Geometry is usually taught as the study of objects: points, lines, planes, shapes that exist independently “out there.” Relational ontology dissolves this presumption. Geometry is not a catalogue of things; it is the formalisation of relational potential actualised through perspectival cuts.
To think geometrically is to think in terms of structured distinctions, not substances. In this first post, we examine the simplest and most fundamental geometrical forms — points, lines, and lattices — and reinterpret them as expressions of relational coherence.
Points: The Limit of a Cut
A point is traditionally imagined as a location, an indivisible object in space. In relational ontology, a point is nothing of the sort.
A point is a perspectival limit — the place where relational differentiation is maximal. It is the boundary of a cut, where the act of distinction produces a localised singularity of coherence.
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It is not “there” in space; it is the act of stabilising a relational potential at a maximal focus.
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Points are the first manifestation of patterned attention, the seeds of all higher geometrical forms.
Lines: Minimal Continuities
Once points are actualised, lines emerge naturally. Lines are not collections of points. They are the minimal continuities that preserve relational coherence between two cuts.
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A line is a path along which potential can be consistently actualised, a stabilisation of continuity.
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It does not “connect points” in space; it preserves the integrity of relational tension across a domain.
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Curvature, later, will appear as a consequence of tension between incompatible cuts along this line.
Lines, then, are habits of relation — the simplest “motion” of relational potential actualised perspectivally.
Lattices: Organising Relational Potential
Lines intersect. Points cluster. From these interactions, lattices arise: regular grids or networks of relational constraints.
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Lattices are stabilised scaffolds of relational differentiation, organising potential into patterns that can be systematically understood.
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They are formal, not physical: no matter, no substance, no independent existence. They exist only insofar as cuts impose coherence.
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In practical terms, lattices underlie classical notions of space and structure, but in relational ontology they precede all representational objectification.
A lattice is, therefore, a network of potential actualisations, the grammar through which complex relational patterns may later emerge.
From Points to Lattices: A Relational Insight
Points, lines, lattices — these are not objects, but structured moments of potential made coherent by our cuts. Geometry, at its most basic, is a language of relational constraints, not a description of a pre-existing spatial theatre.
This reframing has consequences beyond abstraction:
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Classical physics assumes points and lines as primitive; relational topology shows them as emergent from pattern-stabilising cuts.
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Quantum and cosmological structures, when interpreted relationally, are built on lattices of potential, not fields of substance.
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Our intuition of “space” is thus the echo of relational patterns actualised in a particular perspective.
Next Steps
In the next post, we will explore curvature and relational tension, showing how the geometry of constraints produces the phenomena we traditionally call surfaces, manifolds, and curved space.
Geometry, in relational ontology, is not discovered. It is performed, and its deepest truths are revealed not in objects, but in the logic of relational potential itself.
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