The Simplex Ladder

1 · 2 · 3 · 4 — Troolean relaxation across dimensions

Oneness · vertices. Twoness edges. Threeness triangles. And fourness the tetrahedron — four is one plus three, a triangle plus an apex. Every simplex carries a three-valued Troolean truth (vfalse / ish / vtrue), and relaxation labeling settles them into agreement across dimensions through the boundary coupling. Click any piece — a dot, an edge, a face — to anchor a truth and watch it climb the ladder.

vtrue (green) ish (blue) vfalse (red)
What this is (honestly). A real illustration of higher-order relaxation labeling on a simplicial complex. Each k-simplex relaxes toward agreement with its faces (dimension k−1) and cofaces (dimension k+1) — the coupling encoded by the boundary operator ∂ and the Hodge Laplacian, the honest generalization of graph diffusion to shapes. The labels are Troolean — three-valued logic (Łukasiewicz/Kleene), real. So threeness (triangles) genuinely interacts with twoness (edges) and oneness (vertices), and the tetrahedron lifts the ladder to four. The theology is Manny's; the topology is the mathematics'.

← Simplicial (3D) · The Four Poles · Genesis