Constructive Figures

Constructive Figures

Jan 7, 2026

Constructive Figures in GRG: Dyadic Quarter, Mirrored Circle, and Sector Area

This page collects three minimal, constructive drawings used throughout GRG: a dyadic quarter fan, a fully mirrored circle built from four quarters, and a sector-based area accumulator. All figures are π-free inside the loop; conversions happen only at setup or I/O.

1) Dyadic quarter fan (turns-first sweep)

Angle advances in turns. Refinement uses powers of two so that a quarter turn has N = 2^b rays with a step \Delta u = 2^{-b-2}. This aligns with binary hardware.

$$ N = 2^{\,b}, \qquad \Delta u = 2^{-\,b-2} \quad \text{(quarter sampling)} $$
b = 9 → N = 512 rays per full turn (128 per quarter)
Dyadic quarter fan Rays from the center advance by Δu turns (N = 2^b per quarter). Built by recurrence; no π inside the loop.
Quarter sweep with N = 2^b rays. Mirror to complete the circle.

2) Full circle via mirrored quarters (total symmetry)

Build one quarter, then mirror/rotate it three times to assemble the full circle with exact bilateral and rotational symmetry.

Full circle from four mirrored quarters Quarter geometry is cloned with 90° rotations to assemble a full circle with exact symmetry.
Four identical quarters enforce perfect symmetry and cache-friendly construction.

3) Sector-based area accumulation (π-free loop; classical limit)

Area is the sum of radial sectors. The loop aggregates in turn units; the classical value appears in the limit.

$$ \Delta A = \tfrac12 r^2\,\Delta\theta \qquad \text{with}\quad \Delta\theta = 2\pi\,\Delta u $$ $$ A \;=\; \sum \Delta A \;\xrightarrow[\Delta u \to 0]{}\; \pi r^2 $$
Sector area accumulation Radial sectors of angle Δθ tile the disk. Each sector contributes ½ r² Δθ. Summation converges to π r².
Sum of ½ r² Δθ over dyadic steps converges to the classical area.

Notes on determinism and implementation

  • π outside the loop: the runtime loop uses recurrences; π appears only in calibration or reporting.
  • Bit-shift counts: N = 1 << b replaces divisions with a native shift.
  • Mirrored quarters: memory and symmetry benefits; identical update sequences aid reproducibility.