A beautiful equation relating the Golden Ratio phi to pi, e and i !

 


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 This blog contains new findings that I have on the Golden Ratio (phi). In the world of numbers, an enchanting sequence unfolds—one that has captivated mathematicians, nature enthusiasts, and artists for centuries. This sequence, known as the Fibonacci sequence, is a series of numbers where each term is the sum of the two preceding ones: 1, 1, 2, 3, 5, 8, 13, and so on.     Named after the Italian mathematician Leonardo of Pisa, who was known as Fibonacci, this sequence emerges from a simple yet profound recursive pattern. Fibonacci introduced it to the Western world in his book "Liber Abaci," where he described its application to problems of growth and reproduction in rabbits.     The sequence's charm extends beyond its numerical progression. As you divide consecutive terms, you find that the ratios converge toward a constant value, approximately 1.61803398875. This value, known as the golden ratio φ (phi), has earned its place in art, architecture, and nature....
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Golden Identity and a Geometric Proof

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Merchandise:  https://www.teepublic.com/t-shirt/39715980-beautiful-golden-ratio-phi-equation The Golden Identity or Equation is an exact and beautiful equation that  relates 4 fundamental numbers  used in maths and physics:  i (imaginary number), e (Euler’s number), pi and phi (golden ratio). i is critical in quantum physics, e is related to power laws and fractals, pi is related to circles and phi is related to Fibonacci sequence. The left side of equation is phi and based on its defining formula, equates to a value which is about 1.618. The right side of the 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. Another interesting property is that the equation has 5 arithmetic operations: addition, subtraction, multiplication, division and exponentiation.
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A diagram to explain why Golden Rectangle appears balanced to human eyes!

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Merchandise:  https://www.teepublic.com/t-shirt/50015094-golden-rectangle The image formed in each human eye is a circular patch, much like a pinhole camera image. The 2  circular mages from both eyes overlay to form a combined circular image that has a circular region that is more focused, and then less focused towards the outer part of the circle. Human eyes are aligned horizontal like other bipedal animals. This is either due to intrinsic body symmetry, or it allows us to scan movements and positions of preys and predators more easily. Because of horizontal eyes alignment, and left-right eyeballs movement, the less focused regions tends to form a horizontal shape rather than vertical. Finally, the golden rectangle appears balanced because the center circle which is the focused area, has the exactly same total area of the non-circle area  which is the unfocused area.
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