The Golden Equation

The Golden Equation, or Golden Identity, is an equation that contains 4 famous numbers, used frequently in maths and science. 

1) phi (golden ratio) is related to Fibonacci sequence, and both are seen in many structures from galaxy shapes to fractal patterns in plants.

2) pi (Archimedes’ constant) is the ratio of a circle circumference to its diameter and used in trigonometry and physics whenever circles and oscillations are involved.

3) e (Euler’s number) is related to power laws and exponential growth/decay.

4) i (imaginary number) is the square root of -1 and is found to be essential in quantum physics, one of the two main pillars of physics. 

5) The number 5 has interesting relations within the equation as well as other properties.

a) The equation has 5 arithmetic operations: addition, subtraction, multiplication, division and exponentiation. It has 5 terms, which consist of 3 variables and 2 operators (= and +). 

b) Humans have 5 fingers and 5 toes on each hand and foot. We also have 5 appendages (2 arms, 2 legs and 1 head) and 5 senses (sight, smell, taste, sound, touch). 

c) The symbol of the Pythagoreans is the pentagram, a star formed by drawing the diagonals of a regular pentagon. 

d) 5 is the sum of the smallest even prime number (2) and the smallest odd prime number (3).

Equation Explanation: The left side of the Golden Equation is phi and base on its defining formula, equates to a value which is about 1.618. The right side of the Golden Equation which contains e, i and pi is also found to equate to this value, which equals phi (see algebraic proof). The geometric proof shows that phi equals 2 x cos(pi/5), and also that in a regular pentagon with sides = 1, the diagonal is phi.

For a detailed description of the golden ratio phi, please see the link below. 

https://thegoldenequation.blogspot.com/2023/02/golden-ratio.html?m=1