The Golden Ratio

Here are shown various ways to write the Golden Ratio, a favorite theme of mathematical recreation and investigation:

  • The equality of two ratios of simple algebraic quantities in two unknowns
  • The infinite limit of the ratio of successive Fibonacci numbers
  • In an infinite continued fraction representation
  • As a specific irrational number involving the square root of five (but cross multiply and 4 = 4 is explicit)

These instances remind me a bit of the Four Causes in their representations:

  • Relative and relational, but indefinite (Efficient)
  • Idealized as an infinite sequence and infinite completion (Formal)
  • Infinitely nested and recursively reductive (Material)
  • Definite but with embedded irrationality (Final)

Further Reading:

https://en.wikipedia.org/wiki/Golden_ratio

https://en.wikipedia.org/wiki/Fibonacci_number

https://en.wikipedia.org/wiki/Irrational_number

[*12.92]

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