Web extra · not in the printed book · Part III territory

The Spectrum Finds the Rooms

Picture three rooms full of people, connected by a few open doors. Now imagine each person hums, and their hum is shaped by who they can reach through those doors. The three coordinates you are looking at are those hums — the graph’s three quietest standing waves, called its Laplacian eigenvectors — recomputed live every time you touch the slider. When the doors are wide open the whole crowd hums as one room and the constellation huddles; slam most of the doors shut and the second-quietest hum — called λ₁, the spectral gap or Fiedler value — collapses toward zero, and the three rooms physically tear apart on screen. The layout isn’t illustrating the math; the layout is the math.

The constellation above lives in GIGI as data. Every node becomes one record — 72 in total, 24 per community — and every edge becomes a relation between two records. Each node record carries five fields: id, community (A, B, or C), degree (how many neighbors it currently has), x_spec (its coordinate along the graph’s second-quietest vibration), and bridge_incident (0 = interior, 1 = touches a cross-community edge).

Because GIGI is a fiber-bundle database, the bundle’s curvature (κ) and the connectivity gap ride along with every read — no sidecar service to compute them. When you drag the slider or hit "Cut every bridge", the layout below is recomputed from the graph itself, and the readout on the right shows the same connectivity number the engine’s SPECTRAL verb returns for any live bundle.

layout = the graph’s three quietest standing waves (eigenvectors 1, 2, 3 of the graph Laplacian) · recomputed live (Jacobi, 72×72)
drag to orbit · wheel to zoom

Levers & receipts

community A community B community C bridge edge

The number the engine’s SPECTRAL verb reports — how loud that second-quietest hum is, the spectral gap λ₁ — is a connectivity thermometer. λ₁ near zero means "this bundle’s relation graph is about to disconnect" — communities that barely talk — and the same hums that detect it turn out to be good coordinates for seeing it, which is the whole idea of drawing a graph at its own spectral embedding. Watch the moment you drag to one bridge per pair: the rooms are already visibly separate while λ₁ is still nonzero. The spectrum hears the wall before the wall is finished.

The same thing in GQL — run it against an engine

Point at the public read-only instance (no key needed) or your own engine (start with GIGI_CORS_ORIGIN=* — dev only). SPECTRAL rides along on any bundle; the tetmesh bundle is live right now.

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