%
\label{def:archimedes-polygon}
The \textbf{Archimedes polygon sequence}
in the $X$-direction is the sequence of pairs
\[
    \boxed{%
    \bigl(\,P_k^{\mathrm{in}},\; P_k^{\mathrm{out}}\,\bigr)_{k \geq 1},}
\]
where $P_k^{\mathrm{in}}$
is the perimeter of the regular inscribed
$Q_k^X$-gon
and $P_k^{\mathrm{out}}$
is the perimeter of the regular circumscribed
$Q_k^X$-gon,
both computed using the earned Euclidean geometry
(Chapter~\ref{ch:pasch-parallel})
and the denotation map
(Chapter~\ref{ch:orthodox-bridge}).
Explicitly:
\begin{align*}
    P_k^{\mathrm{in}}
    &\;=\;
    Q_k^X \cdot 2\,\sin\!\Bigl(\frac{\pi}{Q_k^X}\Bigr),
    \\[4pt]
    P_k^{\mathrm{out}}
    &\;=\;
    Q_k^X \cdot 2\,\tan\!\Bigl(\frac{\pi}{Q_k^X}\Bigr).
\end{align*}
(The $\sin$ and $\tan$ here denote
the Archimedean trigonometric functions
on the denotation-map image of the solenoidal circle.)