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Among history, artist, designers and architects used some mathematical ratios and equations to help them with their work. Between those ratios one got famous in western tradition and other got famous in the East.
The one that got famous in the West is called golden ratio. This mathematical ratio is believed to be some kind of divine proportion (as called by Paciolli and Da Vinci in their “De Divina Proportione”) that regulate the form of natural things and the human body. We can see the use of the golden ratio in the Euclid’s Elements, in the Renaissance art and architecture and even in modern architecture. The other that got renowned in East is called silver ratio or Japanese ratio (“Yamato-hi” for Japanese). They used the silver ratio in Buddha statues, in architecture and in anime characters, like Doraemon.
Each of those ratios are defined by a geometrical relation. The golden ratio is defined by dividing a section of line in two parts, part a and part b, in a way that the total length of the line (a+b) divided by the greater segment (a) is equal to the division between the two new segments (a and b) as we can see in the equation in the side and the image below, resuming: . The value of this ratio is always an irrational number that we round to 1.618. The silver ratio is quite similar, however, instead of being the total length of line (a+b) divided by the greater segment (a) that is equal to the division of the between the two segments (a and b), it’s the double of the greater segment added by the smaller segment (2a+b) divided by the greater segment (a) that is equal to the division of the between the two segments (a and b), resuming: . The value of this ratio is or, rounded, 1.414.
Golden and silver ratio line
Le Corbusier and the Golden Ratio
The famous architect Le Corbusier was one of the architects that believed and used the golden ratio in his works and advocates for its application in everyday life. To endorse the use of this ratio in any type of architectural work, interior design or product design, and inspired by Leonardo da Vinci’s Vitruvian Man, he created what he calls “Modulor”.
Image 3 – Modulor
A Le Corbusier’s sketch with relation between heights and the Modulor with distances in cm
The Modulor used the golden ratio to calculate relations between parts of the human body. For this, he used, for a base, a six-foot tall man (182 cm) with one of the arms raised. However, preserving the adequate proportions, anyone could transpose it to make it work with people that are taller or smaller than the one Le Corbusier used in the Modulor. Based on this, an architect or a designer could know the best height and angle to put or to better design some furniture, doors, windows and even door handles.
One example of Le Corbusier’s architecture project designed with the help of the golden ratio is the United Nations Secretariat Building. In this building design, Le Corbusier worked with Oscar Niemeyer. Niemeyer is a famous Brazilian architect that were heavily influenced by Le Corbusier since he came to Brazil in 1935 to help Lúcio Costa (another Brazilian architect and also professor at Rio de Janeiro Federal University) and his students to design a Modernist building to the local government and teach Modernist urbanism to them. Being Niemeyer one of Costas’ student, he also learned a lot from Le Corbusier. After this, Niemeyer developed his own language, but still was influenced by what he learned from Le Corbusier.
On the building they designed together, we can see golden rectangles in the façade. We also can see the golden proportion being used in the window configuration of the building and, according to the website The Golden Number, we can also see the use of the golden ratio in the front entrance of the building and in the interior floor plans.
UN Secretariat Building and Golden Rectangles
UM Secretariat Window Configuration
The silver ratio in japanese architecture
In Japan they don’t believe much in the use of the Golden Ratio on design. To them, the use of the Silver Ratio create a design that is more beautiful and serene than the Divine Proportion. Since the Silver Ratio derives a smaller proportion (1.414), the objects that are done based on it are closer to a square than the ones done based on the western proportion.
One famous example of the use of the silver ratio is the Horyu-ji Temple in Ikagura, Nara Prefecture, Japan. This temple is a Buddhist temple with one of the oldest wooden building of the world. As we can see in the image below, the relation between the ground floor of the left building and its second floor is 1.414, and the relation between the first roof of the right floor and its last roof is also 1.414, the valor of the silver ratio.
Horyu-ji Temple with Pagoda
As happened with our ratio, they were more used in the past, but we can also see contemporary works that used it, like the Tokyo Skytree. The Tokyo Skytree is one of the world’s tallest tower, having two observatories and a digital broadcasting antenna at the top. According to the image bellow, we can see a silver ratio relation between the distance of the floor to the second observatory and the distance between the floor and the top of the tower.
Today, neither of the two proportions is seen as a divine relation or a design rule, but we can still see projects that used them as a way to create a guidance or a feeling of harmony and serenity.
 PACIOLLI, Luca; DA VINCI, Leonardo. 2014. De Divina Proportione (On the Divine Proportion): Facsimile in Full Color of the Original Version of 1509. USA: CreateSpace Independent Publishing Platform.  EUCLID. 2017. Euclid’s Elements. Kindle edition. USA: LRP.  EPIC RAP BATTLES OF CARTOONS WIKI. “Doraemon”. Retrieved June 07, 2018 (http://epic-rap-battles-of-cartoons.wikia.com/wiki/Doraemon).  WIKIPEDIA. 2007. “Golden Ratio”. Retrieved June 06, 2018 (https://en.wikipedia.org/wiki/Golden_ratio).  THE GOLDEN RATIO. “Le Corbusier”. Retrieved June 06, 2018 (https://divinumproportione.weebly.com/le-corbusier-and-modulor.html).  ICON. “Modulor Man by Le Corbusier”. Retrieved June 06, 2018 (https://www.iconeye.com/opinion/icon-of-the-month/item/3815-modulor-man-by-le-corbusier).  THE GOLDEN NUMBER. “The UN Secretariat Building, Le Corbusier and the Golden Ratio”. Retrieved June 07, 2018 (https://www.goldennumber.net/un-secretariat-building-golden-ratio-architecture/).  THE GOLDEN NUMBER. “Phi and the Golden Section in Architecture”. Retrieved June 06, 2018 (https://www.goldennumber.net/architecture/).  THE GOLDEN NUMBER. “Phi and the Golden Section in Architecture”. Retrieved June 06, 2018 (https://www.goldennumber.net/architecture/).  FANCLIP. “キティーちゃんの顔バランス。そうだったのか！！「白銀比」「黄金比」の話”. Retrieved June 08, 2018 (https://www.fanclip.jp/blog/archives/614)  FANCLIP. “キティーちゃんの顔バランス。そうだったのか！！「白銀比」「黄金比」の話”. Retrieved June 08, 2018 (https://www.fanclip.jp/blog/archives/614)
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