THE GOLDEN MEAN

Let's start with an introduction of a technique that is well known for many centuries now: The "Golden Mean" (sometimes called "Golden Section") is a geometric formula by the ancient Greeks. A composition following this rule is thought to be "harmonious". The principal idea behind it is to provide geometric lines which can be traversed when viewing a composition. The Golden Mean was a major guideline for many artists/painters so it is certainly worth to have in mind for modern day photographers as well.

Theory - Part I

Well, let's begin with some words about the theory. The formula starts with a perfect square (marked blue in illustration A). Now we devide the base of the square into two equal parts. We take point x as the middle of a circle with a radius of the distance between point x and y. Thereafter we expand the base of the square till it hits the circle at point z. Now the square can be transformed to a rectangle with a proportion ratio of 5:8. The ratio of A to C is the same as the one from A to B. Luckily the 5:8 ration fits pretty close to the ratio of the 35mm format (24x36mm = 5:7.5).

Illustration A

Theory - Part II

So now we've something which is thought to be a "perfect" rectangle. What's next ? We draw a line from the upper left to the lower right edge of the rectangle (see illustration B) and another line from the upper right directed towards point y' (taken from illustration A) till it hits the first cross line. Obviously this divides the rectangle into three different sections.
In principal we're finished with the "Golden Mean" now. Just try to find objects/parts in your scene that fit roughly into these three sections and ... you have a "harmonious" composition.
You can vary the formula by flipping and/or mirroring the scematic rectangle from illustration B.

Illustration B

RULE OF THE THIRDS

The "Rule of the Thirds" is actually nothing else than a simplification of the "Golden Mean". The basic philosophy behind it is to avoid a symmetric compositon which is usually pretty boring because the view is centered. The connection to the "Golden Mean" are the 4 possible crossings of the dividing lines (see the examples in illustration C1 and C2).
To counteract symmetry the "Rule of the Thirds" can follow two concepts:

First we can divide the image into two distinctive areas which cover 1:3 and 2:3 of the size of the picture.

Illustration C1

The second possible application is directly based on the crossing points of the Golden Mean. e.g Let's assume that we a landscape that is pretty charming but lacks a major feature or interesting geometric structure. The resulting image is a boring picture of an empty landscape. So what can we do here. Try to find an object which provides a contrast to the otherwise "monotonious" surrounding and place it at one of these crossing points. This object is an anchor for the first look and invites to a further observation of the scene.

Illustration C2

FRAMING
Sometimes you've a object of huge dominace within a scene. While breathtaking on-location the final picture looks often much less impressive due to uninteresting space around the object.
Try to find a frame which can eliminate the unimportand surrounding and focus the view.
The following picture uses the surrounding trees as a sort of portal to frame the mountain in the center.

Natural holes like the one in the next picture are excellent frames.

CROSS-LINES
Crossing Lines/diagonals are actually again another simplyfication of the golden mean. The basic idea is to provide a sort of "guideline" for the eyes to follow. It is a good idea to place the start or end of such a line to one of the extreme edges. The classical approach states that the upper left edge is the best starting point because most humans start to traverse a picture from here on. However, it cannot hurt to break this rule (see 2nd picture). Just a straight line would be pretty boring thouhg so there should be some sort of disturbance in the picture.
The following picture shows a focus point where many lines find together so there are enough of directions for the eyes to follow making the picture interesting.

by Horst Scheider
The next picture has two anchors - the boat and the sun shade. These provide just the right amount of disturbance for this otherwise very symmetrical composition.

by Horst Schneider

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