Deriving the Pentagon Fold Using the Golden Ratio To fold a regular pentagon from paper, the key quantity is the exact value ofcot⁡72∘\cot 72^\circcot72∘, which is deeply connected to the golden ratio. Exact Value of cot⁡72∘\cot 72^\circcot72∘ We start with the identity:cot⁡72∘=5−255\cot 72^\circ = \sqrt{\frac{5 – 2\sqrt{5}}{5}}cot72∘=55−25​​​ Notice that:(5−1)2=6−25(\sqrt{5} – 1)^2 = 6 – 2\sqrt{5}(5​−1)2=6−25​ […]

Steven’s notebook from 2026 January 12th. He decided to use a new notation to express the sumation of terms. Below includes the extracted info and the original hand writen notes Basic Sums of Powers Definition We define a summation operator for a function f(x)f(x)f(x):S(f)=∑x=1nf(x)S(f) = \sum_{x=1}^{n} f(x)S(f)=x=1∑n​f(x) Our goal is to compute sums of the