Geometric Pi

The circumference-to-diameter ratio of the earned circle S^1, defined as the common limit of inscribed and circumscribed Archimedes polygon perimeters normalized to unit diameter.

Geometric Pi

Summary

The circumference-to-diameter ratio of the earned circle S^1, defined as the common limit of inscribed and circumscribed Archimedes polygon perimeters normalized to unit diameter.

Statement

%
\label{def:geometric-pi}
The \textbf{geometric~$\pi$}
is the common limit of the inscribed and circumscribed
Archimedes polygon perimeters,
normalized to unit diameter:
\[
    \boxed{%
    \pi_{\mathrm{geo}}
    \;:=\;
    \lim_{k \to \infty}\;
    \frac{P_k^{\mathrm{in}}}{2}
    \;=\;
    \lim_{k \to \infty}\;
    \frac{P_k^{\mathrm{out}}}{2}.}
\]
(The factor $1/2$ normalizes to a circle
of diameter~$1$, i.e., radius~$1/2$,
so that circumference equals~$\pi$.)

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-02.jsonl line 66
• Manuscript source: 2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part05/ch25-pi-earned.tex lines 253-273

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookII.Transcendentals.PiEarned
• Name: pi_circumference

Dependencies

• Canonical: II.D29, II.D26

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.