piFromOctagon
plain-language theorem explainer
piFromOctagon supplies the inscribed-octagon lower bound for π by halving the perimeter of the regular 8-gon in the unit circle. Workers on 8-tick derivations of constants in Recognition Science cite it as the concrete starting point for the inequality piFromOctagon < π. The definition is a direct one-line reduction of the octagon-perimeter expression.
Claim. Define the 8-tick approximation by $8_8 := 8 sin(π/8)$, obtained by dividing the perimeter of the inscribed regular octagon (side length $2 sin(π/8)$) in the unit circle by 2.
background
The module MATH-002 derives π from RS 8-tick geometry, where the circle is discretized into eight phases and π appears as the continuous limit of that 8-fold symmetry. octagonPerimeter is the upstream definition giving the perimeter of the inscribed regular octagon in the unit circle: 8 × 2 × sin(π/8). This quantity is the discrete proxy for the circumference 2π under the eight-tick constraint.
proof idea
One-line definition that applies division by 2 to the octagonPerimeter expression.
why it matters
The definition feeds the parent theorem octagon_approximates_pi, which proves the strict inequality piFromOctagon < π. It occupies the first concrete step in MATH-002, linking the eight-tick octave (T7) of the forcing chain to the emergence of π from 8-fold geometry.
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