Time vs Distance: a Space-Time Conundrum

from Google AI:

The Principle of Least Time (Fermat's Principle) states that light travels between two points along the path that requires the minimum amount of time, whereas the older Principle of Least Distance (Hero's Principle) states that light follows the shortest physical path length. [1, 2, 3]

Key Differences

Hero's Principle of Least Distance: Proposed around 100 A.D. by Hero of Alexandria, this concept assumes light takes the path of absolute shortest length. It correctly explains simple phenomena like the Law of Reflection, where the angle of incidence equals the angle of reflection on a flat surface. [1, 2, 3]
Fermat's Principle of Least Time: Proposed in 1662 by Pierre de Fermat, this modern concept corrects and expands the distance rule by factoring in propagation speed. It states time—not distance—is minimized, which correctly explains why light bends (Snell's Law) when passing through different mediums (like air into water) where its speed changes. [1, 2, 3, 4, 5]

When They Agree vs. Disagree

Same Medium: When light stays within a single uniform medium, its speed is constant everywhere. Minimizing the distance automatically minimizes the travel time, making the least distance path identical to the least time path. [1, 2]
Different Mediums: When light crosses a boundary into a slower medium, a straight line (least distance) spends too much time in the slow zone. Bending the path shortens the time spent in the slow medium by traveling a slightly longer physical distance through a faster medium—proving that the actual path minimizes time, not distance.

More from Google AI:

The Dzhanibekov effect (also known as the Intermediate Axis Theorem or Tennis Racket Theorem) cannot be explained by the Principle of Least Time (Fermat's Principle), which governs how light paths travel. Instead, it is beautifully explained by the Principle of Stationary Action (commonly referred to as the Principle of Least Action) mapped onto rigid body rotational dynamics. 

1. The Energy and Momentum Constraints

To understand why the object behaves this way using foundational physical principles, we look at two values that must remain strictly conserved during torque-free flight: 

Kinetic Energy (

$E_k$
):




Total Angular Momentum (L²): L² = L₁² + L₂² + L₃² 

Where I₁ < I₂ < I₃ are the moments of inertia, and L₁, L₂, L₃ are the components of angular momentum along those axes. 

According to the Principle of Least Action, a physical system will always follow a path where the "action" (the time integral of the system's kinetic energy minus potential energy) is stationary. In pure rotation with no external forces, this translates to the path of least resistance on the intersection boundaries: 

Axis Type	Energy State	Intersection Geometry	Rotational Behavior
Minimum Axis (I₁) (e.g., spinning a phone vertically)	Maximum Energy allowed for that momentum.	The ellipsoid and sphere touch only at tiny, isolated points.	Stable: Perturbations have nowhere to go. The object just wobbles tightly around the pole.
Maximum Axis (I₃) (e.g., spinning a phone flat)	Minimum Energy allowed for that momentum.	The ellipsoid and sphere touch at isolated points on the opposite end.	Stable: Lacking excess energy, the object cannot escape this tight track.
Intermediate Axis (I₂) (e.g., spinning a phone flipping end-over-end)	Intermediate Energy.	The intersection forms a saddle point with sweeping, continuous loops that wrap all the way around the sphere.	Unstable (Dzhanibekov Effect): Even a microscopic misalignment sends the system sliding down a massive trajectory loop.

 

3. The Geometry of the Flip

Because the intermediate axis sits at a saddle point, a slight push does not result in a tiny orbit (like the stable axes). Instead, the path of stationary action dictates that the system must traverse the entire length of the intersection loop to satisfy conservation laws. 

As it traverses this loop, the angular velocity vector rapidly migrates away from the original axis, passes the other poles, and lands exactly on the inverse side—producing the sudden 180-degree flip. Once it completes the loop, it flips back, repeating the cycle indefinitely until an external force or energy dissipation disrupts it

Two Hemispheres, Two Types of Attention

Why "Decolonization" Matters.... The Multi-Cult-y AI Tower of Babel

from Google AI:

Optimality Theory (OT)

Developed alongside Alan Prince in 1993, Optimality Theory radically changed linguistics. While not a traditional "neural network," it was heavily inspired by the way connectionist networks settle conflicting constraints. [1]

• Universal Constraints: OT states that languages don't use strict, unbreakable rules. Instead, the human brain possesses a universal set of conflicting constraints.
• The "Limits" of Language: Grammatical variation between different languages is entirely determined by how a specific culture ranks these constraints. An output is "optimal" if it violates the fewest high-ranking limits

 More from Google AI:

Optimality Theory (OT) is a linguistic framework developed by Alan Prince and Paul Smolensky in 1993 that models human language through ranked, violable constraints rather than rigid, unbreakable rules. [1, 2]

Core Architecture

Generator (GEN): Takes a lexical input and generates an infinite or wide set of potential surface-form candidates. [1]
Evaluator (EVAL): Assesses the candidate set based on a language-specific strict domination ranking of universal constraints and selects the optimal output. [1, 2]
Constraints (CON): Universal constraints divided into Markedness (demanding structural simplicity or well-formedness in outputs) and Faithfulness (demanding identity between input and output). [1, 2]

Key Properties

Violability: Constraints can be violated, but violations are minimized based on the hierarchy.
Strict Domination: A higher-ranked constraint takes absolute priority over any combination of lower-ranked constraints.
Typological Variation: Differences between human languages emerge purely from different rankings of the same universal constraint set