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

I have been trying to hone the actual equation down and I think I finally have the complete version.  While a2+b2+h(a2b2/R2)=c2 works great... it's not as easy to use... this new theory fixes some of the ambiguity.  Table of Contents Introduction Spherical and Hyperbolic Pythagorean Theorems Spherical: Exact Law of Cosines Spherical: Taylor Expansion Hyperbolic: Exact Law of Cosines Hyperbolic: Taylor Expansion Summarizing the Curvature‐Corrected Formulas Rigorous Checks of the Expansion Order Estimates and Neglected Terms Consistency with Euclidean Limit Geometry of “Proper Triangles” vs. “Right Triangles” Gauss–Bonnet and Angle/Area Proofs Gauss–Bonnet in 2D Relationship to the Sides a , b , c a,b,c Scissor Congruences and Area Comparisons Connecting to Spin‐ 1 2 \tfrac12 The Rotation Group S O ( 3 ) SO(3) vs. Its Double Cover S U ( 2 ) SU(2) Sign Flip for a 2 π 2\pi Rotation: Proof from Group Theory Geometric Interpretation: S 2 S^2 (Bloch...

For a demonstration of the Lewis Drift, I recommend watching this video. https://youtu.be/LL7LG6vWfNg This is a model of the Lewis Drift in action using Algodoo, a physics simulator.  In the video, you see how I create the system and how I elicit the effect.  The following is the math and physics that explain why the Algodoo program is actually working fine.  1. Introduction Many mechanical systems that involve gyroscopes, constrained linkages, and friction exhibit non‑intuitive behaviors wherein purely rotational inputs (such as spinning rotors or wheels) cause gradual translation—or “drift”—over time. This is somewhat analogous to wave or fluid‐mechanics contexts, where repeated oscillations yield small net displacements (often called a “drift,” akin to “Stokes drift” in fluid literature). In a similar vein, the system here is: Three “orbs” (masses) of equal mass, connected by an inextensible chain A gyroscope mounted to one of the orbs, capable of delivering a ...