Golden Ratio
Written by Abin Alex and Anu Joseph.
Abstract
The golden ratio, about 1.618, is the proportion in which the whole is to the greater part as the greater is to the lesser, a relation defined in Euclid's Elements as extreme and mean ratio. This guide explains its link to the Fibonacci sequence, the golden rectangle, spiral, phi grid and triangle, and shows how crop overlays and logo grids use it. It then reviews critical evidence, which finds little support for claims that the ratio is universally preferred or that classical artists used it, and presents it as one optional compositional tool.
Keywords: golden ratio; phi grid; Fibonacci sequence; golden spiral; proportion
What is the golden ratio?
Short answer. The golden ratio is the proportion in which a line is cut so that the whole is to the longer part as the longer part is to the shorter. Its value is about 1.618.
- Euclid described it about 2,300 years ago as a line cut in extreme and mean ratio. The whole line is to the greater part as the greater part is to the less (Euclid, Elements VI, def. 3).
- If the longer part is a and the shorter part is b, then (a + b) / a = a / b. That ratio is about 1.618.
- Set b to 1 and the rule becomes a x a = a + 1, or a squared minus a minus 1 equals 0. The answer is (1 + the square root of 5) / 2, which is 1.6180339887 and so on.
- The number goes on forever without repeating, so 1.618 is a rounded value. It is written with the Greek letter phi.
- Other names are the golden mean, the golden section, the golden number and the divine proportion.
- A full popular history of the number is Livio's book (Livio, 2002). Read it with Markowsky's review (see the last chapter), which disputes several of its claims.
- Your slides also call it the golden spiral, the golden triangle and the phi grid. These are not different ratios. They are different ways of using the same ratio to divide a frame.
For a photographer the ratio is a way to place a horizon or a subject. It sits between the centre and the edge, a little closer to the centre than the rule of thirds.
A cinematographer meets the same ratio in framing guides and in the proportions of sets and logos. It is one more way to divide a frame.
Cham tipRemember one number: 1.618. The shorter part is about 0.382 of the whole and the longer part is about 0.618.
Cham taskDraw a line 16.18 cm long on paper. Mark a point 10 cm from one end. Check that 16.18 / 10 and 26.18 / 16.18 are both close to 1.618.
Check yourself: What is the value of the golden ratio?
About 1.618.
What is the Fibonacci sequence?
Short answer. The Fibonacci sequence starts 1, 1, 2, 3, 5, 8, and each number is the sum of the two before it. The ratio between neighbours moves toward 1.618.
- Start with 1 and 1. Add them to get 2. Add 1 and 2 to get 3, then 2 and 3 to get 5, then 3 and 5 to get 8. The sequence goes 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 and on.
- It is named after Fibonacci, the Italian mathematician Leonardo of Pisa. He set out the number problem behind it in his book Liber Abaci in 1202 (Sigler, 2003).
- Divide each number by the one before it and the answers swing above and below 1.618, getting closer each time. Table 1 shows this.
- The golden ratio is the number the ratios settle on. The sequence is the whole numbers that come nearest to it.
- This is why you see 3:2 and 8:5 as everyday near golden proportions. Both are Fibonacci pairs.
- A pattern that only needs a number, a sum and a ratio can look like magic. Treat it as mathematics first and art second.
| Pair of numbers | Division | Result |
|---|---|---|
| 1 and 1 | 1 / 1 | 1.0000 |
| 1 and 2 | 2 / 1 | 2.0000 |
| 2 and 3 | 3 / 2 | 1.5000 |
| 3 and 5 | 5 / 3 | 1.6667 |
| 5 and 8 | 8 / 5 | 1.6000 |
| 8 and 13 | 13 / 8 | 1.6250 |
| 13 and 21 | 21 / 13 | 1.6154 |
| 21 and 34 | 34 / 21 | 1.6190 |
| 34 and 55 | 55 / 34 | 1.6176 |
| 55 and 89 | 89 / 55 | 1.6182 |
Table 1. Ratios of successive Fibonacci numbers, to four decimal places.
You do not need to calculate while shooting. Use the sequence as a reminder that near golden proportions are common, such as 3:2, which is the shape of a full frame sensor.
When a film shot is built from nested frames or doors, each opening can be about 0.62 of the one around it. This is a design choice, not a rule.
Cham tip3:2 is near the golden ratio. A 3:2 frame is already a close fit for golden composition.
Cham taskWork out the next five Fibonacci numbers after 89. Divide each by the one before. How close to 1.618 are you?
Check yourself: How is each Fibonacci number made?
By adding the two numbers before it.
What are the golden rectangle and the golden spiral?
Short answer. A golden rectangle has sides in the golden ratio. Remove a square from it and the part left is again a golden rectangle.
- A golden rectangle is about 1.618 times as long as it is wide.
- Cut a square off one end and what is left is again a golden rectangle. You can do this over and over, each rectangle smaller than the last.
- Join the corners of the squares with quarter circles and you get a spiral. This is the golden spiral. Drawn through Fibonacci squares, it is the Fibonacci spiral.
- The two spirals are close but not identical. The Fibonacci squares are whole numbers, and the golden rectangles are not.
- In a picture the spiral tells you where to put the most important detail. It ends in a tight curl at one point, which becomes the focal point.
- A spiral has a direction. Flip or rotate the overlay so that the curl lands on your subject.
Crop a landscape so that the curl of the spiral lands on a person, a tree or a building. The rest of the frame leads the eye toward it.
A cinematographer can use the spiral to plan the move of a camera, the way a dolly arcs in toward a subject. This is a reading of the shape, not a rule.
Cham tipPut the focal point at the tightest part of the spiral.
Cham taskOverlay a golden spiral on three of your own photographs. Say whether the main subject sits near the curl.
Check yourself: What is left when a square is removed from a golden rectangle?
A smaller golden rectangle.
What are the phi grid and the golden triangle?
Short answer. The phi grid divides the frame at 0.382 and 0.618. The golden triangle splits the frame with a diagonal and two perpendicular lines.
- The phi grid is like the rule of thirds, but its lines sit at 0.382 and 0.618 of the frame instead of 0.333 and 0.667. The crossing points are a little closer to the centre.
- On a 3:2 frame the difference between the two grids is small, about 5 per cent of the width. In practice you often cannot tell them apart.
- The golden triangle works on the diagonal. Draw a line from one corner to the opposite corner. Draw a line from each of the other two corners that meets the diagonal at a right angle.
- The frame is then cut into four triangles, two large and two small. The two points where the lines meet the diagonal are places to put key details.
- The perpendicular-to-the-diagonal construction is the reciprocal method taught in Hambidge's dynamic symmetry (Hambidge, 1967). Note that the two points where the lines meet the diagonal depend on the shape of the frame. On a 3:2 frame they fall at about 0.31 and 0.69 of the diagonal, not at golden positions. The name golden triangle is a teaching label, not a proven rule.
- The golden mean is the same idea applied to proportions across a whole design, such as the width of a column against the width of a margin.
Use the phi grid when you want a subject slightly closer to the centre than the rule of thirds puts it. Use the triangle when your scene has a strong diagonal.
When a shot has a strong diagonal, such as a staircase or a street, the golden triangle shows where actors and objects can sit along it.
Cham tipIf the thirds and phi grids both fit, choose the one that feels better. The difference is small.
Cham taskTake one photograph. Crop it twice, once by the thirds and once by the phi grid. Describe the difference.
Check yourself: Where does the phi grid put its lines?
At 0.382 and 0.618 of the frame.
How do I use the golden ratio when I shoot?
Short answer. Use it as one more way to place the horizon, the subject and the crop. Check it in the viewfinder or when you crop.
- Place a horizon on a phi line so that the sky or the land fills about 62 per cent of the frame.
- Place the eye of a portrait, a face or the main object near a crossing point of the phi grid.
- Many photo editors offer golden ratio, spiral and triangle overlays in the crop tool. In Adobe Lightroom Classic you can press O to cycle through the overlays.
- Use the overlay while you crop. Do not force a picture into it if the picture looks right without it.
- Look for the ratio in the world as well: window frames, doors, and the way a path turns. Use it to see, not to prove a point.
- Your slides group the uses into four areas: photography of every category, cinematography, graphic design and everywhere else. The use is the same in each: it is a guide to division.
Pick a scene and shoot it with the horizon on the phi line, then on the thirds line. Keep the one you prefer.
The same division can guide lower thirds, titles and frame lines on a film. Keep them consistent across a project.
Cham tipUse an overlay to check a crop, then ask whether the picture works without it.
Cham taskCrop five of your photographs with a golden overlay. Mark each one as improved, same or worse.
Check yourself: Where can the horizon go in a golden ratio composition?
On a phi line, so that the sky or land takes about 62 per cent of the frame.
Is the golden ratio really more beautiful?
Short answer. The evidence is mixed. Careful reviews find no strong proof that the ratio is universally preferred or that classical artists used it.
- Slides and popular articles often say the ratio is found in all beautiful things, in nature and in famous art. That claim needs care.
- Markowsky reviewed the claims in 2005. He found no credible evidence that Greek artists or architects used the ratio, and measured the Parthenon at close to 9 to 4, not 1.618 (Markowsky, 2005).
- In informal audience tests described in the same review, people did not choose golden rectangles more often than other rectangles, and the most chosen rectangle was about 1.83 to 1 (Markowsky, 2005).
- The first experiments were run by Fechner in the 1870s. He reported a peak of liking for golden section rectangles. Later attempts to repeat them gave a mixed picture (Phillips et al., 2010).
- A 2012 study asked viewers to rate triangles of different shapes. Ratings did not peak at the golden ratio. They fell as triangles became thinner, and compact shapes were rated higher (Friedenberg, 2012).
- The name golden ratio is a nineteenth-century creation, not an ancient one (Markowsky, 2005).
- The ratio is often drawn over Leonardo's paintings, including the Mona Lisa. Markowsky concludes that it does not appear Leonardo used it (Markowsky, 2005).
- Green reviewed the psychology experiments in 1995. He found real effects in some studies, but also found that weak methods easily produce or hide them (Green, 1995).
- The honest summary: the golden ratio is a neat piece of mathematics and a reasonable proportion for a crop. It is not a proven law of beauty.
If you use the ratio, say that you use it as a tool. Do not claim a photograph is good because it follows the ratio.
Film frames are chosen for story and effect. A frame can be fine without the ratio and weak with it.
Cham tipTest a claim before you teach it. Measure the thing yourself.
Cham taskPick a famous painting or building that is said to use the golden ratio. Measure it from a good image and report what you find.
Check yourself: What did Markowsky find about the Parthenon?
Its measured proportions are close to 9 to 4, not 1.618.
How is the golden ratio used in logo design?
Short answer. Designers build logo grids from circles inscribed in Fibonacci squares, so that the parts of the mark relate by a fixed proportion.
- Draw a Fibonacci tiling of squares. Place a circle inside each square. The circles have radii in the Fibonacci ratio.
- Use the circles as the construction grid for a logo: join arcs, trim shapes and keep the curves tangent to the circles.
- The result has parts that relate to each other by the same ratio, which gives the mark an ordered look.
- Check the ratio by overlaying circles on the final logo. If they do not fit well, adjust the logo and not the circles.
- A logo built this way still has to work at small size, in one colour and on different backgrounds.
- The ratio does not make a logo good. It is a way to organise decisions about size and spacing.
Photographers who design a logo for their own studio can use the grid to place the lettering and the symbol.
A film title card and a production logo can use the same circle grid so that the mark and the lettering relate.
Cham tipBuild the logo on a grid, then remove the grid and judge it by eye.
Cham taskDesign a simple monogram with your initials using the Fibonacci circle grid. Show the grid and the final mark.
Check yourself: What shapes make the golden ratio logo grid?
Circles drawn inside Fibonacci squares.
Questions beginners ask
What is the golden ratio?
It is the proportion of about 1.618 in which the whole is to the longer part as the longer part is to the shorter.
What is the difference between the golden ratio and the rule of thirds?
The rule of thirds puts its lines at 0.333 and 0.667 of the frame. The phi grid puts them at 0.382 and 0.618. The difference is small.
What is the Fibonacci sequence?
It is the series 1, 1, 2, 3, 5, 8, 13 and on, where each number is the sum of the two before it. The ratio of neighbours moves toward 1.618.
Is the golden ratio the best composition rule?
There is no strong evidence that it is. Studies are mixed, so treat it as one useful option.
Did the Greeks use the golden ratio in the Parthenon?
Careful measurements do not support it. One review measured the proportions at close to 9 to 4.
How do I use the golden ratio in Lightroom?
Open the crop tool and cycle through the overlays with the O key. Choose the golden ratio, spiral or triangle overlay.
References
- Euclid (c. 300 BCE) Elements, Book VI, Definition 3. Available at: https://mathcs.clarku.edu/~djoyce/elements/bookVI/defVI3.html (Accessed: 31 October 2026).
- Friedenberg, J. (2012) 'Aesthetic judgment of triangular shape: compactness and not the golden ratio determines perceived attractiveness', i-Perception, 3(3), pp. 163-175. Available at: https://pmc.ncbi.nlm.nih.gov/articles/PMC3485848 (Accessed: 31 October 2026).
- Green, C.D. (1995) 'All that glitters: a review of psychological research on the aesthetics of the golden section', Perception, 24(8), pp. 937-968. doi:10.1068/p240937.
- Hambidge, J. (1967) The elements of dynamic symmetry. New York: Dover Publications. (Republication of the 1926 edition, New York: Brentano's.)
- Livio, M. (2002) The golden ratio: the story of Phi, the world's most astonishing number. New York: Broadway Books.
- Markowsky, G. (2005) 'Book review: The golden ratio by Mario Livio', Notices of the American Mathematical Society, 52(3), pp. 344-347. Available at: https://www.goldennumber.net/wp-content/uploads/Livio-Golden-Ratio-book-review-by-Markowsky.pdf (Accessed: 31 October 2026).
- Phillips, F., Norman, J.F. and Beers, A.M. (2010) 'Fechner's aesthetics revisited', Seeing and Perceiving, 23, pp. 263-271. doi:10.1163/187847510X516412.
- Sigler, L.E. (trans.) (2003) Fibonacci's Liber abaci: a translation into modern English of Leonardo Pisano's Book of calculation. New York: Springer.
How to cite this guide. Alex, A., & Joseph, A. (2026). Golden ratio: A guide for photographers and cinematographers. Creative Hut Academy. https://creativehut.org/library/golden-ratio/
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