Tripagy

Description This work introduces Tripagy, a proposed geometric framework based on the concept of hole-image geometry. Tripagy investigates geometric objects through admissible transformation processes rather than through shape alone or ordinary topological equivalence. The central idea is that geometric objects may be related by sequences of transformations involving penetration, hole formation, hole expansion and contraction, stretching, folding, bending, and other admissible deformations. In particular, a hole created during a transformation is treated as a persistent geometric structure: it may be deformed, enlarged, or reduced, but its complete elimination is not regarded as an admissible transformation. Tripagy therefore distinguishes between the geometric state of an object and the transformation history through which that state was obtained. The order of transformations is also significant. For example, penetration followed by hole formation need not be equivalent to hole formation followed by penetration. This leads naturally to a directed transformation structure in which transformations are not necessarily reversible. A central concept introduced in this framework is the "same-image relation." Two geometric objects are regarded as same-image when there exists a geometric state that can be reached from both objects through admissible Tripagy transformations. This provides an alternative way of relating geometric objects without requiring ordinary topological equivalence or mutual reversibility. The framework also introduces restrictions on admissible shape transformations, including a distinction between Tripagy-convex and Tripagy-concave states. These terms are defined internally within the proposed framework and are not intended to have exactly the same meanings as the conventional mathematical definitions of convexity and concavity. The present work provides a preliminary axiomatic and constructive formulation of Tripagy. It develops the concepts of geometric states, persistent holes, penetration, hole formation, admissible deformation, transformation barriers, directed reachability, same-image relations, and candidate Tripagy invariants. Several foundational problems remain open, including the rigorous mathematical definition of penetration, the quantitative characterization of shape-state transitions, the conditions under which the same-image relation becomes transitive, explicit constructions between representative geometric objects, the classification of Tripagy invariants, and the algebraic and differential-geometric structure of the resulting framework. Tripagy is presented as a proposed mathematical framework rather than as an established theory. The purpose of this work is to establish its basic definitions and principles and to provide a foundation for further mathematical investigation into transformation-based geometry, persistent geometric holes, and alternative notions of geometric equivalence.