Tiling: The Pattern Repeats Across Scale
A tiling is a way of filling a space with a repeated pattern. In ART, the point is not decorative. The theory claims that recursive identity produces a self-similar structure: the same kind of relation appears again and again across scale.
This gives the theory a way to move from local relation to a larger ordered field without inventing a new rule at every level.
ℂP³: The Space That Holds the Relations
ℂP³ is the projective geometric space ART uses for the complete structure. Its role in the argument is to hold the theory's relational identities in a form that can later be read as phase space.
In the physics volume, this becomes important because ℂP³ can be read as a kind of phase space: a structure pairing position-like and momentum-like aspects before ordinary spacetime is derived.
Symmetry: What the Pattern Preserves
Symmetry means that something can transform while preserving what makes it itself. In ART, the complete tiling selects a symmetry called U(3). The full derivation belongs in the primary text; here, the essential point is that the geometry has a lawful structure that constrains what can happen next.
Once the geometry and symmetry are established, the physics volume has somewhere to begin. The next question is how physical spacetime and forces emerge from them.