The Prime Gear Geometry: A Mechanical Sieve of Survivors

In the world of numbers, prime numbers are the atoms. Just as every molecule is built from specific elements, every integer greater than 1 is either a prime or built by multiplying primes together. This is the Fundamental Theorem of Arithmetic. Despite their importance, primes have long been shrouded in mystery. They appear to be scattered randomly across the number line, yet they follow a rigid, "exact" logic. This article explores a new way to visualize this logic: Prime Gear Geometry.