[MarkJerling]

As background: The Golden Ratio, also called phi, or Φ is the ratio of length to width of 1 to 1.6180339887… It can be algebraically expressed as x/1=(x+1)/x or x²-x-1=0 or 1+√5)/2.
So, we're talking, in essence, about any rectangle with the very specific height / width ratio of 1/1.6180339887… (Often expressed as 1.618.)

Here's a close proximation of the "true" Golden Ratio. I say "close", because, of course, we're limited with pixel size and scaling to something that works on the forum.
From here, we can superimpose the Golden Spiral which is not a true logarithmic spiral, but closely approximates to the Fibonacci spiral.



Next, we can mirror that over, and we get this:



And, from that, we can also take the main divisions, and get to an approximation of a "Golden Ratio inspired" "Rule of Thirds".



1.618. is an incommensurable number, which means it is an irrational number, which is why it’s shown with the dot or dots after the number. This means that, no matter to how many decimal places you calculated the number, you’ll never reach a finite number. Here it is, calculated to many more places:

1·618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137484754088075386891752126633862223536931793180.

Anyway, it so happens that, since time immemorial, the Golden Ratio has fascinated artists, mathematicians and Architects. Euclid wrote about it 2000 years ago, in his Elements; Leonardo Fibonacci, in 1202 CE in his book, Liber abaci, and in 1509 CE Luca Pacioli wrote at length about the number, although largely copying earlier writings by others. Phi went on to fascinate Leonardo de Vinci, Michelangelo and later Le Corbusier.

Of late, I've seen several photography webpages and blogs writing about it, and, they mostly get it wrong. Without raining on anyone specific's parade, here's a few examples:
Why The Golden Ratio Is Better Than The Rule Of Thirds | PetaPixel Petapixel gets it spectacularly wrong as none of their images are cropped to the Golden Ratio.
The Golden Ratio in Photography: A Comprehensive Guide likewise completely mucks it up.
https://photographyhero.com/golden-ratio-photography/ starts off well and explains things well, and then goes off the rails.
https://www.apogeephoto.com/how-to-use-the-golden-ratio-to-improve-your-photography/ mucks it up badly.
https://expertphotography.com/golden-ratio-photography/ gets more things right than most, but then hammers home on the golden spiral like most everyone else.
Using the Golden Ratio in Photography - Focus Camera is almost like Expert Photography but likewise loses the plot on the spiral. But, they do raise an interesting point with regard to the Fibonacci spiral - more about that below.
https://www.phototraces.com/golden-ratio-in-photography/ is more of the same, perhaps with slightly fewer errors. I almost got excited because their text does a good job but then, surprise surprise, they muck it up with the image examples.

So, to spare you all the deeper delving (unless you want to of course!) here's the take-out:

Most sites seem to fixate on the Golden Ratio spiral. After that, some sites appear to see merit in using the Golden Ratio as a revised "rule of thirds" grid.
But, the key problem I see is that they ALL ignore the basic premise of 1.618.

So, about the spiral: The Golden Ratio spiral is not, of course, a true logarithmic spiral as is found in nature. It's very close to the Fibonacci spiral, predominantly because Fibonacci's work closely relates to (and works around the) Golden Ratio. Peculiarly, not only do most of sites draw the Golden Ratio incorrectly, but some of them completely distort the spiral.

I include some overlays to show what I mean. (All taken from the sites mentioned above.)
No attempt to follow the true Golden Ratio:



Likewise:



Close, but no cigar:



Another complete stretch:



Likewise:



And, lastly, the image that looked "off" to me and prompted this whole investigation.



I think, if a photographer is going to attempt to work to the Golden Ratio, then as a first principle, they need to get the proportion of the image right. After that, the main divisions can be used, but the better way to do it would be to evaluate the sizes of the main elements in the image and look at the relationship in size between those main elements in terms of the Golden Ratio. I think the spiral is misleading at best and very possibly of minor importance in the scope of scale and proportion. The spiral is, after all, simply the result of overlaying ever decreasing groups of the Golden Ratio. The spiral is not the Ratio and not the proportion relationship. And, it bears no resemblance to the logarithmic spiral seen in nature.

Lastly, here's some of the worst examples I've found:
https://www.lifepixel.com/photo-tutorials/what-is-the-golden-spiral-and-how-to-use-it
https://cameraharmony.com/what-is-the-golden-ratio-and-how-to-use-it-to-your-advantage/
https://medium.com/billeasy-artists/designing-golden-ration-style-25be3178e604
https://blog.hubspot.com/marketing/golden-ratio

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Last edited by MarkJerling; 09-05-2022 at 08:44 PM. Reason: More added.

[pschlute]

Hi Mark, thanks for that. Very interesting.

I have always been fascinated by the Golden Ratio and the Fibonacci sequence, and not only because of my interest in photography.

I was a commodities/forex trader for many years, and relied heavily on Elliott-Wave analysis, which uses the Fibonacci sequence, and especially the numbers 0.382 0.618 and 1.618. Technical analysis in general has always been regarded by some to be dubious, and Elliott-Wave analysis akin to witchcraft ! But it worked for me.

[StiffLegged]

Thanks for the entertaining links, Mark! You might have missed the most egregious of them all, a photo including Rover doing what has to be done, duly overlaid with the famous spiral. As said, if the entire image isn’t arranged per the actual proportions then the ratio is meaningless as an explanation. There should also be a significant element of your composition at the spiral’s centre, not Auntie Mabel’s empty flower bed, but PetaPixel et al don’t get that.

[house]

You are wrong to assume that the image should be cropped to the golden ratio.

Proportional systems can be used to linearly divide any surface. Not all elements including "gaps" will conform in all directions. You need to be selective. You also need to determine tolerance. When does it stop being golden ratio. 0,1,2,3,4 decimals? With blobby features how do you determine where it should fall to conform? Don't expect precision.

Now as you say, the numerical ratio is what matters, stretching the spiral so that the proportions are wrong is a mistake. You can however correctly fit it inside or allow it to go outside the frame.

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Last edited by house; 09-05-2022 at 11:10 PM.

[biz-engineer]

As background: The Golden Ratio, also called phi, or Φ is the ratio of length to width of 1 to 1.6180339887… It can be algebraically expressed as x/1=(x+1)/x or x²-x-1=0 or 1+√5)/2.

At that point in the text, after the math equation, you already lost 90% of photographers, including myself

---------- Post added 06-09-22 at 11:42 ----------

Phi went on to fascinate Leonardo de Vinci, Michelangelo and later Le Corbusier.

Vilfredo Pareto came in and said the law of nature is such that 80% of your success come from 20% of your paintings. Say, no matter how hard you try to nail your composition using the golden ratio, Pareto will come and screw 80% of your shots anyway. Damn it!

[Michail_P]

As an artist and architect, I am truly fascinated by the whole existence of the golden ratio. It really is something magical, when you study the various aspects and applications of it. I am sure though that an approximation of the ratio in photography (and even architecture - yes, we don't live in Classical Greece any more) can be generally accepted, as long as the tolerance is controlled. I love the absolute and precise nature of mathematics and geometry, but our brain simply can't tell the optical difference between 1,6 and 1,7. What I find so interesting is the comparison of a linear and a spiral-based composition, the golden-thirds grid and the golden spiral. Yes, the irrational 1.618. is in there, but the compositions are so different...

[GUB]

So we need to be shaving 360 pixels off the vertical of our K-1 s? Seems a bit like sacrilege !

Haven't thought about this stuff since school days - tended to rely on instinct (for what its worth!)

How about now we try and shoot a few good examples for this thread?

[robgski]

Haven't thought about this stuff since school days - tended to rely on instinct (for what its worth!)

I do the same.

[AgentL]

What someone else said: we need to see some good examples, purpose-created to show what things should be like. Every single golden ratio article I've seen is just as the OP showed: totally shoehorned in and often absurdly off-topic on what parts of the image it puts in the "important" spots. I don't pretend to "get" the golden ratio because I've almost never seen it done well and in a way that is definitive. I'd like to see lots of examples of how it can be worked through, approximated and made into variations.

[GUB]

What someone else said: we need to see some good examples, purpose-created to show what things should be like. Every single golden ratio article I've seen is just as the OP showed: totally shoehorned in and often absurdly off-topic on what parts of the image it puts in the "important" spots. I don't pretend to "get" the golden ratio because I've almost never seen it done well and in a way that is definitive. I'd like to see lots of examples of how it can be worked through, approximated and made into variations.

OK here is an example. A file photo from a few years ago that did well in my Photo club confines and also one that I like.
First of all it is a 4 x 28mm on FF stitch and I originally cropped a little off each end of this original for my final print.
My instincts at the time said a little vegetation to the left gave some substance to the image and connected with the people.
And I felt it was critical the sand on the right broke the horizon.
I feel the golden ratio crop would fail this image.
But I feel the take home here is the spiral concept has a place - I had originally only thought of the sand tonings pointing to the subjects for emphasis..

[AgentL]

OK here is an example. A file photo from a few years ago that did well in my Photo club confines and also one that I like.
First of all it is a 4 x 28mm on FF stitch and I originally cropped a little off each end of this original for my final print.
My instincts at the time said a little vegetation to the left gave some substance to the image and connected with the people.
And I felt it was critical the sand on the right broke the horizon.
I feel the golden ratio crop would fail this image.
But I feel the take home here is the spiral concept has a place - I had originally only thought of the sand tonings pointing to the subjects for emphasis..

So, am I understanding the concept correctly that, for the image to be at maximum impact in terms of using this mechanism, the people would be at the smallest part of the "swirl" a little above where they are in this photo?

[AggieDad]

Thanks for bringing up the subject and your thoughts and illustrations.

The golden mean – also known as the golden ratio, golden section, or devine proportion – has been around seemingly forever, and no-one really knows exactly from whence it came. However, it can be traced back as far as Euclid (300 BCE). The Greeks found it to have proportions that were pleasing to them. The Parthenon is the classic example of the golden ratio in Greek architecture. During the Renaissance, daVinci and others revived interest in it, including a treatise, De divina proportione by Pacioli. But reach back even further than the Greeks to ancient Egypt and the golden ratio pops up in the pyramids of Giza where the ratio of the slant height to 1/2 the base is 1.618!

The formula for positive solution of x = (1 + square root of√5)/2. Who cares? What is interesting is that a golden rectangle (1.618 : 1) will resolve itself into two parts – a square and another golden rectangle. Whether you add a square to an existing g.r. or remove a square from it, the result will be a new golden rectangle. See the illustration. The big rectangle is golden and the pink rectangle is golden. You can readily see that adding (or subtracting) a square creates a new golden rectangle. The golden spiral that some photographers use is created by drawing an arc through each succeeding square.



A Fibonacci spiral has also been mentioned. To refresh the memories of anyone who has forgotten (), Fibonacci numbers are a sequence in which the next number in a sequence is the sum of the previous two numbers.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,...

The ratio of two consecutive terms (i.e. 8 and 13) approaches the golden mean. By the 6th iteration of the Fibonacci sequence, the ratio between these two Fibonacci numbers is a mere 0.007 difference from the golden ratio – 1.625 vs 1.618. It continues to get closer with each succeeding iteration.

So while the Fibonacci Spiral is not an exact overlay of the golden spiral, it is very close – certainly more than close enough for anyone's composition.

There is a lot of art and architecture which relate to the golden mean. In architecture Notre Dame, the Canadian National (CN) Tower, and the United Nations Building come to mind. In art, there are many examples, but think of the Girl With the Pearl Earring or Mona Lisa.

From our standpoint as photographers, I submit that the golden mean – whether in the form of the golden rectangle, golden triangle, or golden spiral – is a nice guideline for composition, but really no more than the rule of thirds. It is just a guideline; a suggestion.

[GUB]

So, am I understanding the concept correctly that, for the image to be at maximum impact in terms of using this mechanism, the people would be at the smallest part of the "swirl" a little above where they are in this photo?

I am trying to understand - just like you - not presenting an argument for or agin.
But the spiral to the focal point is what I was thinking as well as the spiral being a construction to connect parts of the image.
Here is another image - one where I have always enjoyed the juxtaposition of the cow against the yachting fraternity. (It is a screenshot to show how the golden ratio is close but not quite.)
What I am thinking if you are for instance a painter then this golden ratio would be a useful tool to craft your image. But we as photographers are stuck with what we are given. I feel the goldern ratio may be for us just confirmation that we are onto something.

[Des]

Great post Mark.

Phi went on to fascinate Leonardo de Vinci, Michelangelo and later Le Corbusier.

There is a lot of art and architecture which relate to the golden mean. In architecture Notre Dame, the Canadian National (CN) Tower, and the United Nations Building come to mind. In art, there are many examples, but think of the Girl With the Pearl Earring or Mona Lisa.

Source: Leonardo and Mathematics - The Mona Lisa Foundation

[MarkJerling]

Hi Mark, thanks for that. Very interesting.

I have always been fascinated by the Golden Ratio and the Fibonacci sequence, and not only because of my interest in photography.

I was a commodities/forex trader for many years, and relied heavily on Elliott-Wave analysis, which uses the Fibonacci sequence, and especially the numbers 0.382 0.618 and 1.618. Technical analysis in general has always been regarded by some to be dubious, and Elliott-Wave analysis akin to witchcraft ! But it worked for me.

Thank you Peter. Now, of course, when you're talking about Elliot-Wave analysis, that's pure wizardry!