# Geometry of Prime Spaces An explanation of how the prime-coordinate representation induces a rich geometric structure that reveals deep patterns in object composition and relationships. The prime-coordinate representation induces a rich geometric structure that reveals deep patterns in object composition and relationships. The prime-coordinate space inherits a natural Euclidean structure from the coherence norm and inner product: - The coherence norm ||φ(x)|| measures the intrinsic complexity of an object - The inner product ⟨φ(x),φ(y)⟩ quantifies the degree of prime-factor similarity - Angles between coordinate vectors represent compositional relationships - Orthogonal objects have no common prime factors Objects with integer exponents form a discrete lattice in prime-coordinate space: - Each lattice point represents a unique factorization - Lines through the origin correspond to powers of the same object - Convex regions define families of objects with bounded prime exponents - Walking along lattice paths corresponds to sequential multiplication by primes The prime-coordinate representation can be viewed as a spectral decomposition: - Each prime contributes a 'frequency' component to the object - The exponent vector forms the object's spectrum across prime frequencies - Composition combines spectra through vector addition - Decomposition separates an object into its spectral components The coherence metric defines notions of distance and proximity between objects: - d(x,y) = ||φ(x) - φ(y)|| measures compositional distance - Neighborhoods contain objects with similar prime factorization - Geodesics represent minimal paths between objects in prime-coordinate space - Curvature appears when considering non-Euclidean coherence measures These geometric structures provide powerful tools for analyzing patterns, relationships, and transformations across mathematical domains. ## References - [[uor-c-020|prime-space-geometry]] - [[uor-c-021|euclidean-structure]] - [[uor-c-022|lattice-geometry]] - [[uor-c-152|coherence-metric]] ## Metadata - **ID:** urn:uor:resource:geometry-prime-spaces - **Author:** UOR Framework - **Created:** 2025-04-22T00:00:00Z - **Modified:** 2025-04-22T00:00:00Z