclusively prove the use of the Golden Ratio. The oldest confirmed evidence of the Golden Ratio's use in architecture is attributed to the buildings of ancient Greece and Rome.
With the invention of printing came the need to develop uniform design principles for letters. Letters are individual metal or wooden characters that are arranged on a printing plate to form texts and then printed. Design principles for the letters were intended, on the one hand, to ensure the readability and aesthetics of the printed characters while at the same time allowing for variation. On the other hand, these rules were meant to ensure that the individual letters of the printing plates, which wore out quickly, could be reproduced as easily as possible. Elaborate and particularly artfully designed letters with individual filigree ornaments, which could only be produced by specially trained craftsmen, were therefore not practical.
A first significant step toward uniform rules for the appearance of letters was taken by Luca Pacioli (1445–1517). As a pupil of the outstanding Renaissance painter Piero della Francesca, who, in addition to his paintings, was also known for his mathematical treatises, Pacioli became one of the outstanding mathematicians of his time; among other things, he is known for the first publication of the groundbreaking method of double-entry bookkeeping. At the Milanese court he worked closely with Leonardo da Vinci.
In 1509 Pacioli published a book still significant today about the golden ratio “De Divina Proportione” (German “On the Divine Proportion”), for which Leonardo created illustrations. The book contains, among other things, a translation of Vitruvius’ description of human proportions, which inspired Leonardo to the famous drawing of the Vitruvian Man.
Particularly noteworthy is that Pacioli, in the appendix to his book, proposed a typography based on geometry. He recommended that the design of the letters should follow the known proportions of classical geometry, such as halving, thirding, and the golden ratio. Pacioli’s proposals for integrating geometric principles into typography had a lasting effect. His design rules can still be found today in textbooks on typography.
The golden ratio is used in art wherever something is to be divided. In addition to the examples shown here, it can, for example, divide the length of a film to highlight certain scenes or shots. In music, for example, the number of certain bars can stand in the golden ratio to one another or the frequency of certain tones (such as the notes C and G-sharp). In poetry, the number of syllables in the lines can emphasize important content, for example through the use of the Fibonacci numbers 3, 5, and 8 for the number of letters of consecutive words or the number of syllables in three consecutive lines.
The golden ratio is rarely used alone in art but almost always in conjunction with other frequently used proportions such as halving, thirding, and quartering. Designing the interplay of the proportions so that the whole appears harmonious on the one hand and on the other underlines the respective idea of the artwork is a high art and requires a certain degree of understanding of geometry as well as the ability to express a narrative through it. Because from a mathematical point of view the golden ratio is the most beautiful proportion, it usually emphasizes one or more special features of an artwork.
Conversely, this does not mean that only that is art which uses the golden ratio or is otherwise particularly harmoniously proportioned.
The division ratio of the golden ratio (0.618 or 1.618) is quite close to another commonly used division ratio, the ratio of 2:3, which is 2/3 (0.6666...).
Since 0.618, the value of the golden ratio, is difficult to measure in an instant, but photography is the art of the moment, the rule of thirds is often applied instead of the golden ratio. To avoid the need to measure 0.618, photographers often use 2/3 and 1/3 instead. Digital cameras, such as smartphones, are particularly helpful for this, as their displays often show three grids next to or on top of each other, representing the thirds. To achieve interesting compositions, it can be useful to position the subjects along these lines rather than in the center.
However, the rule of thirds is not a universally applicable rule in photography, as 2/3 and the golden ratio are just two of several possible proportions. In professional photography, images are composed using various proportions, similar to painting. This may involve selecting proportionate backgrounds, furniture, or accessories in artistic portrait photography, for example, all of which are photographed from a fixed tripod to ensure that the final image is harmoniously proportioned overall.
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Construction of the Golden Ratio using the method of the ancient mathematician Euclid in a given line segment of length 1. A golden rectangle is formed when the smaller part of the Golden Ratio is rotated by 90 degrees (Mouseover)
For the construction of a regular pentagon, you will need the perpendicular bisector and the Golden Ratio of a line segment
Approximate construction of the Golden Spiral. It originates at the intersection of the diagonals of the largest surrounding golden rectangle (Mouseover)
There are various methods to construct the golden ratio. The most well-known is that of the ancient mathematician Euclid (3rd century BC).
When the minor segment is rotated by 90 degrees, it forms a golden rectangle, meaning that the side lengths are in the ratio of the golden ratio (Mouseover, orange rectangle).
The golden ratio is necessary to construct a regular pentagon. In a regular pentagon, all sides are of equal length.
Given a line segment AB, divided in the golden ratio at point D. Additionally, the perpendicular bisector of the line segment is needed (dashed vertical line through point M). For the sake of clarity, the construction of the perpendicular bisector is omitted here.
Connecting these five corner points creates a regular pentagon (dark blue lines). The five diagonals of the pentagon form a five-pointed star, and these lines always intersect at the golden ratio points (light blue and orange lines).
The golden spiral is often represented in an approximation construction.
The golden number indicates the division ratio of the golden section. It is usually denoted by the 21st Greek letter φ (Phi) and has the rounded value of 1.618. The golden number is an irrational number and therefore infinite. It is obtained by adding the length of the major of a line of length 1 divided in the golden section and the number 1, i.e. ~0.618 + 1 = ~1.618.
The numerical value of the golden ratio is derived from the geometric construction of the golden section according to Euclid and is represented by the length of segment AE (left figure).
This length can be calculated using the Pythagorean theorem since points ABC form a right-angled triangle.
The lengths of segments AB and BC are known since the rectangle is composed of two adjacent squares. Therefore, the height of the rectangle is exactly half of its width. Here, the width of the rectangle is 1 unit, so its height is 0,5 units.
The length of diagonal AC is determined by the Pythagorean theorem:
AC2 = AB2 + BC2
AC = 12 + 0,52 = 1,25 ≈ 1,118
The length of the golden section is obtained by subtracting segment DC from AC because DC, like BC, represents the radius of the circle centered at point C, and thus, it has a length of 0,5.
So, the length of AD and thus the value of the golden ratio is:
1/φ = AC - DC ≈ 1,118 - 0,5 ≈ 0,618
Confusingly, the width of the rectangle is often not given as 1, but as 2. This also changes the height of the rectangle from 0,5 to 1. The same calculation with the new values:
AC = 22 + 12 = 5
AD = 5 - 1 = ~1,236
To get to the original numerical ratio of the golden section for a rectangle with the width of 1, this value must be divided by 2 (mouseover/tap). Although the distance AE divides the distance AB in the golden section, the latter has the length 2 and the golden section of length 1 is searched for. Consequently, AE must be divided in the point M, which leads to the searched size of 0,618 (Mouseover/Tap, yellow dot). Mathematically, this results in the following:
AD = (5 - 1) / 2 = ~0,618
From the geometrical consideration it is obvious that this method needs one step more. That it is nevertheless the more popular variant can only be explained by the fact that
1/φ = (5 - 1) / 2 is more pleasant to read than
1/φ = 1,25 - 0,5 from the previous calculation
Representation of the golden ratio on a number line from 0 to 3. It's evident that φ is derived from 1 and the value of the golden ratio, which is approximately 0,618
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The relationship between Fibonacci numbers and the golden ratio. The sides of a rectangle with two consecutive numbers of the Fibonacci sequence form a rectangle that approaches more and more the proportions of the golden rectangle
Pascal's triangle contains the Fibonacci sequence.This becomes clearest if the triangle is displayed left-justified instead of centered (mouseover). The Fibonacci numbers result from the sum of the numbers that lie on the 45° angle
The Fibonacci sequence is named after the Italian mathematician Leonardo Fibonacci (c. 1170-1240). The sequence of numbers is obtained by adding the last two known numbers of the sequence together and adding this sum to the sequence, starting with zero and one.
The Fibonacci sequence is closely related to the Golden Ratio because the result of dividing two consecutive numbers in this sequence steadily approaches the Golden Ratio's division ratio, which is approximately 0,618.
| Calculation | Fibonacci sequence | Division ratio |
| 0 + 1 = 1 | 0,1 | 0/1 = 0 |
| 0 + 1 = 1 | 0,1,1 | 1/1 = 1 |
| 1 + 1 = 2 | 0,1,1,2 | 1/2 = 0,5 |
| 1 + 2 = 3 | 0,1,1,2,3 | 2/3 ≈ 0,666 |
| 2 + 3 = 5 | 0,1,1,2,3,5 | 3/5 = 0,6 |
| 3 + 5 = 8 | 0,1,1,2,3,5,8 | 5/8 = 0,625 |
| 5 + 8 = 13 | 0,1,1,2,3,5,8,13 | 8/13 ≈ 0,615 |
| ... | ||
| 89 + 144 = 233 | 0,1,1,...,144,233 | 144/233 ≈ 0,6180 |
The term "Pascal's Triangle" is attributed to the scientist Blaise Pascal (1623-1662).
In this representation, the numbers in the triangle are generated such that each number is the sum of the two numbers directly above it. If there is only one number, the other is considered to be 0 (for the numbers on the outer edges of the triangle). The triangle starts with the number 1 at the top.
The centered representation of Pascal's Triangle was primarily used to expand binomial powers, such as
(a+b)2 = 1a2 + 2a1b1 + 1b2
Here, 1, 2, and 1 correspond to the numbers in the third row of Pascal's Triangle. The numbers in the subsequent fourth row (1, 3, 3, 1) can be used to expand the third power of the binomial (a+b), like so:
(a+b)3 = 1a3 + 3a2b1 + 3a1b2 + 1b3
and so on.
Pascal's Triangle can also be represented in a left-aligned format. This makes it easier to read the sequences of numbers contained within.
2nd Column: Natural numbers
3rd Column: Triangle numbers
4th Column: Tetrahedral numbers
The Fibonacci number sequence arises from the sum of the numbers lying on a diagonal at a 45° angle when Pascal's Triangle is represented in the left-aligned format.
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