Your Sound.
Made Visible.

Robert Edward Grant describes integers plotted around a 24-sided polygon (icositetragon) forming “circular complements” that sum to 360°, grouping into quaternion sets of four values with the form Q = w + xi + yj + zk, and framing these as a kind of quadripolar symmetry.

This quadripolar/quaternion framing inspired REACT's choice to treat 24-fold symmetry as a first-class organizing grid with 15° anchors, and to treat orientation as something that should “snap” to symmetry-consistent axes rather than drift. REACT does not implement the paper's full quaternion-set enumeration; instead it uses four quaternion ports as a compact, performance-safe way to map camera orientation to four symmetry-related axes.

• 24-fold layer with 15° precision (discrete angular anchors)
• 4 quaternion ports mapping camera orientation to symmetry axes {w, π−w, π+w, τ−w}
• Polarity tracking with quadrant-based detection (aligned with quadripolar thinking)

In the “numbers ↔ sound ↔ geometry” framing, the octave's 12 notes are treated as a 12-part periodic system that can be overlaid onto geometric/angle references, emphasizing consistent angular partitions rather than arbitrary continuous motion.

This “12-as-geometry” idea informed REACT's decision to make the 12-fold layer foundational, using a stable 30° partition to gate intensity from pitch-class chroma. REACT keeps the relationship honest and bounded: chroma influences which partitions and radii are emphasized, while stability comes from REACT's own adaptive smoothing and harmonic confidence weighting, not from any claim of a perfect theoretical mapping.

• 12-fold layer: gates visual intensity at 30° intervals from pitch-class chroma
• Chroma-to-radius mapping with stability-adaptive smoothing
• Harmonic confidence weighting to privilege stable musical structure over jitter

The work formalizes torus knots T(p,q) using integer winding counts, with closure and structure governed by integer relationships (e.g., the GCD rule) and a “winding interval” describing angular advance per cycle. It highlights a “Golden Toroidal Knot” configuration T(55,144) and extends into families via metallic-mean sequences for coverage and variation.

This body of work shaped REACT's view that certain “musical” structures read best when expressed as stable integer-driven topology archetypes, rather than continuously morphing geometry. REACT reflects that philosophy by treating knot-like cymatic forms as a discrete topology state that can be activated, then held, then crossfaded out-specifically via bar-edge switching and timed crossfades, not continuous deformation.

• Topology state: TorusKnotCymatics
• Topology Router: bar-edge quantized switching, 1.6s crossfade, dwell timers, cooldown; only one topology active
• 24-fold layer as the symmetry container in which topology states remain coherent

The wave-matrix approach treats digital roots as repeating sequences that expose a periodic “wave” pattern, and uses interference as a model for emergent structure, producing self-similar fractals with prominent 12-based periodicity.

This perspective provided theoretical basis for REACT's preference for structured repetition (symmetry partitions, interference-like shells) over noisy freeform motion. In REACT, interference is expressed visually through topology choices and quantized angular seams, while timing remains controlled by REACT's own phrase/energy logic (not by wave theory directly).

• Topology state: RadialInterferenceShell
• 12-fold layer periodic gating (stable repetition over time)
• Harmonic confidence weighting to keep interference structures coherent under changing energy

Constants are framed as integer amplitude + fractional “phase” remainder, and the system emphasizes quadrant / angular complementarity on the unit circle-relationships that are stable under symmetry operations.

This amplitude/phase + quadrant-complement viewpoint informed REACT's separation of “what is stable” (symmetry anchors, ports, partitions) from “what is reactive” (intensity, modulation), and its reliance on quadrant-based polarity tracking rather than arbitrary camera drift. REACT also exposes a 432Hz / 440Hz reference tuning selector in the Harmonic Field menu, which can shift the reference tuning used by the harmonic field.

• 432Hz / 440Hz reference tuning selector (user-facing pitch reference control)
• Polarity tracking with quadrant-based detection
• 4 quaternion ports as a compact symmetry-respecting orientation model

Robert Edward Grant's projection framework relates 2D right-triangle invariants to the generation/classification of 3D “harmonic solids,” including the explicit vertex-count relationship V = a + 2b + c, and derived mean cascades used as structural radii.

This work inspired REACT's emphasis on discrete topology states that behave like stable “solids” with predictable structure, rather than ad-hoc meshes. In REACT, that shows up as a foundation scaffold and a polyhedral resonance state that can be selected by the topology state machine, then held long enough for viewers to read the geometry clearly.

• Topology state: Scaffold24 (foundation)
• Topology state: DodecahedralResonance
• 24-fold layer topology state machine (explicit, mutually exclusive topology selection)

The “fractal root” framing treats repeated digit-reduction / digital-root processes as generating self-similar, repeating structures that remain recognizable across scale and iteration.

This notion of structure persisting through reduction shaped REACT's Afterglow philosophy: even as energy decays, the system aims to preserve a coherent residual form rather than collapsing into noise. REACT achieves this through its own hold logic, hysteresis, and smoothing, expressed in the Afterglow phase and the AfterglowResidual topology-not via direct implementation of a “fractal root” computation pipeline.

• Topology state: AfterglowResidual
• Hero's Journey: Afterglow act (residual coherence persistence driven by phrase detection + energy envelopes)
• Stability-adaptive smoothing (preserve readable structure under decay)