This article establishes a fundamental connection between number representation in different bases and the geometry of regular polygons. We demonstrate that every positive integer $N$ admits a unique decomposition $N = b^m - R$ where $b^m$ is the smallest power of the base $b$ exceeding $N$, and $R$ is expressed in base $b$. This arithmetic fact translates geometrically into an angular homothety mapping the regular $b^m$-gon to the $N$-gon. Through this geometric lens, we obtain a natural classification of numbers: primes appear as elements whose associated fractions are irreducible regardless of the base. Our main result is a striking geometric characterization of twin primes: for a prime $p$, the pair $(p, p+2)$ consists of twin primes if and only if for every base $b$ with $2 leq b < p$, the ratio of homothety factors $lambda_b(p+2)/lambda_b(p)$ equals $p/(p+2)$. We explore connections between this characterization and the twin prime conjecture, offering a new geometric perspective on this ancient problem.