THE GOLDEN RATIO
In mathematics two quantities are in the golden ratio if the ratio of the sum of t e quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. E xpressed algebraically:
Where the Greek letter ‘phi’ () represents the golden ratio. Its value is:
Two quantities aandb are said to be in the golden ratioφ if:
One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ,
it is shown that
Multiplying by φ gives
which can be rearranged to
Using the Quadratic Formula, tw o solutions are obtained:
Because of the fact that φ is the ratio between length and width of a rectangle, which are non- zero, the positive solution must b e chosen:
The Golden Ratio in Architecture
The Golden Ratio has appeared in ancient architecture. The examples are many, such as the Great Pyramid in Giza, Egypt w hich is considered one of the Seven World Wonders of the ancient world, and the Greek Parthenon that was constructed between 447 and 472BC. Not only did the ancient Egyptians and Greeks know about the magic of Golden Ratio, so did the Renaissance artists, who used th e Golden Ratio in the design of Notre Dame in b etween the 12th and 14th centuries. Some moder n architecture are also influenced by Golden Ratio as well, such as the United Nations Building.
The Great Pyramid at Giza
Half of the base, the slant height, and the height from the vertex to the center create a right triangle. When that half of the ba se equal to one, the slant height would equal to the value of Phi and the height would equal to th e square root of Phi.
The CN Tower in Toronto, the tallest tower and freestanding structure in the world, 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!
The UN Building
In the United Nations building, the width of the building compared with the height of every ten floors is a Golden Ratio.