Here (a + b) / a = a/b=Φ≈1.6180339887Golden ratio is (a+b):a or a:b which is 1: Φi.e. 1: 1.6180339887…

The Greek used it extensively for beauty and balance in the design of the Parthenon and other architecture:

dimensions: 27 feet long, 18 1/2 feet wide, and 23 feet tall. The ratio of height to diameter of the columns is 7:1,

DORIC ORDER

Examples of the Doric order are Parthenon the temple for the goddess of wisdom.

                                

                                       

LONIC ORDER

Some examples of the Lonic order are the temple of Apollo at Didyma/Turkey.

The temple of Athena Nike was also the Lonic order.

                                  

CORINTHIAN ORDER

The Corinthian order was the latest order.

The oldest example of the Corinthian order was the temple of Apollo at Bassae.

The Greeks made little use of this order.

                                

Great Mosque of Kairouan

The grand Imam Mosque

The carved stucco of the ceiling in the Hall of the Two Sisters in The Alhambra palace in grenda

Φ was used in the construction of the Taj Mahal,

Indians employed planning principles and proportions that rooted the buildings to the cosmos, considering the movements of sun, stars, and other heavenly bodies. vastu shastra , the ancient Indian canons of architecture and town planning employs mathematical drawings called mandalas

Jaipur was designed using the pithapada mandala as the basis. In this mandala of nine squares that represent the universe, earth occupies the central square. In the city, which consists of nine large squares, the central square is assigned to the royal palace. They calculated and planned city map using recursion:Length of City / Length of Middle Town : 771.1/340.5 ≈ 2.26; Length of Middle Town / Length of Castle: 340.5/151  ≈  2.26

During the Indus Valley Civilization at the sites of Harappa and Mohenjo-Daro constructions took place following precise mathematical calculation

x x

 Pythagoras theorem used in architecture: The Samrat Yantra, at Jaipur, designed by Jai Singh, measuring 147′ at its base and 90′ high could calculate time within two seconds accuracy per day. . 

pillar of mehrauli

Φ was used it in the design of Notre Dame in Paris, which was built in the 1163 and 1250

 Christian church in china

Chichen Itza, Mexico has fifty two panels on each side of the pyramid represent the number of years in the Mayan cycle, the stairways dividing the eighteen tiers correspond to the Mayan calendar of eighteen months and the steps within El Castillo mirror the solar year, with a total of 365 steps, one step for each day of the year.

STRUCTURAL ENGINEERING

Structural engineering. What shapes are rigid or flexible, how they respond to forces and stresses. Statics (resolution of forces) is essentially geometry. This goes over into all levels of design, form, and function of many things.

Development of the Tower 1 level 43 floor plan from a Rub el Hizb symbol( Islamic symbol)

The CN Tower in Toronto has contains the golden ratio in its design. The ratio of observation deck at 342 meters to the total height of 553.33 is 0.618 or

phi, the reciprocal of Phi!

where   is the half-width of the Tower at height  ,   is the half-width of the Tower at the ground and    is the maximum wind pressure the Tower can withstand at a height  . 

WILLIS

TOWER

Φ was used in the UN building

The Gherkin’s(UK) unusual design features – the round building, bulge in the middle, the narrow taper at the top and spiraling design – create an impact in more ways than you might think. The cylindrical shape minimizes whirlwinds that can form at the base of large buildings, something that can be predicted by computer modeling using the math of turbulence.

The Eden Project, Cornwall, UK

the building structure, is derived from phyllotaxis, the mathematical basis for most plant growth (opposing spirals are found in many plants, from pine cones to sunflower heads).