Binary Relations as Primitive Notions for Hyperbolic Three-Space and the Inversive Plane

In the foundations of geometry, selecting appropriate primitive notions is crucial for developing coherent mathematical theories. Traditional approaches to hyperbolic geometry and inversive geometry have relied on multiple foundational concepts such as points, lines, circles, and distance functions. However, emerging research suggests that a single binary relation can serve as a sufficient primitive notion to express both hyperbolic three-space (H) and the inversive plane.

The Concept of Primitive Notions

Primitive notions are concepts that are not defined in terms of previously defined concepts but are accepted as undefined terms within a formal system. They form the bedrock upon which mathematical structures are built through axioms and definitions. In Euclidean geometry, for instance, points, lines, and the relation of incidence are typically taken as primitive notions, with axioms establishing their properties and relationships.

The choice of primitive notions is not unique; different systems can use different primitives while ultimately describing the same mathematical structures. The elegance and power of a particular choice of primitives is often measured by how they can serve as a foundation for rich and complex mathematical theories with minimal complexity and maximum conceptual clarity.

Binary Relations as a Geometric Foundation

A binary relation R on a set S is formally defined as a subset of S S. We write R(a,b) to indicate that the ordered pair (a,b) belongs to R. In geometric contexts, elements of S can represent various geometric objects, and the relation R captures fundamental interactions between these objects.

The power of binary relations as primitive notions lies in their remarkable expressiveness. Through appropriate axiomatic constraints, a single binary relation can encode intricate geometric structures. By carefully choosing axioms for R, we can characterize entire geometric frameworks without introducing additional primitive concepts, thereby achieving both conceptual economy and formal elegance.

Formulating Hyperbolic Three-Space

Hyperbolic three-space, denoted H, is a complete Riemannian manifold with constant negative curvature of -1. It can be represented through multiple equivalent models, including the Poincar ball model, the upper half-space model, and the hyperboloid model. In the Poincar ball model, H is represented as the interior of the unit ball in , with a specific metric that causes "lines" (geodesics) to appear as circular arcs orthogonal to the boundary sphere.

When approaching H through binary relations, we begin with a set S of elements (representing geometric objects) and a single binary relation R. In one powerful formulation, elements might represent oriented geodesic surfaces (hyperbolic planes) in H, with R representing orthogonality between these surfaces. The essential properties of H can then be encoded in axioms such as:

Through such axioms, we can derive the complete structure of H, including its geodesics, isometric transformations, and curvature properties, all within a framework that uses only a single binary relation as a primitive notion.

Formulating the Inversive Plane

The inversive plane is the Euclidean plane extended by adding a "point at infinity." This extension allows circles and lines to be treated uniformly as "circles" (with lines understood as circles passing through the point at infinity). The primary transformations in this geometry are inversions in circles and reflections in lines, which together generate the group of Mbius transformations.

Using binary relations as primitive notions, the inversive plane can be constructed with a set S of circles and a binary relation R representing orthogonal intersection. In this framework, the axioms might include:

These axioms capture the essential structure of the inversive plane, including circle geometry, inversion transformations, and conformal properties, all expressed using only a single binary relation.

The Natural Connection Between Geometries

Perhaps the most profound aspect of using binary relations as primitive notions is how it reveals the natural connection between hyperbolic three-space and the inversive plane. This connection can be understood through the following elegant construction:

For hyperbolic three-space formulated via binary relations, we can define ideal boundary elements as limit points of geodesic sequences. The collection of these boundary elements, with the appropriate relation induced by R, can be shown to be isomorphic to the inversive plane structure.

This correspondence is intimately related to the Poincar ball model of H, where the boundary of the ball can be identified with the sphere at infinity. Through stereographic projection, this sphere relates naturally to the inversive plane. The binary relation framework makes this correspondence almost trivial: the elements of the inversive plane can be identified with certain boundary elements of H, and the relation R in the inversive plane is simply the restriction of the relation from H.

This construction reveals that the inversive plane is naturally the conformal boundary of hyperbolic three-space, a relationship that emerges transparently when both geometries are built from the same primitive notion.

Group-Theoretic Perspective

From a group-theoretic standpoint, the binary relation approach illuminates deep connections between the symmetry groups of these geometries. The group of isometries of hyperbolic three-space, denoted Isom(H), is naturally understood as acting on the boundary space.

In the binary relations framework, this correspondence becomes a direct consequence of how the relations extend from H to its boundary. The transformations that preserve the binary relation in H must also preserve the induced relation on the boundary, giving a natural identification between Isom(H) and a subgroup of the conformal transformations of the inversive plane.

This perspective highlights that hyperbolic geometry and inversive geometry are not merely analogous but intrinsically connectedthe inversive plane is essentially the "shadow" cast by hyperbolic three-space on its boundary.

Theoretical and Computational Advantages

Beyond its theoretical elegance, the binary relations approach offers several practical advantages:

• Conceptual Unity: The use of a single primitive notion provides a unified framework for studying seemingly disparate geometries.
• Computational Simplicity: Algorithms for geometric operations can be expressed directly in terms of the fundamental relation, potentially simplifying implementations.
• Formal Verification: The streamlined axiomatic system reduces complexity for formal verification and automated theorem proving.
• Algorithmic Generalization: By varying the axioms governing the relation, the same algorithmic framework can potentially be used for different geometries.
• Computer Graphics Applications: The unified framework provides a natural way to implement non-Euclidean geometries in graphics systems.

Conclusion

The use of a single binary relation as a primitive notion for both hyperbolic three-space and the inversive plane represents a significant development in the foundations of geometry. This approach provides a more economical foundation while revealing deep connections between these geometries.

By viewing these geometries through the lens of binary relations, we gain new insights into their nature and relationships. The unified framework offered by this approach has both theoretical significance, advancing our understanding of geometric foundations, and practical value, potentially simplifying computational implementations of non-Euclidean geometries.

This approach exemplifies a broader trend in mathematics toward using relational frameworks as foundations, connecting geometric insights with algebraic, logical, and computational perspectives. As research continues in this direction, we can expect further unification of diverse geometric structures under common relational frameworks, deepening our understanding of the nature of mathematical space.