Two of the most common composition methods employed by photographers are the "Rule of Thirds" and "Golden Ratio." Both methods can be often found in all sorts of mainstream visual mediums - from your favorite instagram photographer to your high-budget Hollywood movie. I am going to go over some of the basics of these methods, and how they can maximize the potential of capturing your viewer's interest.
Pictured above is a common sight on your camera's live view - be it your DSLR or even your smart phone. The Rule of Thirds dictates that by aligning your subject matter along these lines or at the points of intersection creates "points of interest" that will attract your viewer's eyes. This gives way to an easy method of composing your shots and subject matter. Often times in landscape photography - I find myself cropping the top horizontal third to fit the sky, the middle with the subject, and the bottom third with the foreground.
Pictured above is an example of the "rule of thirds" in action in "The Office." During many of the show's character insights - you'll see this commonly used in their composition whether it is a single or even two people. Here Dwight loses his shit composed along the left vertical line.
The "Golden Ratio" is another common method of composition. First appearing in works by the Greek architect, Phidias, then popularized with Leonardo Fibonacci's famous sequence (1, 1, 2, 3, 5, 8, 13...) valued at 1.618 number and math - lets get to visuals.
Visualized with the "golden rectangle" above, each spiral intersection of the rectangles cuts smaller rectangles and squares in proportion to the "golden ratio." Known as "God's Fingerprint," this ratio randomly appears all throughout nature, from the formation of sea shells to the arrangement of flower petals. You can see mankind's obsession with this ratio in the architecture of the Greek Pantheon or Da Vinci's Mona Lisa.
By composing your images with a particular focus point within the smallest squares in the spiral - visual presentations are said to be naturally "aesthetically pleasing." Now this method is absolutely harder to pull off on the go without drawing your own focus point on your viewfinder - but that doesn't necessarily stop others from trying. Below is an example of a few shots from the movie "There Will be Blood." It can be argued some of the shots are kind of stretch - but you'll find others must have been composed with the ratio in mind. Perhaps the music's dramatic tone makes up for it.