Finding Wedges to Drive thru the Cracks in General Relativity...

Goedel Goes Swimming in the Dirac Sea...

Notes from the Underground....

 Hilbert and Peony curves are Infinity limiting curves (therefore quantum) not classical?...

<i>Euclidean, spherical (or elliptic), and hyperbolic geometry are sometimes called the “classical geometries”. The Euclidean plane and the 2-dimensional sphere are familiar since they can be embedded in the 3-dimensional space in which we live. However, <b>there is no smooth isometric embedding of the hyperbolic plane in Euclidean 3-space, as proved by David Hilbert more than 100 years ago [6]</b>. Thus we must rely on nonisometric models of it. This is probably the reason for the late discovery of hyperbolic geometry by Bolyai, Lobachevsky, and Gauss almost 200 years ago. And it wasn’t until the late 1860’s that Eugenio Beltrami discovered what are now called the Poincar´e disk and half-plane models of the hyperbolic plane. Almost a century later Escher received a copy of a paper from the Canadian mathematician H.S.M. Coxeter[1]. The paper contained the hyperbolic triangle pattern shown in Figure 2. Escher said that the Figure 2 pattern gave him “quite a shock” since it showed him how to make a repeating pattern with a circular limit (hence the name for his “Circle Limit” prints); he was already familiar with patterns with point limits (with dilation symmetries) and “line limits”. Thus inspired, Escher created Circle Limit I in 1958, a rendition of which is shown in Figure 3. Over the next two years Escher created three more “Circle Limit” prints:

Limits in geometry and physic represent the crucial boundary where one mathematical or physical system transforms into another, often bridging the gap between discrete approximations and continuous, exact, or macroscopic reality. In geometry, this involves approaching curves or infinite shapes, while in physics, it describes the transition from probabilistic quantum systems to deterministic classical behavior (classical limit). 

1. Limits in Geometry

• Circle Limits (Polygonal Approximation): A circle is defined as the limit of a regular polygon as the number of sides approaches infinity (
  
  
  
  ). As side lengths decrease towards zero, the perimeter approaches the circumference (
  
  
  ) and the area approaches 
  

  $$
  .
• Line Limits (Geodesics): In hyperbolic geometry, as seen in M.C. Escher's "Circle Limit" works, "straight" lines (geodesics) appear curved as they approach the boundary of a Euclidean disk. These are lines within a restricted domain that operate as straight paths.
• Infinite Limits/Limits at Infinity:
  • Unbounded Functions: As a variable approaches a boundary (e.g., 
    
    
     for 
    
    
    
    
    
    
    ), the function value may approach positive or negative infinity (
    
    
    
    
    
    
    ), which is often visualized as a vertical asymptote.
  • Horizontal Asymptotes: As a variable approaches infinity (
    
    
    
    ), the function may approach a finite value, like a damped oscillation approaching zero, representing a limit at infinity. 

2. Classical vs. Quantum Physics Applications

The transition from quantum to classical physics is a major application of limits, often termed the Classical Limit. 

Concept	Classical Physics	Quantum Physics	Limit Application
State	Deterministic (exact trajectory)	Probabilistic (wave function)	(Classical Limit)
Energy	Continuous	Quantized (Discrete steps)	Large quantum numbers ( )
Size/Scale	Macroscopic	Sub-micron/Microscopic	or   (Thermodynamic)
Motion	Rigid/Definite	Tunneling/Wave-like	High energy/mass ( )

Key Limits in Physics:

• Planck’s Constant 
  
  
  : This is the most common mathematical limit used to recover classical Newtonian mechanics from quantum wave equations. When quantum action is much smaller than the system's total action, quantum effects become negligible.
• High Energy/Quantum Number Limit (
  
  
  
  ): According to Bohr's correspondence principle, as quantum energy levels (n) increase, quantum probability distributions become increasingly uniform, matching classical predictions.
• The Thermodynamic Limit (
  
  
  
  ): Used to connect microscopic quantum behavior to macroscopic classical thermodynamics, where the number of particles (
  

  $$
  ) and volume (
  

  $$
  ) go to infinity while density stays constant.
• Infinite Potential Walls: In quantum mechanics, a particle in a box with an "infinite" potential boundary (
  
  
  
  ) forces the wavefunction to zero at the walls, defining the boundaries of the particle's movement, similar to geometrical constraints in classical physics. 

Classical vs. Quantum Examples:

• Particle in a Box: A quantum particle at 
  
  
   has a high probability in the center, whereas a classical particle is uniformly distributed. As 
  
  
  
  , the quantum distribution becomes uniform, reaching the classical limit.
• Scattering: A hard sphere acts differently in quantum vs. classical scenarios, but they merge when particles are large enough compared to the de Broglie wavelength

Universal Expansion of SpaceTime = Time (locality's intelligibilty "illusion")

More Google AI:

In Book X of Laws, Plato (via the Athenian Stranger) identifies 10 categories of motion to prove that soul is the primary source of all change, existing before body and matter. These motions, which classify all physical and metaphysical activity, are central to his argument for the existence of gods and against atheism.

The 10 Motions (Laws 893c–894c)

Plato divides motion into two main types: those that move only other things, and those that can move themselves.

Physical/Derivative Motions (Motions 1–8): 

1. Circular motion around a fixed center.
2. Locomotion (gliding or rolling).
3. Combination (joining together).
4. Separation (breaking apart).
5. Increase (growth).
6. Decrease (decay).
7. Becoming (generation/birth).
8. Perishing (destruction/death).

Primary/Self-Motions (Motions 9–10):

9. Other-affecting motion: Motion that can move others but cannot move itself.
10. Self-and-other-affecting motion: Motion that is capable of moving itself as well as others.

Context in Book 10

• Soul as Self-Mover: Plato argues that the 10th motion—the self-moving motion—is the definition of "soul" (or psyche). Therefore, soul is the prime mover that initiates all other motions.
• Theological Purpose: The Athenian uses this to argue that because the soul is the cause of all change, the orderly motions of the cosmos (sun, stars, seasons) must be directed by a divine, rational soul.
• Critique of Materialism: Plato is refuting pre-Socratic philosophers who claimed that inanimate matter (fire, air, earth, water) is the original cause of motion, arguing instead that soul is older and superior to body.