OBSERVATION CHAMBER / ACTIVE

OBSERVATION ROUTE

CURRENT: 78 MODULAR GEODESIC TESSELLATION OBSERVATORY

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14 HYPERBOLIC
33 MODULAR
52 HYPERBOLIC / ISOMETRY
78 GEOMETRY / MODULAR FAREY TESSELLATION

Modular Geodesic Tessellation Observatory

Read PSL(2,Z) reduction, Ford circles, Farey adjacency, and continued-fraction itineraries as one exact upper half-plane observation.

modular / Farey tessellation Observation Model

ROUTE 78 / GEOMETRY / MODULAR / FAREY TESSELLATION

ARITHMETIC UPPER HALF-PLANE

A tessellation read as integer arithmetic

Track an upper half-plane point through PSL(2,Z) reduction. The same matrices expose Ford circles, Farey neighbors, geodesic edges, and a canonical continued-fraction itinerary.

READY

Focus the canvas: ←/→ change MODE, ↑ REDUCE STEP, ↓ EXTEND, P PLAY, O PAUSE, A AUTO, R RESET. The picture is finite arithmetic, not a physical map.

Ready. REDUCE STEP reveals the deterministic word.

MATRIX: I
det: 1
REDUCED z: —
WORD: —
FORD: —
FAREY: —
CF: —
ITINERARY: —

OBSERVATION CONTROLS

Choose an arithmetic lens

DOMAIN

MODE

4 views

The modes share one integer state. No decorative tile layer is used as the observation.

FAREY DEPTH

q ≤ 5

denominator ceiling5

EXTEND increases the exact finite fraction set. Ford tangency uses |ps−qr| = 1.

REDUCTION

0 / 0

REDUCE STEP applies T⁻ⁿ or S. PLAY visits the same finite word; AUTO starts a short local sequence.

KEYBOARD / 44PX TARGETS

←→ MODE · ↑ REDUCE STEP · ↓ EXTEND · P PLAY · O PAUSE · A AUTO · R RESET. Reduced motion disables timed playback; hidden tabs pause it.

ARITHMETIC READOUT

One state, four witnesses

EXACT

matrix M: I
det M: 1
residual: —
Ford count: —
Farey edges: —
CF / L-R: —
contract: Integer PSL(2,Z) matrices, reduced upper half-plane point, tangent Ford circles, determinant-one Farey neighbors, and canonical regular-CF itinerary remain synchronized.

RESET clears the report, download link, ledger, and committed readout. The initial matrix and self-tests remain visible.

DETERMINISTIC SELF-TEST

Numerical acceptance

PASS

FIXED S / T / FORD / FAREY / CF FIXTURES

CHECK	VALUE	LIMIT	RESULT

Observation report

WORD LEDGER

Reduction and continued-fraction itinerary

PSL(2,Z)

STEP	OPERATION	MATRIX	POINT	INVERSE RESIDUAL

Each row is a left multiplication by an exact generator. The inverse readback applies the reverse word to the reduced point.

OBSERVATION CONTRACT

Arithmetic before ornament

DOMAIN is the upper half-plane and its fundamental region. FORD draws rational horocycles. GEODESIC draws determinant-one Farey edges. ITINERARY couples a canonical continued fraction to L/R interval choices.

WHAT TO WATCH

One matrix, many traces

REDUCE STEP alternates translation T⁻ⁿ and inversion S until the point is in F.
EXTEND adds exact p/q witnesses; tangent circles satisfy |ps−qr| = 1.
CF and L/R rows are a deterministic rational fixture, not a sampled estimate.

MODEL NOTE

Projective integer action

g·z = (az+b)/(cz+d)
det(g)=1, g ~ −g
C_p/q: |ps−qr|=1

The relations S² = (ST)³ = I are checked projectively. Rational endpoints are reduced with q > 0 and the final continued-fraction digit is canonical.

Formula note

F = {z : |Re z| ≤ 1/2, |z| ≥ 1}

Ford C(p/q) has center p/q + i/(2q²) and radius 1/(2q²). Farey neighbors are reduced fractions with determinant one; their geodesic is the circle orthogonal to the real axis, or a vertical line at infinity.

OBSERVATION GUIDE

Read the coupling

Start in DOMAIN, press REDUCE STEP, then compare FORD, GEODESIC, and ITINERARY. REPORT and PNG stay disabled until the fixed checks pass.

Use EXTEND to increase q and reveal new rational contacts.
Use PLAY or AUTO for a short local sequence; PAUSE stops it.
RESET removes stale report text and generated download state.

Runs locally in the browser. No upload and no registration.

OBSERVATION POLICY

This chamber computes deterministic integer matrices and finite Farey data in your browser. Points, words, reports, and images stay local. It does not infer a physical surface or upload a trajectory.

Route

This chamber belongs to Shape & Space.

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