On Geometry and Physics Aspects of The Generalized Newton's Laws Triplet (G, h, k_B)

The Generalized Newton's Laws triplet (G, h, k_B) is constructed by three monodromy varieties: (1). Newton's gravity constant monodromy G=2/3 =0.666 666 666...(CODATA value G= 6.674 30× 10^-11 m³ kg^-1s^-2), (2). Planck reduced constant h =2π√3 = 1.08827961854×10 .... (CODATAvalue h= 1.054571817... ×10^-34 J.m), (3). Boltzmann constant k_B = 8√3 = 1.38564064606×10 ... (CODATA value k_B = 1.380649 × 10^-23 J K^-1). It is naturely that the triplet (G, h, k_B) is invariant monodromy variety and has a deep geometry and physics meanings.

In this paper, I will present an extended Fano geometry and its topological structures: (1). Newton's constant is extended the Newton gravity variety G_i = (m_iV_i)^-1 , i = 1,2,3,..., ...n, (2). extended Planck variety h^ν = 2π×ν, where ν is special real number, such as, 0, ..., √5^-1, √3^-1, 1, √3, √5, ... ∞. (3). extended Boltzmann variety k_Bⁿ = 2√3×n, n =1,2,3,4,..., ∞. Notes that: gcd(h, k_B) = gcd(2√3×π, 2√3×4) = 2√3, i.e., P¹ project. The classical Fano geometry includes only a circle with radial √3, a envelope triangle with boudary length l= 3×6=18 and a square with boundary length 2√3×4 = 8√3= 19.5959179423. This new Quantum Fano geometry has rich topological structure:

the generic enveloped length l_n =2√3n tan(π/n)|_n→∞ = 2√3π = h. One can obtains that the enveloped area A_n= 2√3×√3 tan(π), then that the enveloped ratio η= l_n /A_n= 2√3×n tan(π/n) / 2√3 ×√3n tan(π/n) = 2/√3 = 2/3√3 = G√3. For a circle, its enveloped ratio η_?= 2π√3 / π√3² =2/√3. It means that all polytopes have an exact same enveloped ratio, which is governed strongly by gravitational constant G.