Phi - not to be confused with pie!
By Thomas Kuijlaars (Yr 9)
I’ll start with an example: Draw a straight line 10cm long, then make a mark on the line at 6.18cm. The line is now in two parts, a longer side (6.18cm) and a shorter side (3.82cm). Now divide the whole length of the line by the length of the longer side (10/6.18= 1.618). Now divide the length of the long side by the short side (6.18/3.82=1.618). You will find that the answers will not be exactly the same but would be very close to 1.618. This is because the numbers have been simplified for you. This number is called phi or Φ , and is an ancient Greek letter. Phi has strange properties, for example Φ 2 = Φ+1 and 1/ Φ= Φ-1 . The Fibonacci sequence is also very closely related to the number. If you divide any number in the Fibonacci sequence by the number which precedes it, you will get a ratio close to phi. This ratio gets closer and closer as you move down the sequence but it will never quite reach the exact number. In fact phi is an irrational number meaning if the value is written as a decimal it would have an infinite amount of decimal places. Phi has been also used in art. Leonardo Da Vinci and many other artists, called the number “the golden ratio” because the ratio had such aesthetically pleasing proportions. The Ancient Greeks and Romans also used the ratio in their architecture, one such example is the Parthenon. However, some architects and artists used the “golden rectangle”, which has two opposite sides of 1 and two opposite sides of phi. This is meant to be the “most beautiful rectangle”. Using this rectangle you can also form the “golden spiral” as seen below.
References The Magic of Mathematics and Mathematical Footprints by Theoni Pappas
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