About rule of the third

-- martin tai (martin.tai@capcanada.com), November 15, 1999

Answers

Response to Rule of the third

Euclid's Golden Mean

One of the composition principle in photography is "rule of the third", which states that the main subject of interest in a picture should not be right at the middle of the picture, but should be positioned at two third postion. More specifically, divide a frame into tic tac to style grid, then position the main object at one of the thirds intersection.

Rule of the third has long history, dated back to ancient Greece.

Euclid discussed how to partition one straight line segment into two parts A and B, and asked what was the most pleasing proportion ? He found that if A/B = B/(A+B) then the partition , which he called phi, is uniquely defined. And he postulated that this proportion was the most pleasing porportion, and called it GOLDEN MEAN.

This Golden Mean proportion influenced the Greek architecture.

Mathematically let A/B =phi, then A/B= B/(A+B) can be reduced to phi=1/(1+phi)

Solving for phi resulted in phi =0.5(sqrt(5)+1) =0.618

This 0.618 number is roughly = 2/3. Hence the Euclidean Gold Means when applied to photography becomes "Rule of the third".

However the ratio 3/8 is closer to 0.618

-- martin tai (martin.tai@capcanada.com), November 15, 1999.

Response to Rule of the third

Fibonacci Numbers

A thirteen century mathematician Leonardo Fibonacci discovered the series:

0,1,1,2,3,5,8,13,21,34,55... in which a number is formed by summing two preceding numbers.

Seventeen century astromomer Kepler when studied the arrangement of leaves in plants found that many arrangements in nature followed the pattern of Fibonacci series. A closed form equation for Fibonacci series was discoverd by French mathematician de Moivre, and he proved that the limit of the ration of two adjacent Fibonacci numbers is none other than the Euclidean Golden Mean 0.618

2/3 = 0.666, 3/5 = 0.6 5/8= 0.625, 8/13= 0.615 .... the further it goes, the closer the proportion of two Fibonacci numbers to 0.618 the golden mean.

-- martin tai (martin.tai@capcanada.com), November 15, 1999.

Response to Rule of the third

Fibonacci Numbers and Newton Equation

Newton equation 1/f=1/d+1/D is a fundamental law of geometrical optics, in which f is the focal length, d is the object distance and D is the subject distance.

The Fibinacci series has a mysterious relation with Newton's equation:

if A,B, C are three consecutive Fibonacci numbers, then

1/A=1/B+1/C.

For example

1. 5,8,13: 1/5 = 1/8 + 1/13
2. 8,13,21: 1/8= 1/13+1/21
3. 34,55,89: 1/34=1/55+1/89 etc

There is really something divine about the Fibonacci and Gold Mean, no wonder in Renainssance, the Golden mean is also knon as "Divine ratio"

There is something mysterious about the rule of the third !

-- martin tai (martin.tai@capcanada.com), November 16, 1999.

Response to Rule of the third

The Nine Cell Grid

There is no equivalent of the Fibonacci number 0.618 in Chinese visual art.

However the use of Nine Cell Grid in caligraphic grid is commonplace.

The nine cell grid

is also often used in Feng Sui and architecture

Recent thread on Golden Mean in Photo.net

-- martin tai (martin.tai@capcanada.com), December 02, 1999.

Response to Rule of the third

While the "Nine cell grid" had impact on caligraphy, Fung Sui and architecture (The most prominant example has being the Ming Dynasty Imperial Palace at Beijing, the Forbidden City is positioned at the center cell or No 5 cell of Nine cell grid, Nine-Five was the symbol of Son of Heaven: I Ching: Nine-Five: Flying Dragon in the Sky ).

But the main format of Chinese painting was long scroll, so the influence of Nine Cell Grid (derived from the River Diagram and Book of Lou ) was absent.

However, in landscape paintings, so called "three section partitions" method was often used, the top one third for distant mountains, the middle third for clouds, the bottom third for the land and nearby scenes.

-- martin tai (martin.tai@capcanada.com), December 04, 1999.

Fibonacci number and Golden Mean

-- martin tai (martin.tai@capcanada.com), July 31, 2001.

Nothing mysterious about 1/a = 1/b + 1/c at all.

1/5 = 1/8 + 1/13 8/40 = 5/40 + 1/13 104/520 = 65/520 + 40/520

104/520 = 105/120

It's close, but it's not a rule.

1/1 = 1/2 + 1/3? Nope, doesn't work....

1/3 = 1/5 + 1/8? 40/120 = 24/120 + 15/120 40/120 = 39/120 close again, but not correct

1/8 = 1/13 + 1/21? 273/2184 = 168/2184 + 104/2184 Close again, but not a rule.

-- Lawrence Nyveen (nyveen@videotron.ca), October 08, 2001.