pith. sign in
module module high

IndisputableMonolith.Mathematics.Pi

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The Mathematics.Pi module supplies definitions and bounds for approximating π from the perimeter of a regular octagon whose vertices align with the eight discrete phases. Researchers tracing the forcing chain from T7 eight-tick octave to spatial geometry would cite it. The module proceeds by defining side lengths via sine at π/8, computing perimeters, and stating explicit approximation theorems.

claimThe module centers on the octagon approximation $8 sin(π/8) ≈ π$ obtained from a regular octagon inscribed in the unit circle with vertices at the phases $0, π/4, …, 7π/4$.

background

The module imports the 8-tick structure whose phases are multiples of π/4 and the RS time quantum τ₀ = 1 tick. These phases fix the vertices of the inscribed octagon. The supplied doc-comment states the side length equals 2 sin(π/8) and the resulting perimeter yields the rough numerical estimate π ≈ 3.06.

proof idea

The module organizes its argument as a sequence of perimeter definitions (octagonPerimeter, inscribedPerimeter, circumscribedPerimeter) followed by bounding lemmas and the explicit approximation theorem octagon_approximates_pi.

why it matters in Recognition Science

This module supplies the geometric content required by the eight-tick octave (T7) in the forcing chain. It links the discrete phases to the circle constant π, a prerequisite step before deriving D = 3. No downstream declarations are recorded.

scope and limits

depends on (2)

Lean names referenced from this declaration's body.

declarations in this module (18)