Geometry doesn't encode properties directly — it encodes frequency behavior, which produces properties. Every crystal structure is a circuit diagram written in geometric shapes.
The Alphabet
Foundational cavity shapes — the alphabet of the circuit language. Interactive 3D viewer with per-shape geometry and dimensional frequency-sweep heatmaps. Pair this with the seven foundational expressions below to read what each shape does.
The Words
When two shapes connect — face-sharing, edge-sharing, or vertex-sharing — the combination produces a new signal type. Spinel is a gate. Perovskite is a switch. Diamond is an insulator. The connection geometry determines the function.
The Sentences
Three connected shapes form the minimum functional circuit. Capacitors, gates, filters, memory cells, and polarization rotators — all from geometry alone. Crystal structures read as complete circuit diagrams.
Atomic frequency-expression archetypes
Non-negative matrix factorization on 27,830 symbol-instances finds seven additive archetypes: CONCENTRATE, DISTRIBUTE, RADIATE, POLARIZE, TRIANGULATE, CUBIFY, FRUSTRATE. Every instance is a weighted mixture; the geometry rarely commits to a single archetype. The phonemes of the language.
Identity · ratio · purity · breadth — at the 1% activity threshold
The next layer up from the seven phonemes. Every symbol-instance is catalogued by which foundationals participate (in dominance order), in what 5%-binned ratio, and at what signal purity. Empirically the vocabulary uses only 55 of 13,699 possible identity signatures, and 98 behaviors form the stable core (≥50 instances each). 3D and 4D held separately; higher dimensions pending.
A separate cosine-distance clustering of the same atlas
1,417 hierarchical clusters of the 27,830 symbol-instances by cosine-distance similarity on their raw 44-feature signatures. This is a parallel view from the v1pct Behavior Vocabulary above — it groups by overall fingerprint similarity rather than by identity·ratio·purity of the foundational mix. Useful for browsing instance neighbourhoods.
The substrate — the alphabet is still emerging from this
Every shape in the library, rendered as a circuit-language symbol at every
decoherence depth (d=0.10 deep → d=0.50 surface)
per dimension. Each symbol is one (shape, dim, depth) instance —
the indexing coordinate from which foundational behaviors are identified.
The component alphabet on the cards above is provisional; the true alphabet
emerges from clustering this atlas.
The outer shape defines the cavity. The cavity's standing wave patterns form a second geometry inside. That internal geometry — measured, not imposed — determines how the shape processes frequency. The frequency behavior produces the measurable material properties.