Quantum Consepts - that can be wrong-but Usually ain't

Overview: In curved (non-Euclidean) geometry, the classic Pythagorean relation a 2 + b 2 = c 2 a^2 + b^2 = c^2 for a right triangle must be modified to account for space curvature. One proposed correction is: a 2 + b 2 + h   a 2 b 2 R 2 = c 2 , a^2 + b^2 + h\,\frac{a^2 b^2}{R^2} = c^2, where R R is the radius of curvature of the space and h h is a chirality factor that can be +1 or –1 . This formula suggests that the sum of the squares of the legs is not exactly equal to the square of the hypotenuse on a curved surface – there is an extra term proportional to the product a 2 b 2 a^2 b^2 scaled by 1 / R 2 1/R^2 . We will analyze how this curvature term works, the role of the sign h h (positive vs. negative curvature), and how it connects to known geometric laws like the Law of Cosines. We’ll also discuss how different sign choices and values of a , b , c , h a, b, c, h lead to different geometric interpretations, and highlight physical/mathematical contexts where this corrected ...

Isaac Newton’s Hidden Work and Intellectual Evolution Introduction Sir Isaac Newton’s genius extended far beyond his published works in physics and mathematics. In private, Newton developed a trove of mathematical innovations and engaged in intense theological and alchemical studies that he deliberately kept from public view. Understanding Newton’s unpublished mathematical work alongside his psychological influences and mystical pursuits reveals a more complete picture of how his ideas evolved. This report examines Newton’s secret equations and manuscripts, the personal experiences and anxieties that shaped his intellectual life, the interplay of his scientific ideas with his theology and alchemy, and a timeline mapping the development of his thought. In doing so, we contextualize Newton’s hidden work within the rich psychological and historical backdrop of his life, drawing on his private notebooks, correspondence, and modern scholarly analysis. Synthesis of Newton’s Unpublish...