Golden Ratio: Math or Art?

By Tracy Man

(Mathematics & School/University Entrance Exam Tutor)

What is the Golden Ratio?

Many of our greatest mathematical minds, spanning across centuries from the times of the Ancient Greeks up to our present day, have dedicated their lives studying this ratio and its properties. Ancient Greek Mathematicians began studying this ratio due to its repeated appearance in Geometry and often referred to its ‘sectioning’ or ‘dividing’ properties. In fact, its common name ‘The Golden Ratio’ was not designated until 2000 years later by a German Mathematician, Martin Ohm in 1835. Its origins, however, can be traced back to Euclid, who refers to the Golden Ratio as ‘extreme and mean ratio’.

The Golden Ratio, represented by φ or ‘phi’, is a special number in Mathematics equating to 1.618. It is calculated when a line is divided into two parts so that the longer segment divided by the shorter segment is equal to the whole line divided by the longer segment.

The irrational number also relates to Pythagoras’s Theorem and other mathematical concepts such as Geometry and the Fibonacci Sequence.

If we look at the property above where   , we can derive its formula:

The given property:   

Rewrite the equation:   

Since   is the Golden Ratio 

The golden ratio can be expressed in terms of itself  

As mentioned earlier, the ratio is an irrational number, and this can be seen clearly when we try to construct the Golden Rectangle.

Step 1: Construct a 1 by 1 square

Step 2: Mark the midpoint, “m”, on one of its sides

Step 3: Connect the midpoint “m” to one of its opposite corners with line “l”

Step 4: Using a compass, construct a curve with “m” as the centre point and “l” as the radius

Step 5: Extend the square

What you can see from the above diagram is that line  can be found using Pythagoras:

 

If, the shorter side is  and the longer side of the rectangle is 

 

The Fibonacci Sequence , where the next term is found by adding the previous two terms, is related to this ratio. The ratio of two consecutive terms in the sequence is remarkably close to the Golden Ratio. In fact, as the terms increase, the ratio between the chosen pair of numbers becomes closer to the approximation.

Applications of the Golden Ratio

Despite its long history and deep roots in Mathematics, The Golden Ratio is not just applied in Mathematics and Geometry; it is a concept used in other numerous fields such as Art and Architecture. In the field of Art, it is more commonly referred to as the Golden Section or the Divine Proportion. As Luca Pacioli, a contemporary of Da Vinci, said:

“Without Mathematics there is no art.”

 

The Golden Rectangle is formed by using the Golden Ratio for its dimensions.  If you look closely, the Golden Rectangle represents a typical frame used for painting. To many artists, the Golden Ratio is believed to be the most aesthetically pleasing and beautiful shape in the world, a reason why artists naturally gravitate towards paintings which are created using this ratio.

 

One of the world’s most renowned artists, Leonardo Da Vinci, believed that the Golden Ratio is portrayed by some bodily proportions. He used the Golden Spiral, which stems from the Golden Rectangle and the Fibonacci Sequence, to create his famed painting the Mona Lisa. The Golden Spiral begins from her left wrist, which then travels round and frames her entire face before the final rotation ends at her nose. This is important as the nose is the ideal place to look as it is central point of the face.

Other fellow Renaissance artists have used the Golden Section to produce their masterpieces: Sandro Botticelli’s ‘The Birth of a Venus’ and Michelangelo’s ‘The creation of Adam’.

 

Though it has been debated and refuted, many people have claimed that we can find traces of the Golden Ratio applied to architectural structures which dates back centuries, with The Great Pyramid of Giza (2570BC) being one of the earliest examples. Pyramidologists, interior designers, and historians have theorized that Egyptians naturally sought the Golden Ratio even without Mathematical techniques. Another well-known structure, the Parthenon (447-432BC) was believed to have been created using the Golden Rectangles, from its floor plan to the spacing between each column, but recent studies have questioned this idea. However, Phidias, and Ancient Greek sculptor, studied ‘phi’ and it is believed that he designed the sculptures of the Parthenon based on this ratio.

 

 

To this day, the Golden Ratio is still applied to designs and art works. Many modern-day companies apply the ratio for the design of their logos. For example, Twitter, if you look at its design you can see that the logo is mainly based on the usage of geometry and perfect circles, which derives from the Golden Rectangle. This concept is also applied to webpage layouts and book cover designs as it is intentionally used to appeal to its consumers.

Did you know? The Golden Ratio is also equal to  .


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